On the tilts caused by equilibrium fluctuations of atmospheric pressure: effect of topography

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Abstract


The influence of the relief on baric tilts is studied. Main attention is focused on two-dimensional problem. It is assumed that variations of baric field are equilibrium (horizontal component of pressure gradient is absent) and the relief is finite. The latter means that the area between the topographic profile and the abscissa axis is finite. It is shown that this area is the only geometric parameter that determines vertical displacements and tilts in the far zone. The asymptotics for the variations in vertical displacements and tilts are written out. The generalization to the three-dimensional case is presented.


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Влияние процессов в атмосфере на наклоны ранее неоднократно обсуждалось, в том числе, в работах расчетно-теоретического плана. Начало положила статья [Darwin, 1882]. В ней рассматривается упругое полупространство, атмосферное давление над которым меняется вдоль оси x по синусоидальному закону. То есть: рельеф отсутствует, распределение давления неравновесное и должно сопровождаться массопереносом. Эти два момента являются основными в постановке задачи. Авторы последующих (часто весьма содержательных) работ их так или иначе воспроизводят.

Показательна статья [Перцев, Ковалёва, 2004]: в ней авторы моделируют нестационарный процесс в атмосфере, а именно MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ прохождение циклона, который представляет собой область пониженного атмосферного давления с характерным размером MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4haaa@3F2D@ 600 км. Давление в центре циклона отличается от нормального атмосферного давления на 40 мбар, т. е. на 4%.

Авторами получены следующие оценки: максимальное значение MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ 5 мсек дуги, или 2 10 8 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGOmaiabgwSixlaaigdacaaIWaWaaWbaaSqabeaa cqGHsislcaaI4aaaaaaa@4419@ рад MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ наклоны достигают на расстоянии MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4haaa@3F2D@ 200 км от центра циклона. На расстоянии 1000 км от центра наклоны составляют 5 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4NaaGynaiaducOHflY1caaIXaGaaGim amaaCaaaleqabaGaeyOeI0IaaGymaiaaicdaaaaaaa@4757@ рад. Для сравнения: амплитуда волны М2 на широте Москвы MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ величина порядка 5 10 8 ; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGynaiabgwSixlaaigdacaaIWaWaaWbaaSqabeaa cqGHsislcaaMc8UaaGioaaaakiaacUdaaaa@4670@ амплитуды суточных волн примерно в 2 раза меньше.

Но эти оценки носят «усредненный» характер, поскольку топографию MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ фактор, обладающий малым (по сравнению с масштабом рассматриваемого явления) «радиусом действия» MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ авторы не учитывают. «На локальном уровне» фактор рельефа может стать решающим.

Второй пример: статья [Латынина, Васильев, 2001]. Авторы анализируют данные, которые были получены на Подмосковной станции Протвино. Метеорологическая обстановка в районе станции известна, она характеризуется чередованием циклонов и антициклонов. Авторы вводят в рассмотрение передаточную функцию «от баровариаций к наклонам». Для периодов от 2 до 15 суток значение этой функции примерно в 7 раз больше теоретической. Авторы считают, что «причиной возникновения аномально больших наклонов земной поверхности является латеральная неоднородность поверхностных слоев земной коры». Однако не исключено, что отчасти это несоответствие обусловлено наличием рельефа, который авторы «не замечают». Ниже будет показано, что даже слабый рельеф может существенно повлиять на барические наклоны.

О влиянии рельефа на атмосферные наклоны говорится только в статье [Широков, Анохина, 2003], но авторы ограничиваются констатацией факта и не приводят количественных оценок. В данной работе этот пробел будет восполнен.

Вариацию барического поля мы будем считать равновесной. Это значит, что давление в каждый момент времени зависит только от высоты над уровнем моря; горизонтальной составляющей у градиента давления нет. При наличии рельефа даже равновесные вариации давления должны сопровождаться изменением наклонов.

Рассмотрение будет вестись в рамках линейной теории упругости. Основное внимание мы уделим двумерной задаче. Это связано с тем, что рельеф, как правило, имеет линейно-протяженную структуру (разломы, цепи холмов, речные русла). С другой стороны, решить двумерную задачу проще, поскольку ее можно свести к краевой задаче теории функций комплексного переменного.

Этот подход был предложен Г. Колосовым [Колосов, 1935]. С его помощью в статье получен упомянутый в аннотации приближенный результат общего характера, а также точное решение для рельефа специального вида MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ образа нижней полуплоскости при конформном отображении:

z w = w a 2 / wic ,ca>0;Imw0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEamaabmaabaGaam4DaaGaayjkaiaawMcaaiab g2da9maalyaabaGaam4DaiabgkHiTiaadggadaahaaWcbeqaaiaaik daaaaakeaadaqadaqaaiaadEhacqGHsislcaWGPbGaam4yaaGaayjk aiaawMcaaaaacaGGSaGaaGjbVlaaysW7caWGJbGaeyyzImRaamyyai abg6da+iaaicdacaGG7aGaaGjbVlaaysW7ciGGjbGaaiyBaiaadEha cqGHKjYOcaaIWaGaaiOlaaaa@5E41@ (1)

Этот рельеф представляет собой впадину, выраженную тем более явно, чем больше отношение a/c . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaWGHbaabaGaam4yaaaacaGGUaaaaa@4058@ Показано, что вклад такого рельефа в барические наклоны может на порядок превышать чувствительность современных наклономеров.

1. ОСНОВНЫЕ СООТНОШЕНИЯ

1.1. Финитный рельеф

Пусть D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (нижняя) полуплоскость с рельефом, D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ линия, ограничивающая ее сверху, то есть собственно рельеф. Предположим, что у бесконечных ветвей рельефа есть общая горизонтальная асимптота. Введем правую систему координат (x,y), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiaadIhacaGGSaGaamyEaiaacMcacaGGSaaa aa@4276@ совместив с этой асимптотой ось абсцисс; ось ординат направим вертикально вверх. Условимся через y(x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiaacIcacaWG4bGaaiykaaaa@4116@ обозначать ординату точки рельефа с абсциссой x; будем считать (хотя это и необязательно), что такая точка только одна.

Назовем рельеф финитным, если при x± MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabgkziUkabgglaXkabg6HiLcaa@440B@

y x =o 1 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGabmyEayaafaWaaeWaaeaacaWG4baacaGLOaGaayzk aaGaeyypa0Jaam4BamaabmaabaGaaGymaaGaayjkaiaawMcaaiaacY caaaa@4640@ y x =O 1/ |x | 1+α , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEamaabmaabaGaamiEaaGaayjkaiaawMcaaiab g2da9iaad+eadaqadaqaamaalyaabaGaaGymaaqaaiaacYhacaWG4b GaaiiFamaaCaaaleqabaGaaGymaiabgUcaRiaaykW7cqaHXoqyaaaa aaGccaGLOaGaayzkaaGaaiilaaaa@4E25@ (2)

где α MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdegaaa@3F61@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ любое положительное число. Кроме того, пусть производная y (x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGabmyEayaafaGaaiikaiaadIhacaGGPaaaaa@4122@ имеет не более конечного числа точек разрыва.

Из (2), в частности, следует сходимость интеграла:

S= y(x)dx . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4uaiabg2da9maapehabaGaamyEaiaacIcacaWG 4bGaaiykaiaadsgacaWG4baaleaacqGHsislcqGHEisPaeaacqGHEi sPa0Gaey4kIipakiaac6caaaa@4BCE@ (3)

Этот интеграл представляет собой «алгебраическую площадь» области, заключенной между линией рельефа и осью абсцисс. Все дальнейшие рассмотрения в полной мере относятся именно к финитному рельефу.

ПРИМЕР 1.

D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ незамкнутая ломаная, такая, как на рис. 1а: вне отрезка, соединяющего крайне правую и крайне левую вершины, ордината y(x)=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiaacIcacaWG4bGaaiykaiabg2da9iaaicda caGGUaaaaa@4388@ У производной y (x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGabmyEayaafaGaaiikaiaadIhacaGGPaaaaa@4122@ конечное число точек разрыва (оно равно числу вершин ломаной), поэтому условия (2) соблюдены.

 

Рис. 1. Примеры финитного рельефа: (а) линия рельефа – ломаная с конечным числом звеньев; (б) линия рельефа – образ действительной оси при отображении (1), где c=2a

 

Терминологическое замечание . Если y(x), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiaacIcacaWG4bGaaiykaiaacYcaaaa@41C6@ как в Примере 1, функция с компактным носителем, рельеф будем называть «совсем финитным».

ПРИМЕР 2. D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ граница области, в которую переходит нижняя полуплоскость комплексной переменной w при простейшем рациональном конформном отображении (1), см. рис. 1б). Оно имитирует ложбину, глубина которой H равна:

H= a 2 /c . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9maalyaabaGaamyyamaaCaaaleqa baGaaGOmaaaaaOqaaiaadogaaaGaaiOlaaaa@431E@ (4)

Интеграл (3) легко вычислить:

S=π a 2 1 a 2 4 c 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4uaiabg2da9iabgkHiTiabec8aWjaaykW7caWG HbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaeyOeI0YaaS aaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaiaadoga daahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaaiOlaaaa@4E13@ (5)

Вклад рельефа (1) в барические наклоны будет вычислен в п. 2.

1.2. Метод Колосова MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbbKqzGgaeaaaaaaaaa8qacaWFtacaaa@39E4@ Мусхелишвили

Обозначим через σ xx , σ xy , σ yy MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiaa cYcacqaHdpWCdaWgaaWcbaGaamiEaiaadMhaaeqaaOGaaiilaiabeo 8aZnaaBaaaleaacaWG5bGaamyEaaqabaaaaa@4AF4@ компоненты тензора напряжений, через (u,v) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiaadwhacaGGSaGaamODaiaacMcaaaa@41C0@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ компоненты вектора упругих смешений, через λ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWgaaa@3F76@ и μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0gaaa@3F78@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ коэффициенты Ламе (модуль сжатия и модуль сдвига). Двумерная задача теории упругости представляет собой систему из семи скалярных уравнений, из которых три уравнения выражают закон Гука:

σ xx =λ u/x+v/y +2μu/x, σ yy = =λ u/x+v/y +2μv/y, σ xy =μ u/y+v/x , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqa aOGaeyypa0Jaeq4UdW2aaeWaaeaacqGHciITcaWG1bGaai4laiabgk Gi2kaadIhacqGHRaWkcqGHciITcaWG2bGaai4laiabgkGi2kaadMha aiaawIcacaGLPaaacqGHRaWkcaaIYaGaeqiVd0MaaGjbVlabgkGi2k aadwhacaGGVaGaeyOaIyRaamiEaiaacYcacaaMe8Uaeq4Wdm3aaSba aSqaaiaadMhacaWG5baabeaakiabg2da9aqaaiabg2da9iabeU7aSn aabmaabaGaeyOaIyRaamyDaiaac+cacqGHciITcaWG4bGaey4kaSIa eyOaIyRaamODaiaac+cacqGHciITcaWG5baacaGLOaGaayzkaaGaey 4kaSIaaGOmaiabeY7aTjaaysW7cqGHciITcaWG2bGaai4laiabgkGi 2kaadMhacaGGSaGaaGzbVdqaaiabeo8aZnaaBaaaleaacaWG4bGaam yEaaqabaGccqGH9aqpcqaH8oqBdaqadaqaaiabgkGi2kaadwhacaGG VaGaeyOaIyRaamyEaiabgUcaRiabgkGi2kaadAhacaGGVaGaeyOaIy RaamiEaaGaayjkaiaawMcaaiaaykW7caGGSaaaaaa@964E@ (6)

два MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ условия равновесия:

σ xx /x+ σ xy /y=0, σ xy /x+ σ yy /y=0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqGHciITcqaHdpWCdaWgaaWcbaGaamiEaiaa dIhaaeqaaOGaai4laiabgkGi2kaadIhacqGHRaWkcqGHciITcqaHdp WCdaWgaaWcbaGaamiEaiaadMhaaeqaaOGaai4laiabgkGi2kaadMha cqGH9aqpcaaIWaGaaiilaaqaaiabgkGi2kabeo8aZnaaBaaaleaaca WG4bGaamyEaaqabaGccaGGVaGaeyOaIyRaamiEaiabgUcaRiabgkGi 2kabeo8aZnaaBaaaleaacaWG5bGaamyEaaqabaGccaGGVaGaeyOaIy RaamyEaiabg2da9iaaicdaaaaa@6580@ (7)

и еще два MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ краевые условия, заданные на границе D: MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraiaacQdaaaa@40AF@

σ xx n x + σ xy n y = F x , σ xy n x + σ yy n y = F y . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiaa d6gadaWgaaWcbaGaamiEaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcba GaamiEaiaadMhaaeqaaOGaamOBamaaBaaaleaacaWG5baabeaakiab g2da9iaadAeadaWgaaWcbaGaamiEaaqabaGccaGGSaGaaGzbVlabeo 8aZnaaBaaaleaacaWG4bGaamyEaaqabaGccaWGUbWaaSbaaSqaaiaa dIhaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaadMhacaWG5baabe aakiaad6gadaWgaaWcbaGaamyEaaqabaGccqGH9aqpcaWGgbWaaSba aSqaaiaadMhaaeqaaOGaaiOlaaaa@60E9@ (8)

В (8) n x , n y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOBamaaBaaaleaacaWG4baabeaakiaacYcacaaM e8UaaGPaVlaad6gadaWgaaWcbaGaamyEaaqabaaaaa@45CD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ компоненты вектора внешней нормали; F x , F y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOramaaBaaaleaacaWG4baabeaakiaacYcacaaM e8UaaGPaVlaadAeadaWgaaWcbaGaamyEaaqabaaaaa@457D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ компоненты вектора внешних усилий. В нашем случае, согласно закону Паскаля:

F x =p n x , F y =p n y , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOramaaBaaaleaacaWG4baabeaakiabg2da9iab gkHiTiaadchacaaMc8UaamOBamaaBaaaleaacaWG4baabeaakiaacY cacaWGgbWaaSbaaSqaaiaadMhaaeqaaOGaeyypa0JaeyOeI0IaamiC aiaaykW7caWGUbWaaSbaaSqaaiaadMhaaeqaaOGaaiilaaaa@5052@ (9)

где p MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ атмосферное давление, меняющееся от точки к точке: p= p 0 ρgy x . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiCaiabg2da9iaadchadaWgaaWcbaGaaGimaaqa baGccqGHsislcqaHbpGCcaaMc8Uaam4zaiaaykW7caWG5bWaaeWaae aacaWG4baacaGLOaGaayzkaaGaaiOlaaaa@4C87@

Здесь p0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ давление «на уровне моря», которому соответствует значение ординаты y=0; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiabg2da9iaaicdacaGG7aaaaa@413F@ g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ ускорение свободного падения; ρ1.2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyWdiNamGfGgIKi7kaaygW7caaIXaGaaiOlaiaa ikdaaaa@45C6@ кг/м3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ плотность воздуха вблизи поверхности Земли.

