Surface impedance of electromagnetic field excited by a grounded horizontal antenna in the Earth–ionosphere waveguide

Abstract


Analytical formulas for the tangential components of extremely-low-frequency (ELF) electromagnetic field in the Earth–ionosphere plane waveguide excited by a grounded linear horizontal antenna are obtained. The behavior of surface impedance is studied as a function of electrodynamic characteristics of the waveguide and the distance from the source. It is shown that surface impedance coincides with the plane wave impedance on the Earth’s surface at distances from the source larger than the skin depth provided that the skin layer is thinner than double the waveguide’s height. The influence of the ionosphere on the amplitude of the ELF and lower-frequency magnetic field and, thus, on the impedance at the distances shorter than two ionospheric heights is theoretically substantiated. This type of effect was observed in the experiments conducted on the Kola Peninsula where the low conductivity of the Earth allowed the detection of the effect of the ionosphere on the amplitude of the magnetic field in the low-frequency band.


ВВЕДЕНИЕ

Электромагнитные методы наряду с сейсмическими играют значительную роль как в исследованиях литосферы, так и в геологоразведке [Жданов, 2012]. Наиболее часто в экспериментальных работах в качестве источника электромагнитного поля используются естественные магнитосферно-ионосферные электромагнитные шумы. В то же время, в связи с развитием в последнее время технических средств излучения и приема, в практику исследований начинают внедряться активные эксперименты с применением сигналов от мощных низкочастотных передающих устройств [Велихов, 1997]. Использование волн крайне низкочастотного (3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ 30 Гц) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ КНЧ, и более низкого диапазона, глубоко проникающих в исследуемую среду, позволяет надеяться на получение новой важной информации о структуре Земли и происходящих в ней процессах. Основным при работе с естественными полями является импедансный подход, использующий отношение тангенциальных составляющих электрического поля к магнитным [Тихонов, 1950; Cagniard, 1953; Ковтун, 2009]. Такой путь удобен при регистрации естественных полей, так как не требует информации как об источнике, так и о трассе распространения поля от источника к точке приема. Естественно, в методе используется ряд предположений, в частности, что поверхностный импеданс на границе раздела определяется электромагнитными параметрами среды под границей раздела и расстоянием от источника до точки наблюдения и совпадает с импедансом плоской волны [Бреховских, 1957] на расстояниях, превышающих величину скин-слоя. Кроме того, в литературе, посвященной анализу полей в волноводе Земля MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ ионосфера [Wait, 1970], сложилось физическое представление, что на расстояниях меньше двух высот ионосферного волновода для длинных волн можно пренебречь влиянием ионосферы на поле, а, соответственно, и на величину импеданса. Такой вывод неявно предполагает, что величина скин-слоя значительно меньше двойной высоты ионосферного волновода. Однако при использовании волн КНЧ и более низкого диапазонов при работе на кристаллических щитах при зондированиях полями искусственных возбудителей (типа длинных электрических линий) величина скин-слоя может превышать двойную высоту волновода, и вопрос о влиянии ионосферы не очевиден. Это показали эксперименты с длинным, заземленным на концах электрическим кабелем в качестве источника поля [Терещенко, 2007], в которых было установлено влияние ионосферы на амплитуду низкочастотного магнитного поля, регистрируемую на расстоянии, сопоставимом с высотой ионосферного волновода.

Ниже на примере плоского волновода Земля MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ ионосфера оценим влияние ионосферы на поверхностный импеданс при низкой проводимости подстилающей поверхности.

ПОЛЕ ЗАЗЕМЛЕННОЙ ГОРИЗОНТАЛЬНОЙ АНТЕННЫ В ВОЛНОВОДЕ ЗЕМЛЯ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuyTjMCPf gaiuaajugWbabaaaaaaaaapeGaa83eGaaa@3AC9@ ИОНОСФЕРА

Рассмотрим возбуждение плоского волновода горизонтальной заземленной антенной. Определим поле в трехслойной среде (рис. 1), формируемое заземленной на концах горизонтальной линией длиной 2L, питаемой током с гармонической зависимостью от времени exp( MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ iωt) и находящейся на границе раздела 0.