Деформации и наклоны линейно зависят от атмосферного давления p. Поэтому вклад рельефа можно представить как сумму двух слагаемых: первое пропорционально p0 (слагаемое I); второе, учитывающее изменение давления с высотой MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ пропорционально ρgy(x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyWdiNaaGjcVlac2b4GNbGaaGzaVlaadMhacaGG OaGaamiEaiaacMcaaaa@47DB@ (слагаемое II). Заметим, что слагаемое I в двумерной постановке рассчитывается так же, как вклад рельефа в приливные наклоны (для слабого рельефа эта задача решена в ст. [Молоденский, 1983]): давление p0 играет роль приливного напряжения.

Соотношение слагаемых I и II заслуживает отдельного рассмотрения, которое выходит за рамки данной статьи. Здесь скажем только, что доля второго слагаемого тем значительнее, чем «масштабнее» рельеф; но главное MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ оно превалирует в дальней от активной части рельефа зоне. Поэтому, на наш взгляд, оно представляет гораздо больший интерес. Именно его мы и будем анализировать.

Итак, будем считать, что:

p=ρgy x . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiCaiabg2da9iabgkHiTiabeg8aYjaaykW7caWG NbGaaGPaVlaadMhadaqadaqaaiaadIhaaiaawIcacaGLPaaacaGGUa aaaa@4AA1@ (10)

Двумерную задачу теории упругости (6) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (8) можно свести к краевой задаче теории функций комплексного переменного, используя метод Колосова. Этот метод с исчерпывающей полнотой изложен в книге [Мусхелишвили, 1966] (см. также [Портон, Перлин, 1981]). Он основан на том, что компоненты тензора напряжений являются производными бигармонической функции, т. н. функции Эри U(x,y): MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyvaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa aiOoaaaa@435E@

σ xx = 2 U y 2 , σ xy = 2 U xy , σ yy = 2 U x 2 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiab g2da9maalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvaa qaaiabgkGi2kaadMhadaahaaWcbeqaaiaaikdaaaaaaOGaaiilaiaa ysW7caaMe8Uaeq4Wdm3aaSbaaSqaaiaadIhacaWG5baabeaakiabg2 da9iabgkHiTmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGa amyvaaqaaiabgkGi2kaadIhacqGHciITcaWG5baaaiaacYcacaaMe8 Uaeq4Wdm3aaSbaaSqaaiaadMhacaWG5baabeaakiabg2da9maalaaa baGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyvaaqaaiabgkGi2k aadIhadaahaaWcbeqaaiaaikdaaaaaaOGaaiilaaaa@6991@

и на теореме Гурса, в силу которой бигармоническую в области D функцию можно представить в виде:

U=Re z φ ¯ + ψdz , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyvaiabg2da9iGackfacaGGLbWaaeWaaeaacaWG 6bGanqjGeA8aQzacucyeaiaducOHRaWkdaWdbaqaaiabeI8a5jaayk W7caWGKbGaamOEaaWcbeqab0Gaey4kIipaaOGaayjkaiaawMcaaiaa cYcaaaa@51EC@

где φ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadQhacaGGPaaaaa@41D7@ и ψ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadQhacaGGPaaaaa@41E8@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ аналитические в этой области функции комплексной переменной z=x+iy. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEaiabg2da9iaadIhacqGHRaWkcaWGPbGaamyE aiaac6caaaa@4444@ Функции φ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadQhacaGGPaaaaa@41D7@ и ψ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadQhacaGGPaaaaa@41E8@ связаны на границе условием:

φ(ξ)+ξ φ (ξ) ¯ + ψ(ξ) ¯ =f(ξ),ξD. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiabe67a4jaacMcacqGHRaWkcqaH +oaEdaqdaaqaaiqbeA8aQzaafaGaaiikaiabe67a4jaacMcaaaGaey 4kaSYaa0aaaeaacqaHipqEcaGGOaGaeqOVdGNaaiykaaaacqGH9aqp caWGMbGaaiikaiabe67a4jaacMcacaGGSaGaaGjbVlaaysW7cqaH+o aEcqGHiiIZcqGHciITcaWGebGaaiOlaaaa@5F12@ (11)

Здесь f(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacaWG6bGaaiykaaaa@4105@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ «комплексный градиент» функции U(x,y): MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyvaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa aiOoaaaa@435E@

f(z)= U x +i U y , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacaWG6bGaaiykaiabg2da9maalaaa baGaeyOaIyRaamyvaaqaaiabgkGi2kaadIhaaaGaey4kaSIaamyAam aalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2kaadMhaaaGaaiilaaaa @4DF2@

он определен с точностью до аддитивной константы. Функцию f(ξ) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacqaH+oaEcaGGPaaaaa@41C9@ будем называть нагрузочной функцией. Ее можно выразить через компоненты вектора внешних усилий:

f(ξ)=i D ( F x +i F y )ds , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacqaH+oaEcaGGPaGaeyypa0JaamyA amaapefabaGaaiikaiaadAeadaWgaaWcbaGaamiEaaqabaGccqGHRa WkcaWGPbGaamOramaaBaaaleaacaWG5baabeaakiaacMcacaWGKbGa am4CaaWcbaGaeyOaIyRaamiraaqab0Gaey4kIipakiaacYcaaaa@51F7@ (12)

где ds MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ элемент дуги контура D. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraiaac6caaaa@40A3@ Интегрирование идет в положительном направлении так, что область D находится от нас слева. Подставив (9) в (12), получим:

f ξ = D p dx+idy . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzamaabmaabaGaeqOVdGhacaGLOaGaayzkaaGa eyypa0JaeyOeI0Yaa8quaeaacaWGWbGaeyyXIC9aaeWaaeaacaWGKb GaamiEaiabgUcaRiaadMgacaaMi8UaamizaiaadMhaaiaawIcacaGL PaaaaSqaaiabgkGi2kaadseaaeqaniabgUIiYdGccaGGUaaaaa@5517@ (13)

Мы учли, что n x = dy/ ds , n y = dx/ ds . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOBamaaBaaaleaacaWG4baabeaakiabg2da9maa lyaabaGaamizaiaadMhaaeaacaWGKbGaam4CaaaacaGGSaGaaGjbVl aaysW7caWGUbWaaSbaaSqaaiaadMhaaeqaaOGaeyypa0ZaaSGbaeaa cqGHsislcaaMc8UaamizaiaadIhaaeaacaWGKbGaam4CaaaacaGGUa aaaa@52CA@ Наконец, пользуясь (10), приводим (13) к виду:

f ξ =f x+iy = =ρg b x y x 1 d x 1 + i 2 y 2 x y 2 b +const. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaWGMbWaaeWaaeaacqaH+oaEaiaawIcacaGL PaaacqGH9aqpcaWGMbWaaeWaaeaacaWG4bGaey4kaSIaamyAaiaadM haaiaawIcacaGLPaaacqGH9aqpaeaacqGH9aqpcqaHbpGCcaWGNbWa aeWaaeaadaWdXbqaaiaadMhadaqadaqaaiaadIhadaWgaaWcbaGaaG ymaaqabaaakiaawIcacaGLPaaacaWGKbGaamiEamaaBaaaleaacaaI XaaabeaakiabgUcaRmaalaaabaGaamyAaaqaaiaaikdaaaWaaeWaae aacaWG5bWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWG4baacaGL OaGaayzkaaGaeyOeI0IaamyEamaaCaaaleqabaGaaGOmaaaakmaabm aabaGaamOyaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaWcbaGaamOy aaqaaiaadIhaa0Gaey4kIipaaOGaayjkaiaawMcaaiabgUcaRiaabo gacaqGVbGaaeOBaiaabohacaqG0bGaaeOlaaaaaa@6F5D@

Здесь: x 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@3FA6@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ абсцисса текущей точки D; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraiacucOG7aaaaa@41CC@ b MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ произвольное действительное число. Если рельеф финитный, мы можем положить b=, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiabg2da9iabg6HiLkaacYcaaaa@41D0@ и, отбросив несущественную константу, написать:

f ξ =ρg x y x 1 d x 1 + i 2 y 2 x . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzamaabmaabaGaeqOVdGhacaGLOaGaayzkaaGa eyypa0JaeqyWdiNaam4zamaabmaabaWaa8qCaeaacaWG5bWaaeWaae aacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaamiz aiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaWcaaqaaiaadM gaaeaacaaIYaaaaiaadMhadaahaaWcbeqaaiaaikdaaaGcdaqadaqa aiaadIhaaiaawIcacaGLPaaaaSqaaiabg6HiLcqaaiaadIhaa0Gaey 4kIipaaOGaayjkaiaawMcaaiaac6caaaa@5B26@ (14)

Решив задачу (11) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (14), мы с помощью формулы, принадлежащей Г. Колосову:

2μ u+iv =κφ z z φ z ¯ ψ z ¯ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGOmaiabeY7aTjaaysW7daqadaqaaiaadwhacqGH RaWkcaWGPbGaamODaaGaayjkaiaawMcaaiabg2da9iabeQ7aRjabeA 8aQnaabmaabaGaamOEaaGaayjkaiaawMcaaiabgkHiTiaadQhadaqd aaqaaiqbeA8aQzaafaWaaeWaaeaacaWG6baacaGLOaGaayzkaaaaai abgkHiTmaanaaabaGaeqiYdK3aaeWaaeaacaWG6baacaGLOaGaayzk aaaaaiaacYcaaaa@5A5E@ (15)

где κ= λ+3μ / λ+μ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOUdSMaeyypa0ZaaSGbaeaadaqadaqaaiabeU7a SjabgUcaRiaaiodacqaH8oqBaiaawIcacaGLPaaaaeaadaqadaqaai abeU7aSjabgUcaRiabeY7aTbGaayjkaiaawMcaaaaacaGGSaaaaa@4DA7@ сможем найти смещения во всей области D. На линии рельефа:

2μ u+iv = κ+1 φ ξ f ξ ,ξD. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGOmaiabeY7aTnaabmaabaGaamyDaiabgUcaRiaa dMgacaWG2baacaGLOaGaayzkaaGaaGjbVlabg2da9maabmaabaGaeq OUdSMaey4kaSIaaGymaaGaayjkaiaawMcaaiaayIW7caaMc8UaeqOX dO2aaeWaaeaacqaH+oaEaiaawIcacaGLPaaacqGHsislcaWGMbWaae WaaeaacqaH+oaEaiaawIcacaGLPaaacaGGSaGaaGjbVlaaywW7cqaH +oaEcqGHiiIZcqGHciITcaWGebGaaiOlaaaa@6429@ (16)

Нас интересует, в основном, наклоны v(z)/x z=ξD . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaqGaaeaacqGHciITcaWG2bGaaiikaiaadQhacaGG PaGaai4laiabgkGi2kaadIhaaiaawIa7amaaBaaaleaacaWG6bGaey ypa0JaeqOVdGNaeyicI4SaamiraaqabaGccaGGUaaaaa@4E24@ Наша ближайшая цель MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ получить асимптотику для наклонов при |x|. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiaadIhacaGG8bGaeyOKH4QaeyOhIuQaaiOl aaaa@44CF@ Мы покажем, что единственный геометрический параметр, от которого она зависит MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ интеграл (3).

1.3. Вертикальные смещения и наклоны в дальней зоне

Обратимся к общей задаче (11) для полуплоскости с рельефом. Как уже было сказано, функция f(ξ) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacqaH+oaEcaGGPaaaaa@41C9@ в правой части (11) определена с точностью до постоянного слагаемого. Если его можно подобрать так, чтобы величина f(ξ) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacqaH+oaEcaGGPaaaaa@41C9@ при ξ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOVdGNaeyOKH4QaeyOhIukaaa@42E3@ достаточно быстро стремилась к нулю (например, как 1/ ξ α , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaaIXaaabaWaaqWaaeaacqaH+oaEaiaa wEa7caGLiWoaaaWaaWbaaSqabeaacqaHXoqyaaGccGGtakilaaaa@46D4@ где α>0), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdeMaeyOpa4JaaGimaiaacMcacaGGSaaaaa@4280@ функция φ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadQhacaGGPaaaaa@41D7@ на бесконечности будет регулярной. Это значит, что:

φ z =o 1/ z ,z. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGafqOXdOMbauaadaqadaqaaiaadQhaaiaawIcacaGL PaaacqGH9aqpcaWGVbWaaeWaaeaadaWcgaqaaiaaigdaaeaadaabda qaaiaadQhaaiaawEa7caGLiWoaaaaacaGLOaGaayzkaaGaaiilaiaa ywW7caWG6bGaeyOKH4QaeyOhIuQaaiOlaaaa@51D4@

Например, для нижней полуплоскости, ограниченной действительной осью, решение гласит:

φ(z)= 1 2πi f(ξ)dξ ξz MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadQhacaGGPaGaeyypa0ZaaSaa aeaacaaIXaaabaGaaGOmaiabec8aWjaaykW7caWGPbaaamaapehaba WaaSaaaeaacaWGMbGaaGPaVlaacIcacqaH+oaEcaGGPaGaamizaiab e67a4bqaaiabe67a4jabgkHiTiaadQhaaaaaleaacqGHEisPaeaacq GHsislcqGHEisPa0Gaey4kIipaaaa@5ACF@

и φ (z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGafqOXdOMbauaacaGGOaGaamOEaiaacMcaaaa@41E3@ убывает как 1/ z 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaaIXaaabaWaaqWaaeaacaWG6baacaGL hWUaayjcSdaaamaaCaaaleqabaGaaGOmaaaaaaa@439D@ [Мусхелишвили, 1966]. Но функция (14) при обходе контура D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ испытывает приращение f D , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaamWaaeaacaWGMbaacaGLBbGaayzxaaWaaSbaaSqa aiabgkGi2kaadseaaeqaaOGaaiilaaaa@43B4@ не равное 0:

f D =ρg y x 1 d x 1 =ρgS. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaamWaaeaacaWGMbaacaGLBbGaayzxaaWaaSbaaSqa aiabgkGi2kaadseaaeqaaOGaeyypa0JaeqyWdiNaam4zamaapehaba GaamyEamaabmaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjk aiaawMcaaiaadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaqaaiabg6 HiLcqaaiabgkHiTiaaykW7cqGHEisPa0Gaey4kIipakiabg2da9iab gkHiTiabeg8aYjaadEgacaWGtbGaaiOlaaaa@5BE3@ (17)

Поэтому применим прием, также описанный в кн. [Мусхелишвили, 1966]: будем искать функции φ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadQhacaGGPaaaaa@41D7@ и ψ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadQhacaGGPaaaaa@41E8@ в виде:

φ z =Aln z z 0 +Φ z , ψ z =Bln z z 0 +C+Ψ z . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqaHgpGAdaqadaqaaiaadQhaaiaawIcacaGL PaaacqGH9aqpcaWGbbGaaGPaVlaaykW7ciGGSbGaaiOBamaabmaaba GaamOEaiabgkHiTiaadQhadaWgaaWcbaGaaGimaaqabaaakiaawIca caGLPaaacqGHRaWkcqqHMoGrdaqadaqaaiaadQhaaiaawIcacaGLPa aacaGGSaGaaGPaVdqaaiabeI8a5naabmaabaGaamOEaaGaayjkaiaa wMcaaiabg2da9iaadkeacaaMe8UaaGPaVlGacYgacaGGUbWaaeWaae aacaWG6bGaeyOeI0IaamOEamaaBaaaleaacaaIWaaabeaaaOGaayjk aiaawMcaaiabgUcaRiaadoeacqGHRaWkcqqHOoqwdaqadaqaaiaadQ haaiaawIcacaGLPaaacaGGUaaaaaa@6D30@ (18)

Здесь: z 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEamaaBaaaleaacaaIWaaabeaaaaa@3FA7@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ точка, лежащая в верхней полуплоскости, выше самой высокой точки области D; Φ(z),Ψ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeuOPdyKaaiikaiaadQhacaGGPaGaaiilaiaayIW7 caaMe8UaeuiQdKLaaiikaiaadQhacaGGPaaaaa@4949@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ аналитические функции, регулярные на бесконечности. Под lnz MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciiBaiaac6gacaWG6baaaa@40A5@ мы будем понимать ветвь логарифма

lnz=ln|z|+iArg z, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciiBaiaac6gacaWG6bGaeyypa0JaciiBaiaac6ga caGG8bGaamOEaiaacYhacqGHRaWkcaaMe8UaamyAaiaaykW7caqGbb GaaeOCaiaabEgacaqGGaGaamOEaiaabYcaaaa@506A@

где Arg z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaeyqaiaabkhacaqGNbGaaeiiaiaadQhaaaa@4207@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ аргумент комплексного числа, который меняется в пределах от π MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOeI0IaeqiWdahaaa@406C@ до π. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiWdaNaaiOlaaaa@4031@ Важно, что приращение логарифма при обходе контура D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ отлично от 0:

ln ξ z 0 D =πi. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGPaVlaaykW7daWadaqaaiGacYgacaGGUbWaaeWa aeaacqaH+oaEcqGHsislcaWG6bWaaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaaacaGLBbGaayzxaaWaaSbaaSqaaiabgkGi2kaadsea aeqaaOGaeyypa0JaeyOeI0IaeqiWdaNaaGPaVlaadMgacaGGUaaaaa@5416@

Подставляем (18) в (11):

Φ(ξ)+ξ Φ (ξ) _______ + Ψ(ξ) ______ = =f(ξ)Aln(ξ z 0 ) B __ ln( ξ __ z 0 ___ )ξ A __ ξ __ z 0 ___ C __ , ξD. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqqHMoGrcaGGOaGaeqOVdGNaaiykaiabgUca Riabe67a4naaxacabaGafuOPdyKbauaacaGGOaGaeqOVdGNaaiykaa Wcbeqaaiaac+facaGGFbGaai4xaiaac+facaGGFbGaai4xaiaac+fa aaGccqGHRaWkdaWfGaqaaiabfI6azjaacIcacqaH+oaEcaGGPaaale qabaGaai4xaiaac+facaGGFbGaai4xaiaac+facaGGFbaaaOGaeyyp a0dabaGamailg2da9iacaYYGMbGaiailcIcacWaGSqOVdGNaiailcM cacWaGSyOeI0IaiaildgeacOaGSiiBaiacaYIGUbGaiailcIcacWaG SqOVdGNamailgkHiTiacaYYG6bWaiailBaaaleacaYIaiailicdaae qcaYcakiacaYIGPaGamailgkHiTmacaYYfGaqaiailcGaGSmOqaaWc bKaGSeacaYIaiailc+facGaGSi4xaaaakiGcaYIGSbGaiailc6gacG aGSiikamacaYYfGaqaiailcWaGSqOVdGhaleqcaYsaiailcGaGSi4x aiacaYIGFbaaaOGamailgkHiTmacaYYfGaqaiailcGaGSmOEamacaY YgaaWcbGaGSiacaYcIWaaabKaGSaaabKaGSeacaYIaiailc+facGaG Si4xaiacaYIGFbaaaOGaiailcMcacWaGSyOeI0Iamaile67a4nacaY YcaaqaiaildGaGSCbiaeacaYIaiaildgeaaSqajailbGaGSiacaYIG FbGaiailc+faaaaakeacaYYaiailxacabGaGSiadaYcH+oaEaSqaja ilbGaGSiacaYIGFbGaiailc+faaaGccWaGSyOeI0YaiailxacabGaG SiacaYYG6bWaiailBaaaleacaYIaiailicdaaeqcaYcaaeqcaYsaia ilcGaGSi4xaiacaYIGFbGaiailc+faaaaaaOGamailgkHiTmacaYYf GaqaiailcGaGSm4qaaWcbKaGSeacaYIaiailc+facGaGSi4xaaaaki acaYIGSaGaiailysW7aeaacWaGacaay9pH+oaEcWaGacaay9VHiiIZ cWaGacaay9VHciITcGaGacaay9=GebGaiaiGaaaw=lOlaaaaaa@E3A9@ (19)

При Reξ± MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciOuaiaacwgacaaMe8UaeqOVdGNaeyOKH4QaeyyS aeRaaGPaVlabg6HiLcaa@49AA@ правая часть (19) должна оставаться ограниченной, более того MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ стремиться к нулю, откуда:

A=B= [f] D 2πi = ρgS 2πi ;C= [f] D 2πi = ρgS 2πi . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyqaiabg2da9iaadkeacqGH9aqpcqGHsisldaWc aaqaaiaacUfacaWGMbGaaiyxamaaBaaaleaacqGHciITcaWGebaabe aaaOqaaiaaikdacqaHapaCcaaMc8UaamyAaaaacqGH9aqpdaWcaaqa aiabeg8aYjaadEgacaWGtbaabaGaaGOmaiabec8aWjaaykW7caWGPb aaaiaacUdacaaMf8Uaam4qaiabg2da9maalaaabaGaai4waiaadAga caGGDbWaaSbaaSqaaiabgkGi2kaadseaaeqaaaGcbaGaaGOmaiabec 8aWjaaykW7caWGPbaaaiabg2da9iabgkHiTmaalaaabaGaeqyWdiNa am4zaiaadofaaeaacaaIYaGaeqiWdaNaaGPaVlaadMgaaaGaaiOlaa aa@6F3E@ (20)

При этих значениях A, B и C правая часть (19) стремится к 0 по степенному закону: если α1, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdeMaeyizImQaaGymaiaacYcaaaa@4281@ не медленнее чем 1/ ξ α , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaaIXaaabaGaeqOVdG3aaWbaaSqabeaa cqaHXoqyaaaaaOGaaiilaaaa@42DC@ а в случае α>1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdeMaeyOpa4JaaGymaaaa@4124@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ не медленнее, чем 1/ξ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaaIXaaabaGaeqOVdGhaaiaac6caaaa@4108@ Следовательно:

φ z = ρgS 2πi ln z z 0 +Φ z , ψ z = ρgS 2πi ln z z 0 ρgS 2πi +Ψ z , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqaHgpGAdaqadaqaaiaadQhaaiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiabeg8aYjaadEgacaWGtbaabaGaaGOmai abec8aWjaaykW7caWGPbaaaiaaykW7caaMc8UaciiBaiaac6gadaqa daqaaiaadQhacqGHsislcaWG6bWaaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaGaey4kaSIaeuOPdy0aaeWaaeaacaWG6baacaGLOaGa ayzkaaGaaiilaiaaykW7aeaacqaHipqEdaqadaqaaiaadQhaaiaawI cacaGLPaaacqGH9aqpdaWcaaqaaiabeg8aYjaadEgacaWGtbaabaGa aGOmaiabec8aWjaaykW7caWGPbaaaiaaykW7caaMe8UaaGPaVlGacY gacaGGUbWaaeWaaeaacaWG6bGaeyOeI0IaamOEamaaBaaaleaacaaI WaaabeaaaOGaayjkaiaawMcaaiabgkHiTmaalaaabaGaeqyWdiNaam 4zaiaadofaaeaacaaIYaGaeqiWdaNaaGPaVlaadMgaaaGaaGPaVlaa ysW7caaMc8Uaey4kaSIaeuiQdK1aaeWaaeaacaWG6baacaGLOaGaay zkaaGaaiilaaaaaa@8AA4@ (21)

где

Φ z =o 1/ z ,z. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGafuOPdyKbauaadaqadaqaaiaadQhaaiaawIcacaGL PaaacqGH9aqpcaWGVbWaaeWaaeaadaWcgaqaaiaaigdaaeaadaabda qaaiaadQhaaiaawEa7caGLiWoaaaaacaGLOaGaayzkaaGaaiilaiaa ywW7caWG6bGaeyOKH4QaeyOhIuQaaiOlaaaa@5192@ (22)

Выразим через Φ(z) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeuOPdyKaaiikaiaadQhacaGGPaaaaa@4194@ вертикальные смещения на границе. Подставив (21) в (16), имеем после отделения мнимой части:

v= λ+2μ μ λ+μ ρgS 2π ln|ξ z 0 |+ImΦ ξ ρg y 2 4μ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamODaiabg2da9maalaaabaGaeq4UdWMaey4kaSIa aGOmaiabeY7aTbqaaiabeY7aTnaabmaabaGaeq4UdWMaey4kaSIaeq iVd0gacaGLOaGaayzkaaaaaiaaykW7daqadaqaaiabgkHiTmaalaaa baGaeqyWdiNaam4zaiaadofaaeaacaaIYaGaeqiWdaNaaGPaVdaaca aMc8UaaGPaVlGacYgacaGGUbGaaiiFaiabe67a4jabgkHiTiaadQha daWgaaWcbaGai4jGicdaaeqaaOGaaiiFaiabgUcaRiGacMeacaGGTb GaeuOPdy0aaeWaaeaacqaH+oaEaiaawIcacaGLPaaaaiaawIcacaGL PaaacqGHsisldaWcaaqaaiabeg8aYjacEc4GNbGaaGjcVlaadMhada ahaaWcbeqaaiaaikdaaaaakeaacaaI0aGaeqiVd0gaaiaac6caaaa@779D@

(23)

Мы учли, что согласно (14),

Imf= ρg y 2 /2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciysaiaac2gacaWGMbGaeyypa0ZaaSGbaeaacqaH bpGCcaWGNbGaiqjGdMhadGaLaYbaaSqajqjGbGaLakacuciIYaaaaa GcbaGaaGOmaaaacaGGUaaaaa@4D20@

Предположим, что рельеф «совсем финитный», т. е. для далеких точек, лежащих на линии рельефа:

y=0,ξ=x. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiabg2da9iaaicdacaGGSaGaaGzbVlabe67a 4jabg2da9iaadIhacaGGUaaaaa@4736@

Тогда в силу (23) вертикальные смещения в этих точках:

v= λ+2μ μ λ+μ ρgS 2π ln|x z 0 |+ImΦ x . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamODaiabg2da9maalaaabaGaeq4UdWMaey4kaSIa aGOmaiabeY7aTbqaaiabeY7aTnaabmaabaGaeq4UdWMaey4kaSIaeq iVd0gacaGLOaGaayzkaaaaaiaaykW7daqadaqaaiabgkHiTmaalaaa baGaeqyWdiNaam4zaiaadofaaeaacaaIYaGaeqiWdaNaaGPaVdaaca aMc8UaaGPaVlGacYgacaGGUbGaaiiFaiaadIhacqGHsislcaWG6bWa aSbaaSqaaiaaicdaaeqaaOGaaiiFaiabgUcaRiGacMeacaGGTbGaeu OPdy0aaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzkaaGa aiOlaaaa@6A26@ (24)

Дифференцируя (24) по x, получаем выражение для наклонов:

v x = λ+2μ μ λ+μ ρgS 2π x |x z 0 | 2 +Im Φ x . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiE aaaacqGH9aqpdaWcaaqaaiabeU7aSjabgUcaRiaaikdacqaH8oqBae aacqaH8oqBdaqadaqaaiabeU7aSjabgUcaRiabeY7aTbGaayjkaiaa wMcaaaaacaaMc8+aaeWaaeaacqGHsisldaWcaaqaaiabeg8aYjaadE gacaWGtbaabaGaaGOmaiabec8aWjaaykW7aaGaaGPaVlaaykW7daWc aaqaaiaadIhaaeaacaGG8bGaamiEaiabgkHiTiaadQhadaWgaaWcba GaaGimaaqabaGccaGG8bWaaWbaaSqabeaacaaIYaaaaaaakiabgUca RiGacMeacaGGTbGafuOPdyKbauaadaqadaqaaiaadIhaaiaawIcaca GLPaaaaiaawIcacaGLPaaacaGGUaaaaa@6E26@

Принимая во внимание (22), находим асимптотику при x: MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabgkziUkabg6HiLkaacQdaaaa@42DB@

v x = λ+2μ μ λ+μ ρgS 2πx +o 1/ x . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiE aaaacqGH9aqpcqGHsisldaWcaaqaaiabeU7aSjabgUcaRiaaikdacq aH8oqBaeaacqaH8oqBdaqadaqaaiabeU7aSjabgUcaRiabeY7aTbGa ayjkaiaawMcaaaaacaaMc8+aaSaaaeaacqaHbpGCcaaMc8Uaam4zai aadofaaeaacaaIYaGaeqiWdaNaaGPaVlaadIhacaaMc8oaaiaaykW7 caaMc8Uaey4kaSIaam4BamaabmaabaWaaSGbaeaacaaIXaaabaWaaq WaaeaacaWG4baacaGLhWUaayjcSdaaaaGaayjkaiaawMcaaiaac6ca aaa@6A79@ (25)

Таким образом, изменение наклона δv/ x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacqaH0oazcaaMc8UaeyOaIyRaamODaaqa aiabgkGi2kaadIhaaaaaaa@45CC@ вследствие равновесных вариаций барического поля в далеких от активной части рельефа точках:

δ v x λ+2μ μ λ+μ δρgS 2πx , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdq2aaSaaaeaacqGHciITcaWG2baabaGaeyOa IyRaamiEaaaaimaacqWF8iIFcqGHsisldaWcaaqaaiabeU7aSjabgU caRiaaikdacqaH8oqBaeaacqaH8oqBdaqadaqaaiabeU7aSjabgUca RiabeY7aTbGaayjkaiaawMcaaaaacaaMc8+aaSaaaeaacqaH0oazcq aHbpGCcaaMc8UaaGPaVlaadEgacaaMc8Uaam4uaaqaaiaaikdacqaH apaCcaaMc8UaamiEaiaaykW7aaGaaiilaaaa@65D7@ (26)

где δρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdihaaa@4127@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ вариация плотности воздуха. Формулы (25) и (26) верны и в случае «просто финитного» рельефа, когда функция y(x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiaacIcacaWG4bGaaiykaaaa@4116@ удовлетворяет условиям (2).

Будем считать температуру неизменной. Тогда:

δρδ p 0 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdihcdaGae8hpI4NaeqiTdqMaaGjc VlaadchadaWgaaWcbaGaaGimaaqabaGccaGGSaaaaa@485E@

где δ p 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaaGjcVlaadchadaWgaaWcbaGaaGimaaqa baaaaa@42D3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ вариация давления вблизи поверхности Земли. Если p0 увеличивается или падает на 50 мбар, т. е. на 5% по отношению к среднему значению, на 5% по отношению к средней плотности 1.2 кг/м3, изменяется плотность ρ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyWdiNaaiilaaaa@4032@ т. е.:

δρ 0.06 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGPaVpaaemaabaGaeqiTdqMaeqyWdiNaaGPaVdGa ay5bSlaawIa7aiaaykW7cqGHijYUcaaIWaGaaiOlaiaaicdacaaI2a aaaa@4D81@ кг/м3. (27)

В дальнейшем мы будем исходить из оценки (27). На первый взгляд, она кажется завышенной: дневная вариация давления редко превышает 15 мбар, плотности (при постоянной температуре) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ 0.018 кг/м3. Однако на промежутке времени длиной MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4haaa@3F2D@ 10 суток вариация давления в данном пункте вполне может составить 50 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ 55 мбар, а «… при последовательном прохождении циклона и антициклона… перепад давления может достигать 100 мбар» [Латынина, Васильев, 2001]. Этому перепаду давления отвечает вдвое большая, чем (27), вариация плотности: δρ =0.12 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGPaVpaaemaabaGam4jGes7aKjadEciHbpGCcaaM c8oacaGLhWUaayjcSdGaeyypa0JaaGimaiaac6cacaaIXaGaaGOmaa aa@4D94@ кг/м3.

Но плотность воздуха ρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyWdihaaa@3F82@ не только пропорциональна давлению, она обратно пропорциональна абсолютной температуре. Поэтому δρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdihaaa@4127@ может быть еще больше, если температура и давление меняются в противофазе. Рассмотрим ситуацию, типичную для среднеширотной зимы: циклон уступает место антициклону, при этом температура падает на 1520°. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGymaiaaiwdacqGHsislcaaIYaGaaGimaGWaaiab =blaWkab=5caUaaa@4470@ Тогда δρ0.180.20 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGPaVlabes7aKjabeg8aYHWaaiab=XJi+jaaicda caGGUaGaaGymaiaaiIdacqGHsislcaaIWaGaaiOlaiaaikdacaaIWa aaaa@4AD6@ кг/м3.

Модули λ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWgaaa@3F76@ и μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0gaaa@3F78@ мы примем равными:

λ=μ=3 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWMaeyypa0JaeqiVd0Maeyypa0JaaG4maiab gwSixlaaigdacaaIWaWaaWbaaSqabeaacaaIXaGaaGimaaaaaaa@4956@ н/м2. (28)

Значения вариации плотности (27) и коэффициентов Ламе (28) будем называть стандартными.

ПРИМЕР 3. Область D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ протяженный карьер (траншея, русло реки) сечением 100 × MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaey41aqlaaa@3FD8@ 100 м (см. рис. 2). Тогда S= 10 4 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbiqaayZccaWGtbGaeyypa0JaeyOeI0IaaGymaiaaicda daahaaWcbeqaaiaaisdaaaaaaa@4331@ м2. В силу (26), при стандартных значениях δρ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaqWaaeaacqaH0oazcqaHbpGCaiaawEa7caGLiWoa caGGSaaaaa@44F9@ λ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWgaaa@3F76@ и μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0gaaa@3F78@ на расстоянии x=1000M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabg2da9iaaigdacaaIWaGaaGimaiaaicda caaMe8EcLLpacaqGnbaaaa@4672@ от карьера вариация наклона, обусловленная изменением атмосферного давления, равна:

δ v x =0.5 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdq2aaSaaaeaacqGHciITcaWG2baabaGaeyOa IyRaamiEaaaacqGH9aqpcaaIWaGaaiOlaiaaiwdacqGHflY1caaIXa GaaGimamaaCaaaleqabaGaeyOeI0IaaGymaiaaicdaaaaaaa@4DBA@ рад. (29)

Мы получили величину в пределах точности наклономера. Но (29) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ это, скорее, оценка снизу: значения упругих модулей (28) характерны для консолидированной среды с преобладанием гранита и базальта. Там, где преобладают осадочные породы, модули на полпорядка меньше [Латынина, Васильев, 2001]. Кроме того, речь идет о дальней зоне; максимальное значение вариации наклона, очевидно, в несколько раз больше. Следовательно, эффект от равновесной вариации давления при данной геометрии рельефа может составить 5 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4NaaGynaiabgwSixlaaigdacaaIWaWa aWbaaSqabeaacqGHsislcaaIXaGaaGimaaaaaaa@463B@ рад, а это MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ доступная измерению величина.

 

Рис. 2. Протяженный карьер сечением 10x100 м

 

2. ОТОБРАЖЕНИЕ (1).

Рассмотрим подробно рельеф, определяемый отображением (1). Параметры a и c имеют следующий геометрический смысл: пусть H MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ глубина впадины, r MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ радиус кривизны в самой низкой точке дна. Тогда:

a 2 =H 2rH + H 2 ,c= 2rH +H. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyyamaaCaaaleqabaGaaGOmaaaakiabg2da9iaa dIeadaGcaaqaaiaaikdacaWGYbGaamisaaWcbeaakiabgUcaRiaadI eadaahaaWcbeqaaiaaikdaaaGccaGGSaGaaGzbVlaadogacqGH9aqp daGcaaqaaiaaikdacaWGYbGaamisaaWcbeaakiabgUcaRiaadIeaca GGUaaaaa@4FE7@

При фиксированном значении с рельеф представляет собой впадину, выраженную тем резче, чем больше a. Случаю a/c =1, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSGbaeaacaWGHbaabaGaam4yaaaacqGH9aqpcaaI XaGaaiilaaaa@4217@ показанному на рис. 3б, соответствует значение r=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOCaiabg2da9iaaicdacaGGUaaaaa@412B@ Тогда ложбина превращается в каньон, и:

a=c=H, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyyaiabg2da9iaadogacqGH9aqpcaWGibGaaiil aaaa@4319@

где H MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ глубина каньона. Функция y(x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiaacIcacaWG4bGaaiykaaaa@4116@ содержит точку заострения x=0,y=c, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabg2da9iaaicdacaGGSaGaaGjbVlaaysW7 caWG5bGaeyypa0JaeyOeI0IaaGPaVlaadogacaGGSaaaaa@4A5D@ в которой производная y (x) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGabmyEayaafaGaaiikaiaadIhacaGGPaaaaa@4122@ скачком меняется от MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOeI0IaeyOhIukaaa@4020@ до +. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaey4kaSIaeyOhIuQaiqjGc6caaaa@41E3@

Для области D, задаваемой отображением (1), решение задачи о наклонах, обусловленных равновесными вариациями барического поля, выражается через элементарные функции, правда, формула получается весьма громоздкая. Запишем нагрузочную функцию (14) в виде:

f(ξ)=ρg i0 ξ Im ξ 1 d ξ 1 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOzaiaacIcacqaH+oaEcaGGPaGaeyypa0JaeqyW diNaam4zamaapehabaGaciysaiaac2gacqaH+oaEdaWgaaWcbaGaaG ymaaqabaGccaaMe8Uaamizaiabe67a4naaBaaaleaacaaIXaaabeaa aeaacqGHEisPcaaMc8UaeyOeI0IaaGPaVlaadMgacWGIaAyXICTaaG jbVlaaicdaaeaacqaH+oaEa0Gaey4kIipakiaacYcaaaa@5FEF@

где интегрирование идет по контуру D. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraiaac6caaaa@40A3@ Прообразом точки ξD MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOVdGNaeyicI4SaeyOaIyRaamiraaaa@4338@ служит точка η, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4TdGMaaiilaaaa@401E@ принадлежащая действительной оси комплексной плоскости w. Перейдем к (действительной) переменной η. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4TdGMaaiOlaaaa@4020@ Из (1) имеем:

ξ=η a 2 ηic ,Imη=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOVdGNaeyypa0Jaeq4TdGMaeyOeI0YaaSaaaeaa caWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaeq4TdGMaeyOeI0Iaam yAaiaadogaaaGaaiilaiaaysW7caaMe8UaaGzbVlGacMeacaGGTbGa eq4TdGMaeyypa0JaaGimaiaac6caaaa@54B2@ (30)

Отсюда:

Imξ= a 2 2i 1 η+ic 1 ηic ; dξ dη =1+ a 2 ηic 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciysaiaac2gacqaH+oaEcqGH9aqpdaWcaaqaaiaa dggadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaamyAaaaadaqada qaamaalaaabaGaaGymaaqaaiabeE7aOjadSbOHRaWkcaWGPbGaam4y aaaacqGHsisldaWcaaqaaiaaigdaaeaacqaH3oaAcqGHsislcaWGPb Gaam4yaaaaaiaawIcacaGLPaaacaaMe8Uaai4oaiaaywW7caaMe8+a aSaaaeaacaWGKbGaeqOVdGhabaGaamizaiabeE7aObaacqGH9aqpca aIXaGamyhGgUcaRmaalaaabaGaamyyamaaCaaaleqabaGaaGOmaaaa aOqaamaabmaabaGaeq4TdGMaeyOeI0IaamyAaiaadogaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlaaaa@6B90@ (31)

Следовательно:

f η =f ξ η = = ρg a 2 2i η 1 η 1 +ic 1 η 1 ic 1+ a 2 η 1 ic 2 dη 1 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaWGMbWaaeWaaeaacqaH3oaAaiaawIcacaGL PaaacqGH9aqpcaWGMbWaaeWaaeaacqaH+oaEdaqadaqaaiabeE7aOb GaayjkaiaawMcaaaGaayjkaiaawMcaaiabg2da9aqaaiabg2da9maa laaabaGaeqyWdiNaam4zaiaadggadaahaaWcbeqaaiaaikdaaaaake aacaaIYaGaamyAaaaadaWdXbqaamaabmaabaWaaSaaaeaacaaIXaaa baGaeq4TdG2aaSbaaSqaaiaaykW7caaIXaaabeaakiadIbOHRaWkca WGPbGaam4yaaaacqGHsisldaWcaaqaaiaaigdaaeaacqaH3oaAdaWg aaWcbaGaaGPaVlaaigdaaeqaaOGamigGgkHiTiaadMgacaWGJbaaaa GaayjkaiaawMcaaiaaykW7daqadaqaaiaaigdacqGHRaWkdaWcaaqa aiaadggadaahaaWcbeqaaiaaikdaaaaakeaadaqadaqaaiabeE7aOn aaBaaaleaacaaMc8UaaGymaiaaykW7aeqaaOGaaGzaVlabgkHiTiaa dMgacaWGJbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaki aaysW7aiaawIcacaGLPaaacaWGKbGaeq4TdGMaaGPaVpaaBaaaleaa caaIXaaabeaaaeaacqGHEisPaeaacqaH3oaAa0Gaey4kIipakiaac6 caaaaa@8633@

Очевидно, что этот интеграл «берется». Результат можно записать в следующем виде:

f η =ρg S 2πi ln ηic ln η+ic + a 4 η/ c2i 4 ηic 2 , Imη=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaWGMbWaaeWaaeaacqaH3oaAaiaawIcacaGL PaaacqGH9aqpcqaHbpGCcaWGNbWaaeWaaeaadaWcaaqaaiaadofaae aacaaIYaGaeqiWdaNaaGPaVlaadMgaaaWaaeWaaeaaciGGSbGaaiOB amaabmaabaGamigGeE7aOjaaygW7cWGIaAOeI0IaamyAaiaadogaai aawIcacaGLPaaacaaMb8UamilGgkHiTiGacYgacaGGUbWaaeWaaeaa cWawas4TdGMaaGzaVlabgUcaRiaadMgacaWGJbaacaGLOaGaayzkaa aacaGLOaGaayzkaaGaey4kaSIaamyyamaaCaaaleqabaGaaGinaaaa kmaalaaabaWaaSGbaeaacqaH3oaAaeaacaWGJbGaeyOeI0IaaGOmai aadMgaaaaabaGaaGinaiaaykW7daqadaqaaiabeE7aOjabgkHiTiaa dMgacaWGJbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaO GaayjkaiaawMcaaiaaykW7caGGSaaabaGaciysaiaac2gacqaH3oaA cqGH9aqpcaaIWaGaaiOlaaaaaa@8131@ (32)

Здесь S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ «алгебраическая площадь» области, заключенной между линией D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ и осью Imz=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciysaiaac2gacaWG6bGaeyypa0JaaGimaiaac6ca aaa@42F3@ Формула для S приведена в п. 1, см. (5). Задача (11) для области D эквивалентна краевой задаче для нижней полуплоскости с условием, заданным на действительной оси:

φ(η)+ ξ(η) ξ (η) ______ φ (η) _______ + ψ(η) ______ =f(η),Imη=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiabeE7aOjaacMcacqGHRaWkdaWc aaqaaiabe67a4jaacIcacqaH3oaAcaGGPaaabaWaaCbiaeaacuaH+o aEgaqbaiaacIcacqaH3oaAcaGGPaaaleqabaGaai4xaiaac+facaGG FbGaai4xaiaac+facaGGFbaaaaaakmaaxacabaGafqOXdOMbauaaca GGOaGaeq4TdGMaaiykaaWcbeqaaiaac+facaGGFbGaai4xaiaac+fa caGGFbGaai4xaiaac+faaaGccqGHRaWkdaWfGaqaaiabeI8a5jaacI cacqaH3oaAcaGGPaaaleqabaGaai4xaiaac+facaGGFbGaai4xaiaa c+facaGGFbaaaOGaeyypa0JaamOzaiaacIcacqaH3oaAcaGGPaGaaG PaVlaacYcacaaMe8Uaciysaiaac2gacqaH3oaAcqGH9aqpcaaIWaGa aiOlaaaa@7803@ (33)

Здесь φ(η) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiabeE7aOjaacMcaaaa@4284@ и ψ(η) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiabeE7aOjaacMcaaaa@4295@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ граничные значения аналитических в нижней полуплоскости функций φ(w) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadEhacaGGPaaaaa@41D4@ и ψ(w); MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadEhacaGGPaGaai4oaaaa@42A4@ зависимость ξ(η) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOVdGNaaiikaiabeE7aOjaacMcaaaa@428A@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ см. (30). Опуская промежуточные вычисления, приведем решение задачи (33) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (32):

φ w = ρgS 2πi ln wic + + ρg a 4 4 a 2 c 4 c 2 2 a 2 1 wic i wic 2 ; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqaHgpGAdaqadaqaaiaadEhaaiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiabeg8aYjaadEgacaWGtbaabaGaaGOmai abec8aWjaaykW7caWGPbaaaiaaykW7caaMc8UaciiBaiaac6gadaqa daqaaiaadEhacqGHsislcaWGPbGaam4yaaGaayjkaiaawMcaaiabgU caRaqaaiabgUcaRmaalaaabaGaaGPaVlabeg8aYjaadEgacaWGHbWa aWbaaSqabeaacaaI0aaaaaGcbaGaaGinaiaaykW7aaWaaeWaaeaada WcaaqaaiaayIW7caWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaam4y aiaaykW7daqadaqaaiaaisdacaWGJbWaaWbaaSqabeaacaaIYaaaaO GaaGzaVlabgkHiTiaaikdacaWGHbWaaWbaaSqabeaacaaIYaaaaaGc caGLOaGaayzkaaaaamaalaaabaGaaGymaaqaaiaadEhacqGHsislca WGPbGaam4yaaaacqGHsisldaWcaaqaaiaadMgaaeaacaaMc8+aaeWa aeaacaWG3bGaeyOeI0IaamyAaiaadogaaiaawIcacaGLPaaadaahaa WcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaai4oaaaaaa@7FA7@ (34)

ψ w = ρgS 2πi ln wic + ρg a 2 8 c 2 × × c 16 c 4 a 4 2 c 2 a 2 w 2 i 64 c 8 32 c 6 a 2 7 c 2 a 6 +2 a 8 w c 5 32 c 4 +10 a 4 48 c 2 a 2 4 c 2 a 2 2 c 2 a 2 wic 2 + a 2 wic . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacqaHipqEdaqadaqaaiaadEhaaiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiabeg8aYjaadEgacaWGtbaabaGaaGOmai abec8aWjaaykW7caWGPbaaaiaaykW7caaMc8UaciiBaiaac6gadaqa daqaaiaadEhacqGHsislcaWGPbGaam4yaaGaayjkaiaawMcaaiabgU caRmaalaaabaGaaGPaVlabeg8aYjaadEgacaWGHbWaaWbaaSqabeaa caaIYaaaaaGcbaGaaGioaiaadogadaahaaWcbeqaaiaaikdaaaGcca aMc8oaaiabgEna0cqaaiabgEna0oaalaaabaGaam4yaiaaykW7daqa daqaaiaaigdacaaI2aGaam4yamaaCaaaleqabaGaaGinaaaakiabgk HiTiaadggadaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPaaadaqa daqaaiaaikdacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam yyamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaadEhadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGPbGaaGPaVpaabmaabaGaaG OnaiaaisdacaWGJbWaaWbaaSqabeaacaaI4aaaaOGaeyOeI0IaaG4m aiaaikdacaWGJbWaaWbaaSqabeaacaaI2aaaaOGaamyyamaaCaaale qabaGaaGOmaaaakiabgkHiTiaaiEdacaWGJbWaaWbaaSqabeaacaaI YaaaaOGaamyyamaaCaaaleqabaGaaGOnaaaakiabgUcaRiaaikdaca WGHbWaaWbaaSqabeaacaaI4aaaaaGccaGLOaGaayzkaaGaam4Daiab gkHiTiaadogadaahaaWcbeqaaiaaiwdaaaGcdaqadaqaaiaaiodaca aIYaGaam4yamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaigdacaaI WaGaamyyamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaisdacaaI4a Gaam4yamaaCaaaleqabaGaaGOmaaaakiaadggadaahaaWcbeqaaiaa ikdaaaaakiaawIcacaGLPaaaaeaadaqadaqaaiaaisdacaWGJbWaaW baaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOm aaaaaOGaayjkaiaawMcaamaabmaabaGaaGOmaiaadogadaahaaWcbe qaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaGc caGLOaGaayzkaaWaaeWaaeaadaqadaqaaiaadEhacqGHsislcaWGPb Gaam4yaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUca RiaadggadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaadaqada qaaiaadEhacqGHsislcaWGPbGaam4yaaGaayjkaiaawMcaaaaacaGG Uaaaaaa@C0D3@ (35)

В том, что функции (34) и (35) действительно решают задачу (33) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (32) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ (30), можно убедиться непосредственной подстановкой.

 

Рис. 3. Наклоны на границе области задаваемой отображением (1), при λ=μ и различных значениях ε=a2c2: (а) – слабый рельеф, ε=01.; (б) – каньон (линия рельефа содержит особую точку), ε=1; (в) – промежуточный случай, ε=0,5. Толстая линия – линия рельефа; тонкая линия – точное значение 2μv/xδρgH; пунктир – асимптотика (26).

 

Пользуясь явными выражениями для потенциалов φ(w) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadEhacaGGPaaaaa@41D4@ и ψ(w), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadEhacaGGPaGaaiilaaaa@4295@ мы можем найти вектор смещений:

2μ u+iv =κφ w z w z w _______ φ w _______ ψ w ______ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGOmaiabeY7aTnaabmaabaGaamyDaiabgUcaRiaa dMgacaWG2baacaGLOaGaayzkaaGaeyypa0JaeqOUdSMaeqOXdO2aae WaaeaacaWG3baacaGLOaGaayzkaaGaeyOeI0YaaSaaaeaacaWG6bWa aeWaaeaacaWG3baacaGLOaGaayzkaaaabaWaaCbiaeaaceWG6bGbau aadaqadaqaaiaadEhaaiaawIcacaGLPaaaaSqabeaacaGGFbGaai4x aiaac+facaGGFbGaai4xaiaac+facaGGFbaaaaaakmaaxacabaGafq OXdOMbauaadaqadaqaaiaadEhaaiaawIcacaGLPaaaaSqabeaacaGG FbGaai4xaiaac+facaGGFbGaai4xaiaac+facaGGFbaaaOGaeyOeI0 YaaCbiaeaacqaHipqEdaqadaqaaiaadEhaaiaawIcacaGLPaaaaSqa beaacaGGFbGaai4xaiaac+facaGGFbGaai4xaiaac+faaaGccaGGSa aaaa@7180@ (36)

где κ=(λ+3μ)/(λ+μ). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOUdSMaeyypa0JaaiikaiabeU7aSjabgUcaRiaa iodacqaH8oqBcaGGPaGaai4laiaacIcacqaH7oaBcqGHRaWkcqaH8o qBcaGGPaGaaiOlaaaa@4DE6@ Формула (36) равносильна формуле Колосова (15). Она справедлива при всех w с неположительной мнимой частью, а значит, с ее помощью мы можем найти компоненты смещения во всей области D. Для того, чтобы вычислить компоненты тензора деформации u/x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamyDaiaac+cacqGHciITcaWG4baaaa@4338@ и v/x, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamODaiaac+cacqGHciITcaWG4bGaaiil aaaa@43E9@ продифференцируем (36) по x:

2μ u x +i v x = =κ φ w z w 1 z w _______ φ w _______ + ψ w ______ +z w d dw φ w _______ z w _______ ; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaaIYaGaeqiVd02aaeWaaeaadaWcaaqaaiab gkGi2kaadwhaaeaacqGHciITcaWG4baaaiabgUcaRiaadMgacaaMe8 +aaSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiEaaaaaiaawIca caGLPaaacqGH9aqpaeaacqGH9aqpcqaH6oWAdaWcaaqaaiqbeA8aQz aafaWaaeWaaeaacaWG3baacaGLOaGaayzkaaaabaGabmOEayaafaWa aeWaaeaacaWG3baacaGLOaGaayzkaaaaaiabgkHiTmaalaaabaGaaG ymaaqaamaaxacabaGabmOEayaafaWaaeWaaeaacaWG3baacaGLOaGa ayzkaaaaleqabaGaai4xaiaac+facaGGFbGaai4xaiaac+facaGGFb Gaai4xaaaaaaGcdaqadaqaamaaxacabaGafqOXdOMbauaadaqadaqa aiaadEhaaiaawIcacaGLPaaaaSqabeaacaGGFbGaai4xaiaac+faca GGFbGaai4xaiaac+facaGGFbaaaOGaey4kaSYaaCbiaeaacuaHipqE gaqbamaabmaabaGaam4DaaGaayjkaiaawMcaaaWcbeqaaiaac+faca GGFbGaai4xaiaac+facaGGFbGaai4xaaaakiaducOHRaWkcaWG6bWa aeWaaeaacaWG3baacaGLOaGaayzkaaGaaGjbVpaalaaabaGaamizaa qaaiaadsgacaWG3baaamaalaaabaWaaCbiaeaacuaHgpGAgaqbamaa bmaabaGaam4DaaGaayjkaiaawMcaaaWcbeqaaiaac+facaGGFbGaai 4xaiaac+facaGGFbGaai4xaiaac+faaaaakeaadaWfGaqaaiqadQha gaqbamaabmaabaGaam4DaaGaayjkaiaawMcaaaWcbeqaaiaac+faca GGFbGaai4xaiaac+facaGGFbGaai4xaiaac+faaaaaaaGccaGLOaGa ayzkaaGaai4oaaaaaa@9D34@ (37)

мы учли аналитичность функций φ(w), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOXdOMaaiikaiaadEhacaGGPaGaaiilaaaa@4284@ ψ(w) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiYdKNaaiikaiaadEhacaGGPaaaaa@41E5@ и z(w). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEaiaacIcacaWG3bGaaiykaiaac6caaaa@41C8@ Подставив (34) и (35) в (37), положив w=η, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4Daiabg2da9iabeE7aOjaacYcaaaa@4220@ где Imη=0, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaciysaiaac2gacqaH3oaAcqGH9aqpcaaIWaGaaiil aaaa@439E@ и, отделив мнимую часть, получим формулу, которая позволяет вычислить наклоны на линии рельефа:

2μ v x = ερg c 2 8 2ε η η 2 + c 2 η 4 +2 η 2 c 2 1+ε + c 4 1ε 2 × × η 2 κ+1 η 2 ε 2 6ε+8 + c 2 ε 3 10 ε 2 +4ε+16 + + c 4 13κ ε 3 + 13κ11 ε 2 + 1022κ ε+8 κ+1 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaaIYaGaeqiVd0MaaGzaVlaaygW7caaMe8+a aSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiEaaaacqGH9aqpda Wcaaqaaiabew7aLjaaykW7cqaHbpGCcaaMc8Uaam4zaiaaykW7caWG JbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGioamaabmaabaGaiigGik dacaaMb8UaeyOeI0IaeqyTdugacaGLOaGaayzkaaaaamaalaaabaGa eq4TdGgabaWaaeWaaeaacqaH3oaAdaahaaWcbeqaaiaaikdaaaGcca aMb8UamqjGgUcaRiaadogadaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaadaqadaqaaiadybiH3oaAdGawaYbaaSqajGfGbGawakacyb iI0aaaaOGaiGfGygW7cWamaA4kaSIaaGzaVlaaikdacqaH3oaAdaah aaWcbeqaaiaaikdaaaGccaWGJbWaaWbaaSqabeaacaaIYaaaaOGaaG zaVpaabmaabaGaaGymaiaaygW7cWGNaA4kaSIaeqyTdugacaGLOaGa ayzkaaGaaGzaVlad8cOHRaWkcGGNao4yamaaCaaaleqabaGaaGinaa aakiaaygW7daqadaqaaiaaigdacaaMb8Uam4jGgkHiTiabew7aLbGa ayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaa aacqGHxdaTaeaacWaxaA41aq7aaeqaaeaacqaH3oaAdaahaaWcbeqa aiaaikdaaaGccaaMb8+aaeWaaeaacWawasOUdSMaaGzaVlabgUcaRi aaigdaaiaawIcacaGLPaaadaqadaqaaiabeE7aOnaaCaaaleqabaGa aGOmaaaakmaabmaabaGaeqyTdu2aaWbaaSqabeaacaaIYaaaaOGaaG zaVlabgkHiTiaaiAdacqaH1oqzcqGHRaWkcaaI4aaacaGLOaGaayzk aaGaey4kaSIaam4yamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaeq yTdu2aaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaicdacqaH 1oqzdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI0aGaeqyTduMaey 4kaSIaaGymaiaaiAdaaiaawIcacaGLPaaaaiaawIcacaGLPaaacqGH RaWkaiaawIcaaaqaamaabiaabaGaey4kaSIaaGjbVlaadogadaahaa WcbeqaaiaaisdaaaGccaaMb8+aaeWaaeaadaqadaqaaiaaigdacaaM b8UamGkGgkHiTiaaiodacqaH6oWAaiaawIcacaGLPaaacqaH1oqzda ahaaWcbeqaaiaaiodaaaGccaWLa8UaaGzaVladucOHRaWkdaqadaqa aiaaigdacaaIZaGaeqOUdSMam4eGgkHiTiaaygW7caaIXaGaaGymaa GaayjkaiaawMcaaiabew7aLnaaCaaaleqabaGaaGOmaaaakiaaygW7 cqGHRaWkdaqadaqaaiacSbiIXaGaiWgGicdacWaraAOeI0IaaGzaVl aaikdacaaIYaGaeqOUdSgacaGLOaGaayzkaaGaeqyTduMaaGzaVlad OcOHRaWkcaaI4aWaaeWaaeaacWawasOUdSMaaGzaVlabgUcaRiaaig daaiaawIcacaGLPaaaaiaawIcacaGLPaaaaiaawMcaaiaaykW7caGG Saaaaaa@09E9@ (38)

где

ε= a 2 / c 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyypa0ZaaSGbaeaacaWGHbWaaWbaaSqa beaacaaIYaaaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGcca GGUaaaaa@44EB@ (39)

Замечание . Параметр ε MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdugaaa@3F69@ характеризует «крутизну» рельефа: чем ε MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdugaaa@3F69@ меньше, тем меньше углы наклона элементов рельефа к горизонту. Справедлива оценка:

| y x | 3 3 ε 1ε 8+ε , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiqadMhagaqbamaabmaabaGaamiEaaGaayjk aiaawMcaaiaacYhacqGHKjYOdaWcaaqaaiaaiodadaGcaaqaaiaaio dacaaMc8oaleqaaOGaeqyTdugabaWaaOaaaeaacaaIXaGaeyOeI0Ia eqyTdugaleqaaOWaaeWaaeaacaaI4aGaey4kaSIaeqyTdugacaGLOa GaayzkaaaaaiaacYcaaaa@52E0@ (40)

равенство достигается при η=±c 1ε /3 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4TdGMaeyypa0JaeyySaeRaaGjbVlaadogadaGc aaqaamaalyaabaWaaeWaaeaacaaIXaGaeyOeI0IaeqyTdugacaGLOa GaayzkaaaabaGaaG4maaaaaSqabaGccaGGUaaaaa@4B59@

Выразим через ε MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdugaaa@3F69@ и c глубину ложбины H и «площадь» S: из (4), (5) и (39) имеем:

H=εc MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iabew7aLjaaykW7caWGJbaaaa@43AF@ (41)

и

S=πε c 2 1ε/4 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4uaiabg2da9iabgkHiTiabec8aWjabew7aLjaa ykW7caWGJbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey OeI0IaeqyTduMaai4laiaaisdaaiaawIcacaGLPaaacaGGUaaaaa@4E52@ (42)

Целесообразно ввести еще один линейный параметр, ширину ложбины b:

b= S /H . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiabg2da9maalyaabaWaaqWaaeaacaWGtbaa caGLhWUaayjcSdaabaGaamisaaaacaGGUaaaaa@453E@ (43)

В терминах c и ε: MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaiWfGcQdaaaa@410A@

b=πc 1ε/4 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiabg2da9iabec8aWjaaykW7caWGJbWaaeWa aeaadaWcgaqaaiaaigdacqGHsislcqaH1oqzaeaacaaI0aaaaaGaay jkaiaawMcaaiaac6caaaa@4A3D@ (44)

Ниже мы рассмотрим два крайних случая: ε<<1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyipaWJaeyipaWJaaGymaaaa@422C@ (слабый рельеф) и ε=1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyypa0JaaGymaaaa@412A@ (каньон), а также среднее значение ε=0.5. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyypa0JaaGimaiaac6cacaaI1aGaaiOl aaaa@434C@

1). Слабый рельеф (ε<<1). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiabew7aLjabgYda8iabgYda8iaaigdacaGG PaGaaiOlaaaa@4437@ В этом случае x=η+O(ε). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabg2da9iabeE7aOjabgUcaRiaad+eacaGG OaGaeqyTduMaaiykaiaac6caaaa@46D9@ Поэтому из (38) имеем:

2μ v x =ρ κ+1 gε c 2 2 x x 2 + c 2 +O ε 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGOmaiabeY7aTjaaykW7caaMc8+aaSaaaeaacqGH ciITcaWG2baabaGaeyOaIyRaamiEaaaacqGH9aqpcqaHbpGCcaaMc8 UaeyyXIC9aaeWaaeaacqaH6oWAcqGHRaWkcaaIXaaacaGLOaGaayzk aaWaaSaaaeaacaWGNbGaeqyTduMaaGPaVlaadogadaahaaWcbeqaai aaikdaaaaakeaacaaIYaaaamaalaaabaGaamiEaaqaaiaadIhadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGJbWaaWbaaSqabeaacaaIYa aaaaaakiabgUcaRiaad+eadaqadaqaaiabew7aLnaaCaaaleqabaGa aGOmaaaaaOGaayjkaiaawMcaaiaac6caaaa@669B@

Вариация наклонов, с точностью до величин порядка ε 2 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdu2aaWbaaSqabeaacaaIYaaaaOGaaiilaaaa @410C@ равна:

δ v x =δρgε c 2 λ+2μ 2μ λ+μ x x 2 + c 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaaGPaVpaalaaabaGaeyOaIyRaamODaaqa aiabgkGi2kaadIhaaaGaeyypa0JaeqiTdqMaeqyWdiNaaGPaVlaadE gacqaH1oqzcaaMc8Uaam4yamaaCaaaleqabaGaaGOmaaaakmaalaaa baWaaeWaaeaacqaH7oaBcqGHRaWkcaaIYaGaeqiVd0gacaGLOaGaay zkaaaabaGaaGOmaiabeY7aTnaabmaabaGaeq4UdWMaey4kaSIaeqiV d0gacaGLOaGaayzkaaaaamaalaaabaGaamiEaaqaaiaadIhadaahaa WcbeqaaiaaikdaaaGccqGHRaWkcaWGJbWaaWbaaSqabeaacaaIYaaa aaaakiaac6caaaa@6709@ (45)

Правая часть (45) достигает экстремума при x=±c, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabg2da9iabgglaXkaaysW7caWGJbGaaiil aaaa@44D8@ в точках с ординатой y= cε/2 =H/2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiabg2da9iabgkHiTiaaykW7daWcgaqaaiaa dogacqaH1oqzaeaacaaIYaaaaiabg2da9iabgkHiTmaalyaabaGaam isaaqaaiaaikdaaaGaaiOlaaaa@49E3@ В этих точках:

δ v x = λ+2μ 4μ λ+μ δρ gH. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaqWaaeaacqaH0oazcaaMc8+aaSaaaeaacqGHciIT caWG2baabaGaeyOaIyRaamiEaaaaaiaawEa7caGLiWoacqGH9aqpda WcaaqaamaabmaabaGaeq4UdWMaey4kaSIaaGOmaiabeY7aTbGaayjk aiaawMcaaaqaaiaaisdacqaH8oqBdaqadaqaaiabeU7aSjabgUcaRi abeY7aTbGaayjkaiaawMcaaaaadaabdaqaaiabes7aKjabeg8aYbGa ay5bSlaawIa7aiaaykW7caWGNbGaamisaiaac6caaaa@6355@

ПРИМЕР 4. Зададимся стандартными значениями δρ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdiNaaiilaaaa@41D7@ λ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWgaaa@3F76@ и μ; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0Maai4oaaaa@4037@ (см. (27), (28)). Пусть c=1000M,ε=0.1. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4yaiabg2da9iaaigdacaaIWaGaaGimaiaaicda caaMe8EcLzpacaqGnbGccaGGSaGaaGjbVlaaysW7cqaH1oqzcqGH9a qpcaaIWaGaaiOlaiaaigdacaGGUaaaaa@4FB4@ Тогда глубина ложбины H=εc=100M. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iabew7aLjaaykW7caaMe8Uaam4y aiabg2da9iaaigdacaaIWaGaaGimaiaaysW7juM9aiaab2eakiaab6 caaaa@4CEC@ В силу (40), угол наклона элементов рельефа к горизонту не превышает 4°, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGinaGWaaiab=blaWkaacYcaaaa@4120@ т. е. рельеф действительно выражен слабо. Тем не менее, на расстоянии 1000 м от дна ложбины вариация барического наклона согласно (45) равна:

δ v x =7.2 10 10 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaaGPaVpaalaaabaGaeyOaIyRaamODaaqa aiabgkGi2kaadIhaaaGaeyypa0JaaG4naiaac6cacaaIYaGaeyyXIC TaaGymaiaaicdadaahaaWcbeqaaiabgkHiTiaaigdacaaIWaaaaOGa aiilaaaa@5003@

что на порядок превышает точность измерений.

Далее: сравним (45) и асимптотику (26) (см. рис. 3а). При ε<<1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyipaWJaeyipaWJaaGymaaaa@422C@ «площадь» S и эффективная ширина b:

S=πε c 2 ,b=πc, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4uaiabg2da9iabgkHiTiabec8aWjabew7aLjaa ykW7caWGJbWaaWbaaSqabeaacaaIYaaaaOGaaiilaiaaywW7caWGIb Gaeyypa0JaeqiWdaNaaGPaVlaadogacaGGSaaaaa@5062@ (46)

(см. (41), (43)). Учитывая (46), запишем (45) в виде:

δ v x =δρgS λ+2μ 2πμ λ+μ x x 2 + b/π 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaaGPaVpaalaaabaGaeyOaIyRaamODaaqa aiabgkGi2kaadIhaaaGaeyypa0JaeyOeI0IaeqiTdqMaeqyWdiNaaG PaVlaadEgacaaMc8Uaam4uamaalaaabaWaaeWaaeaacqaH7oaBcqGH RaWkcaaIYaGaeqiVd0gacaGLOaGaayzkaaaabaGaaGOmaiabec8aWj abeY7aTnaabmaabaGaeq4UdWMaey4kaSIaeqiVd0gacaGLOaGaayzk aaaaamaalaaabaGaamiEaaqaaiaadIhadaahaaWcbeqaaiaaikdaaa GccqGHRaWkdaqadaqaaiaadkgacaGGVaGaeqiWdahacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaaaakiaac6caaaa@6B01@ (47)

Отношение правых частей (26) и (47) равно 1+ b 2 / (πx) 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGymaiabgUcaRiaadkgadaahaaWcbeqaaiaaikda aaGccaGGVaGaaiikaiabec8aWjaaykW7caWG4bGaaiykamaaCaaale qabaGaaGOmaaaakiaac6caaaa@492F@ При x=±b MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabg2da9iabgglaXkaadkgaaaa@429A@ оно отличается от 1 на 1/ π 2 0.1. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGymaiaac+cacqaHapaCdaahaaWcbeqaaiaaikda aaGccqGHijYUcaaIWaGaaiOlaiaaigdacaGGUaaaaa@466A@ Это значит, что вне интервала x(b,b) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiabgIGiolaacIcacqGHsislcaWGIbGaaiil aiaadkgacaGGPaaaaa@4507@ асимптотика (26) дает ошибку меньше 10 %.

2). Каньон (ε=1). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiabew7aLjabg2da9iaaigdacaGGPaGaaiOl aaaa@4335@ Глубина каньона H=c; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iaadogacaGG7aaaaa@413C@ самая низкая точка является точкой заострения (см. рис. 3б). Заменив c на H и ε MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdugaaa@3F69@ на 1, из (38) имеем:

v x = ρg H 2 16μ η 2 κ+1 3 η 2 +11 H 2 +4 H 4 2κ η η 2 + H 2 η 2 +4 H 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiE aaaacqGH9aqpdaWcaaqaaiaaykW7cqaHbpGCcaaMc8Uaam4zaiaayk W7caWGibWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGymaiaaiAdacqaH 8oqBcaaMe8oaaiabgwSixpaalaaabaGaeq4TdG2aaWbaaSqabeaaca aIYaaaaOWaaeWaaeaacqaH6oWAcqGHRaWkcaaIXaaacaGLOaGaayzk aaWaaeWaaeaacaaIZaGaeq4TdG2aaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaaGymaiaaigdacaaMc8UaamisamaaCaaaleqabaGaaGOmaaaa aOGaayjkaiaawMcaaiabgUcaRiaaisdacaWGibWaaWbaaSqabeaaca aI0aaaaOWaaeWaaeaacaaIYaGaeyOeI0IaeqOUdSgacaGLOaGaayzk aaaabaGaeq4TdGMaaGPaVpaabmaabaGaeq4TdG2aaWbaaSqabeaaca aIYaaaaOGaey4kaSIaamisamaaCaaaleqabaGaaGOmaaaaaOGaayjk aiaawMcaamaabmaabaGaeq4TdG2aaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaaGinaiaadIeadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGL PaaaaaGaaiOlaaaa@8051@

(48)

Если λ=μ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4UdWMaeyypa0JaeqiVd0Maaiilaaaa@42E2@ то κ=2. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOUdSMaeyypa0JaaGOmaiaac6caaaa@41E8@ В этом (и только в этом) случае правая часть (48) при η0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4TdGMaeyOKH4QaaGimaaaa@4215@ остается ограниченной. Формула (48) переходит в:

v x = 3ρg H 2 16μ η 3 η 2 +11 H 2 ( η 2 + H 2 ) η 2 +4 H 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWaaSaaaeaacqGHciITcaWG2baabaGaeyOaIyRaamiE aaaacqGH9aqpdaWcaaqaaiaaiodacaaMc8UaeqyWdiNaaGPaVlaadE gacaaMc8UaamisamaaCaaaleqabaGaaGOmaaaaaOqaaiaaigdacaaI 2aGaeqiVd0gaaiabgwSixpaalaaabaGaeq4TdGMaaGPaVpaabmaaba GaaG4maiabeE7aOnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigda caaIXaGaaGPaVlaadIeadaahaaWcbeqaaiaaikdaaaaakiaawIcaca GLPaaaaeaacaGGOaGaeq4TdG2aaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamisamaaCaaaleqabaGaaGOmaaaakiaacMcadaqadaqaaiabeE 7aOnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdacaWGibWaaWba aSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaiaac6caaaa@6F94@

В вариациях:

δ v x = 3δρg H 2 16μ η 3 η 2 +11 H 2 η 2 + H 2 η 2 +4 H 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaaGjbVpaalaaabaGaeyOaIyRaamODaaqa aiabgkGi2kaadIhaaaGaeyypa0ZaaSaaaeaacaaIZaGaaGPaVlabes 7aKjabeg8aYjaaykW7caWGNbGaaGPaVlaadIeadaahaaWcbeqaaiaa ikdaaaaakeaacaaIXaGaaGOnaiabeY7aTbaacqGHflY1daWcaaqaai abeE7aOjaaykW7daqadaqaaiaaiodacqaH3oaAdaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIXaGaaGymaiaaykW7caaMc8UaamisamaaCa aaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaamaabmaabaGaeq4T dG2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamisamaaCaaaleqaba GaaGOmaaaaaOGaayjkaiaawMcaamaabmaabaGaeq4TdG2aaWbaaSqa beaacaaIYaaaaOGaey4kaSIaaGinaiaadIeadaahaaWcbeqaaiaaik daaaaakiaawIcacaGLPaaaaaGaaiOlaaaa@7626@ (49)

Результаты вычислений по формуле (49), в сравнении с асимптотикой (26), показаны на рис. 3б. Удовлетворительную точность (ошибку < MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpaWdaaa@3EC9@ 10 %) асимптотика дает при |x|>1.6b, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiaadIhacaGG8bGaaGPaVlabg6da+iaaykW7 caaIXaGaaiOlaiaaiAdacaaMc8UaamOyaiaacYcaaaa@4A2C@ где b MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ эффективная ширина (44); в рассматриваемом случае b=3πH/4. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiabg2da9iaaiodacqaHapaCcaaMc8Uaamis aiaac+cacaaI0aGaaiOlaaaa@46A4@

Абсолютная величина вариации наклона достигает максимума при η±H, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4TdGMaeyisISRaeyySaeRaaGPaVlaadIeacaGG Saaaaa@4615@ в точках с ординатой yH/2, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiabgIKi7kabgkHiTiaadIeacaGGVaGaaGOm aiaacYcaaaa@444A@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ как и в случае слабого рельефа. Этот максимум равен 0.25δρgH/μ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyisISRaaGimaiaac6cacaaIYaGaaGynaiaaykW7 cqaH0oazcqaHbpGCcaaMc8Uaam4zaiaaykW7caWGibGaai4laiabeY 7aTjaacYcaaaa@4F32@ что при H=100M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iaaigdacaaIWaGaaGimaiaaykW7 caaMe8EcLbqacaqGnbaaaa@461B@ (и стандартных значениях остальных параметров) составляет 5 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGynaiabgwSixlaaigdacaaIWaWaaWbaaSqabeaa cqGHsislcaaIXaGaaGimaaaaaaa@44CF@ рад.

3). Промежуточный случай (ε=0.5). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiabew7aLjabg2da9iaaicdacaGGUaGaaGyn aiaacMcacaGGUaaaaa@44A5@ Результаты вычислений по формуле (38) при этом значении ε MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTdugaaa@3F69@ и κ=2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqOUdSMaeyypa0JaaGOmaaaa@4136@ представлены на рис. 3в. Асимптотика (26) дает погрешность, не превышающую 10 %, при условии |x|>1.4b, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiaadIhacaGG8bGaaGPaVlabg6da+iaaykW7 caaIXaGaaiOlaiaaisdacaaMc8UaamOyaiaacYcaaaa@4A2A@ где b MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ ширина ложбины: при ε=0.5 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyypa0JaaGimaiaac6cacaaI1aaaaa@429A@ из (44) и (41) имеем b=7πH/4. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiabg2da9iaaiEdacqaHapaCcaaMc8Uaamis aiaac+cacaaI0aGaaiOlaaaa@46A8@

Вариация наклона достигает максимума при |η|1.4H, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiabeE7aOjaacYhacaaMe8UaeyisISRaaGym aiaac6cacaaI0aGaaGPaVlaadIeacaGGSaaaaa@49DF@ в точках с ординатой y0.7H. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaiabgIKi7kabgkHiTiaaicdacaGGUaGaaG4n aiaaykW7caWGibGaaiOlaaaa@4695@ Сам максимум равен 0.37δρgH/μ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyisISRaaGimaiaac6cacaaIZaGaaG4naiaaykW7 cqaH0oazcqaHbpGCcaaMc8Uaam4zaiaaykW7daWcgaqaaiaadIeaae aacqaH8oqBaaGaaiOlaaaa@4E9A@ При H=100M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iaaigdacaaIWaGaaGimaiaaykW7 caaMe8EcLzpacaqGnbaaaa@4710@ и стандартных значениях остальных параметров, наибольшее значение, как и в случае 1), составляет 7.2 10 10 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaG4naiaac6cacaaIYaGaeyyXICTaaGymaiaaicda daahaaWcbeqaaiabgkHiTiaaigdacaaIWaaaaaaa@463F@ рад.

Итак: для рельефа, задаваемого конформным отображением (1), величина эффекта определяется, в основном, глубиной ложбины H= a 2 /c . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9maalyaabaGaamyyamaaCaaaleqa baGaaGOmaaaaaOqaaiaadogaaaGaaiOlaaaa@431E@ В зависимости от крутизны рельефа, которую характеризует параметр ε= a 2 / c 2 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqyTduMaeyypa0ZaaSGbaeaacaWGHbWaaWbaaSqa beaacaaIYaaaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGcca GGSaaaaa@44E9@ максимальное значение вариации наклона меняется незначительно, в пределах от 0.25δρgH/μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGimaiaac6cacaaIYaGaaGynaiabes7aKjabeg8a YjaaykW7caWGNbGaaGPaVlaadIeacaGGVaGaeqiVd0gaaa@4B46@ до 0.37δρgH/μ. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaGimaiaac6cacaaIZaGaaG4naiabes7aKjabeg8a YjaaykW7caWGNbGaaGPaVlaadIeacaGGVaGaeqiVd0MaaiOlaaaa@4BFB@ При H=100M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamisaiabg2da9iaaigdacaaIWaGaaGimaiaaykW7 caaMe8EcLzpacaqGnbaaaa@4710@ и стандартных значениях δρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdihaaa@4127@ и μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0gaaa@3F78@ ( см. (27) и (28)) это MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ величины 5 10 10 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4NaaGynaiabgwSixlaaigdacaaIWaWa aWbaaSqabeaacqGHsislcaaIXaGaaGimaaaakiaac6caaaa@46F7@ Но вариация плотности |δρ| MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiabes7aKjabeg8aYjaacYhaaaa@4327@ может оказаться вдвое и даже втрое больше, а модуль сдвига μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiVd0gaaa@3F78@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ в 3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ 5 раз меньше стандартного значения; тогда эффект составит 7 10 9 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacdaGae8hpI4NaaG4naiabgwSixlaaigdacaaIWaWa aWbaaSqabeaacqGHsislcaaI5aaaaaaa@458B@ рад.

Максимального значения наклоны достигают примерно на половине глубины, в точках с ординатой yH/2. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyEaGWaaiab=XJi+jabgkHiTiaadIeacaGGVaGa aGOmaiaac6caaaa@4407@

Асимптотика (26) «начинает работать» на расстояниях порядка эффективной ширины b|S|/H MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOyaiaacYhacaWGtbGaaiiFaiaac+cacaWGibaa aa@4301@ от самой низкой точки дна. Она дает погрешность, не превышающую 10 %, в случае слабого рельефа MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ при |x|>b; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiaadIhacaGG8bGaaGPaVlabg6da+iaadkga caGG7aaaaa@44F8@ в противоположном случае a=c, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamyyaiabg2da9iaadogacaGGSaaaaa@4146@ когда рельеф содержит точку заострения MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbcKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A5@ при |x|>1.6b. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiiFaiaadIhacaGG8bGaaGjbVlabg6da+iaaigda caGGUaGaaGOnaiaaykW7caWGIbGaaiOlaaaa@48A5@

3. ОБСУЖДЕНИЕ ОСНОВНОГО РЕЗУЛЬТАТА: СВЯЗЬ С РЕШЕНИЕМ БУССИНЕСКА И ВОЗМОЖНОСТЬ ОБОБЩЕНИЯ НА ТРЕХМЕРНЫЙ СЛУЧАЙ

Статья посвящена оценке влияния рельефа на барические наклоны.

Вариации барического поля мы считаем равновесными: это значит, что горизонтальной составляющей у градиента давления нет. Наклоны возникают, по большей части из-за того, что на поверхности Земли атмосферное давление меняется от точки к точке, в зависимости от высоты «над уровнем моря». В отсутствие рельефа, ответные напряжения в земной коре не влияли бы на наклоны. «Равновесная» постановка задачи кажется автору наиболее естественной. К тому же, в ней есть элемент новизны: до сих пор в работах, посвященных влиянию атмосферы на наклоны элементов земной коры, задавалось неравновесное распределение давления, а влияние рельефа игнорировалось.

В данной работе рельеф предполагался двумерным и финитным. Последнее означает, что площадь S области, образованной линией рельефа и общей для обеих бесконечных ветвей рельефа горизонтальной асимптотой, конечна.

Основной результат был получен в п. 1: это формула (26). Из нее следует, что S MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ единственный геометрический параметр, от которого зависит вариация наклонов в дальней от активной части рельефа зоне.

Теперь рассмотрим двумерную задачу Буссинеска: на границу нижней полуплоскости (т. е. на ось x) действует единичная сосредоточенная сила, приложенная к началу координат и направленная вертикально вверх. Требуется найти вертикальную компоненту смещения точки x на границе полуплоскости G(x). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4raiaacIcacaWG4bGaaiykaiaac6caaaa@4196@

Решение этой задачи хорошо известно:

G x = λ+2μ 2πμ λ+μ ln|x|. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4ramaabmaabaGaamiEaaGaayjkaiaawMcaaiab g2da9iabgkHiTmaalaaabaGaeq4UdWMaey4kaSIaaGOmaiabeY7aTb qaaiaaikdacqaHapaCcaaMc8UaeqiVd02aaeWaaeaacqaH7oaBcqGH RaWkcqaH8oqBaiaawIcacaGLPaaaaaGaciiBaiaac6gacaGG8bGaam iEaiaacYhacaGGUaaaaa@5941@ (50)

Сравнивая (26) и (50), видим, что:

δ v x δρgS G x . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdq2aaSaaaeaacqGHciITcaWG2baabaGaeyOa IyRaamiEaaaaimaacqWF8iIFcqaH0oazcqaHbpGCcaaMc8UaaGPaVl aadEgacaaMc8Uaam4uaiaaysW7daWcaaqaaiabgkGi2kaadEeaaeaa cqGHciITcaWG4baaaiaac6caaaa@5655@

Иными словами: вариации наклонов в дальней зоне пропорциональны производной решения Буссинеска G/x; MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaam4raiaac+cacqGHciITcaWG4bGaai4o aaaa@43C9@ роль коэффициента пропорциональности играет величина δρgS. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdqMaeqyWdiNaaGjcVlaaykW7cqGHflY1caaM c8Uaam4zaiabgwSixlaaykW7caaMi8Uaam4uaiaac6caaaa@4FF4@

Естественно предположить, что связь между наклонами в дальней зоне и решением Буссинеска имеет место и в трехмерном случае. Пусть D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ полупространство с рельефом, D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ гладкая поверхность, ограничивающая его сверху (то есть собственно рельеф). Направим ось z вертикально вверх. Предположим, что объем области, заключенной между D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ и горизонтальной плоскостью xOy MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEaiaad+eacaWG5baaaa@4091@ конечен, т. е. интеграл:

V= z(x,y)dxdy MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOvaiabg2da9maapehabaGaaGPaVlaaykW7daWd XbqaaiaadQhacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaadsgaca WG4bGaaGPaVlaadsgacaWG5baaleaacqGHsislcqGHEisPaeaacqGH EisPa0Gaey4kIipaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcq GHRiI8aaaa@5984@

абсолютно сходится (здесь z(x,y) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEaiaacIcacaWG4bGaaiilaiaadMhacaGGPaaa aa@42C5@ высота текущей точки рельефа).

Обозначим через G(x,y) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4raiaacIcacaWG4bGaaiilaiaadMhacaGGPaaa aa@4292@ вертикальное смещение точек на границе «идеального» полупространства z0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEaiabgsMiJkaaicdaaaa@4130@ под действием единичной силы, приложенной к началу координат:

G x,y = 1 4π λ+2μ μ λ+μ 1 r , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4ramaabmaabaGaamiEaiaacYcacaWG5baacaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGinaiabec8aWb aadaWcaaqaaiabeU7aSjabgUcaRiaaikdacqaH8oqBaeaacqaH8oqB daqadaqaaiabeU7aSjabgUcaRiabeY7aTbGaayjkaiaawMcaaaaada WcaaqaaiaaigdaaeaacaWGYbaaaiaacYcaaaa@5623@ (51)

где r= x 2 + y 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOCaiabg2da9maakaaabaGaamiEamaaCaaaleqa baGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaaabe aaaaa@4488@ [Ландау, Лифшиц, 2003]. И сразу, по аналогии с (26), пишем формулы для вариации наклонов w/x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaam4Daiaac+cacqGHciITcaWG4baaaa@433A@ и w/y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaam4Daiaac+cacqGHciITcaWG5baaaa@433B@ в дальней зоне:

δ w x δρgV G x = λ+2μ μ λ+μ δρgVx 4π r 3 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdq2aaSaaaeaacqGHciITcaWG3baabaGaeyOa IyRaamiEaaaaimaacqWF8iIFcqaH0oazcqaHbpGCcaaMc8UaaGPaVl aadEgacaaMc8UaamOvamaalaaabaGaeyOaIyRaam4raaqaaiabgkGi 2kaadIhaaaGaeyypa0JaeyOeI0YaaSaaaeaacqaH7oaBcqGHRaWkca aIYaGaeqiVd0gabaGaeqiVd02aaeWaaeaacqaH7oaBcqGHRaWkcqaH 8oqBaiaawIcacaGLPaaaaaGaaGPaVpaalaaabaGaeqiTdqMaeqyWdi NaaGPaVlaaykW7caWGNbGaaGPaVlaadAfacaaMc8UaamiEaaqaaiaa isdacqaHapaCcaaMc8UaamOCamaaCaaaleqabaGaaG4maaaakiaayk W7aaGaaiilaaaa@78CC@

δ w y λ+2μ μ λ+μ δρgVy 4π r 3 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqiTdq2aaSaaaeaacqGHciITcaWG3baabaGaeyOa IyRaamyEaaaaimaacqWF8iIFcqGHsisldaWcaaqaaiabeU7aSjabgU caRiaaikdacqaH8oqBaeaacqaH8oqBdaqadaqaaiabeU7aSjabgUca RiabeY7aTbGaayjkaiaawMcaaaaacaaMc8+aaSaaaeaacqaH0oazcq aHbpGCcaaMc8UaaGPaVlaadEgacaaMc8UaamOvaiaadMhaaeaacaaI 0aGaeqiWdaNaaGPaVlaadkhadaahaaWcbeqaaiaaiodaaaGccaaMc8 oaaiaac6caaaa@67CD@ (52)

Конечно, приведенное рассуждение не является доказательством. Однако, если рельеф «слабый» (α<<1, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiabeg7aHjabgYda8iabgYda8iaaigdacaGG Saaaaa@4380@ где α MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdegaaa@3F61@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ синус угла между нормалью к поверхности D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ и осью z), формулы (52) следуют из теории возмущений. Эта теория была развита С.М. Молоденским для оценки влияния рельефа на приливные деформации и наклоны [Молоденский, 1983]. Идея состоит в том, чтобы заменить функцию Грина для полупространства со слабым рельефом функцией Грина для «идеального» полупространства, т. е. решением Буссинеска. С.М. Молоденский показал, что вычисленные таким образом смещения и деформации отличаются от истинных значений на величину порядка α 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySde2aaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa @4106@

Пусть на поверхности D MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeyOaIyRaamiraaaa@3FF1@ заданы напряжения:

σ xx = σ yy = σ zz =p; σ xy = σ xz = σ yz =0, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiab g2da9iabeo8aZnaaBaaaleaacaWG5bGaamyEaaqabaGccqGH9aqpcq aHdpWCdaWgaaWcbaGaamOEaiaadQhaaeqaaOGaeyypa0JaamiCaiaa cUdacaaMe8UaaGjbVlabeo8aZnaaBaaaleaacaWG4bGaamyEaaqaba GccqGH9aqpcqaHdpWCdaWgaaWcbaGaamiEaiaadQhaaeqaaOGaeyyp a0Jaeq4Wdm3aaSbaaSqaaiaadMhacaWG6baabeaakiabg2da9iaaic dacaGGSaaaaa@61DC@

где p=δρgz(x,y). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiCaiabg2da9iabes7aKjabeg8aYjaaykW7caWG NbGaaGPaVlaadQhacaGGOaGaamiEaiaacYcacaWG5bGaaiykaiaac6 caaaa@4CD9@ С учетом сделанных выше предположений, вертикальное смещение w на границе с точностью до величин второго порядка по α MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaeqySdegaaa@3F61@ равно:

w=δρg z x 1 , y 1 G x x 1 ,y y 1 d x 1 d y 1 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4Daiabg2da9iabes7aKjabeg8aYjaaykW7caaM c8Uaam4zaiaaykW7daWdXbqaaiaaysW7daWdXbqaaiaadQhadaqada qaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyEamaaBaaa leaacaaIXaaabeaaaOGaayjkaiaawMcaaaWcbaGaeyOeI0IaeyOhIu kabaGaeyOhIukaniabgUIiYdaaleaacqGHsislcqGHEisPaeaacqGH EisPa0Gaey4kIipakiabgwSixlaadEeadaqadaqaaiaadIhacqGHsi slcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaaykW7caWG5bGa eyOeI0IaamyEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaai aaykW7caWGKbGaamiEamaaBaaaleaacaaIXaaabeaakiaaykW7caWG KbGaamyEamaaBaaaleaacaaIXaaabeaakiaac6caaaa@75D5@

Ограничимся рассмотрением ситуации, когда рельеф «совсем финитный»: вне круга достаточно большого радиуса R с центром в начале координат высота z(x,y)=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamOEaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa eyypa0JaaGimaiaac6caaaa@4537@ Тогда координаты x 1 , y 1 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG 5bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaaaa@42FF@ по которым идет интегрирование, по абсолютной величине не превосходят R. Если точка (x,y) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiaadIhacaGGSaGaaGPaVlaadMhacaGGPaaa aa@4351@ находится в дальней зоне (|x|,|y|<<R), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaaiikaiaacYhacaWG4bGaaiiFaiaacYcacaGG8bGa amyEaiaacYhacWaQaAipaWJamGkGgYda8iaadkfacaGGPaGaaiilaa aa@4BB5@ то G(x x 1 ,y y 1 )G(x,y), MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaGaam4raiaacIcacaWG4bGaeyOeI0IaamiEamaaBaaa leaacaaIXaaabeaakiaacYcacaaMc8UaamyEaiabgkHiTiaadMhada WgaaWcbaGaaGymaaqabaGccaGGPaGaeyisISRaam4raiaacIcacaWG 4bGaaiilaiaaykW7caWG5bGaaiykaiaacYcaaaa@5290@ и

w x,y =δρgG x,y z x 1 , y 1 d x 1 d y 1 = =δρgVG x,y . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaacaWG3bWaaeWaaeaacaWG4bGaaiilaiaadMha aiaawIcacaGLPaaacqGH9aqpcqaH0oazcqaHbpGCcaaMc8UaaGPaVl aadEgacaaMc8Uaam4ramaabmaabaGaamiEaiaacYcacaaMc8UaamyE aaGaayjkaiaawMcaaiaaykW7daWdXbqaaiaaysW7daWdXbqaaiaadQ hadaqadaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyE amaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaWcbaaabaaani abgUIiYdaaleaaaeaaa0Gaey4kIipakiaaykW7caWGKbGaamiEamaa BaaaleaacaaIXaaabeaakiaaykW7caWGKbGaamyEamaaBaaaleaaca aIXaaabeaakiabg2da9aqaaiabg2da9iabes7aKjabeg8aYjaaykW7 caaMc8Uaam4zaiaaykW7caaMc8UaamOvaiaaykW7caWGhbWaaeWaae aacaWG4bGaaiilaiaaykW7caWG5baacaGLOaGaayzkaaGaaGPaVlaa c6caaaaa@8301@ (53)

Что и требовалось: формулы (52) следуют из (53) и (51). По-видимому, формулы (52) верны для любого трехмерного финитного рельефа, необязательно слабого.

К сожалению, подходов, сравнимых по эффективности с методом Колосова MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ именно с помощью этого метода была получена асимптотика (26) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ в трехмерном случае нет, по крайней мере, автору они неизвестны. Поэтому подтвердить (или опровергнуть) высказанную выше гипотезу можно только «экспериментально»: то есть, решить трехмерную задачу теории упругости для нескольких финитных рельефов численно и сопоставить результаты вычислений с (52). В ближайших планах автора MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFtacaaa@39A3@ выполнить такую работу.

Благодарности

Автор благодарит Ю.О. Кузьмина, М.С. Кучая и И.А. Широкова за интерес к работе и полезное обсуждение результатов.

About the authors

I. Ya. Tsurkis

Schmidt Institute of Physics of the Earth, Russian Academy of Sciences

Author for correspondence.
Email: tsurkis@ifz.ru

Russian Federation, Moscow

References

  1. Колосов Г.В. Применение комплексной переменной к теории упругости. М.: ОНТИ. 1935. 224 с.
  2. Ландау Л.Д., Лифшиц Е.М. Теоретическая физика, т. VII. Теория упругости. М.: Физматлит. 2003. 264 с.
  3. Латынина Л.А., Васильев И.М. Деформация земной коры под влиянием атмосферного давления // Физика Земли. 2001. № 5. С. 45-54.
  4. Молоденский С.М. О локальных аномалиях амплитуд и фаз приливных наклонов и деформаций // Физика Земли. 1983. № 7. С. 3-9.
  5. Мусхелишвили Н.И. Некоторые основные задачи математической теории упругости. М.: Наука. 1966. 707 с.
  6. Партон В.З., Перлин П.И. Математический аппарат теории упругости. М.: Наука. 1981. 688 с. С. 362-446.
  7. Перцев Б.П., Ковалёва О.В. Оценка влияния колебаний атмосферного давления на наклоны и линейные деформации земной поверхности // Физика Земли. 2004. № 8. С. 79-81.
  8. Широков И.А., Анохина К.М. О связи пространственно-временных вариаций наклонов земной поверхности с вариациями атмосферного давления // Физика Земли. 2003. № 1. С. 84-87.
  9. Darwin G.H. XLVI. On variations in the vertical due to elasticity of the earth’s surface // Phil. Mag. S. 1882. V. 14. № 90. P. 409-427.

Supplementary files

Supplementary Files Action
1.
Fig. 1. Examples of compact relief: (a) the relief line is a broken line with a finite number of links; (b) the relief line is the image of the real axis when displaying (1)

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2.
Fig. 2. An extended quarry with a cross section of 10x100 m

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3.
Fig. 3. Slopes at the boundary of the region

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