При этом будем считать проводимость Земли σg и ионосферы σi постоянными и изотропными. Систему координат выбрали следующим образом. Центр декартовых координат поместили в середину антенны, ось z направили вверх, ось x MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ вдоль антенны, а y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ поперек антенны. Расстояние от центра антенны до произвольной точки наблюдения (x, y, z) обозначили R, а расстояние на плоскости (x, y, 0) от центра антенны обозначили ρ и ρη MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ расстояние от произвольной точки антенны.

Среду в области 0zh считаем практически непроводящей (σ=+0, при этом наличие + у нуля указывает на небольшое поглощение) с диэлектрической проницаемостью ε0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb sr4rNCHbacfaqcLbwaqaaaaaaaaaWdbiab=nKi7aaa@3CA4@ 10-9/36π Ф/м и магнитной проницаемостью μ0=4π10-7 Г/м. Предполагаем, что в области z<0 имеем электромагнитные параметры εg, μ0, σg, а при z>h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ εi, μ0, σi.

Задача возбуждения электромагнитного поля сторонним током J сводится к решению уравнений Гельмгольца для электрического вектор-потенциала A с соответствующими граничными условиями [Wait, 1970; Вешев, 1980]. В дальнейшем удобно использовать уравнения для комплексных амплитуд соответствующих монохроматических компонент (AAexp( MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ iωt), EEexp( MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ iωt), HHexp( MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ iωt), где E и H MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ электрическое и магнитное поле, соответственно).

 

Рис. 1. Геометрия задачи.

 

Принимая во внимание, что источник направлен вдоль оси x (рис. 1), представим A в виде двух составляющих:

A j = A x j e x + A z j e z , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaqefeezVjwzGmuyZX2BUbcuY9gic9gBKbacgmaeaaaa aaaaa8qacaWFbbWdamaaCaaaleqabaWdbmaabmaapaqaa8qacaWGQb aacaGLOaGaayzkaaaaaOGaeyypa0Jaamyqa8aadaqhaaWcbaWdbiaa dIhaa8aabaWdbmaabmaapaqaa8qacaWGQbaacaGLOaGaayzkaaaaaO Gaa8xza8aadaWgaaWcbaWdbiaa=Hhaa8aabeaak8qacqGHRaWkcaWG bbWdamaaDaaaleaapeGaamOEaaWdaeaapeWaaeWaa8aabaWdbiaadQ gaaiaawIcacaGLPaaaaaGccaWFLbWdamaaBaaaleaapeGaa8NEaaWd aeqaaOWdbiaacYcaaaa@59E5@

где j=g, 0, i указывает на среду, ex и ez MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ единичные орты, направленные вдоль осей x и z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ соответственно.

Дальнейшие шаги аналогичны сделанным в работе [Терещенко, 2017]. Это решение уравнения Гельмгольца для компонент вектора-потенциала с соответствующими граничными условиями, но в отличие от работы [Терещенко, 2017] с добавлением дополнительных условий на потенциал на границе ионосферы.

В результате, опуская промежуточные преобразования и вычисления, можем представить A x j MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGbbWdamaaDaaaleaapeGaamiE aaWdaeaapeWaaeWaa8aabaWdbiaadQgaaiaawIcacaGLPaaaaaaaaa@42A7@ и A z j MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGbbWdamaaDaaaleaapeGaamOE aaWdaeaapeWaaeWaa8aabaWdbiaadQgaaiaawIcacaGLPaaaaaaaaa@42A9@ для точечного заземленного горизонтального источника с дипольным моментом jΔx в следующем виде:

A x i = J Δ x 4π 0 α i exp ( ν i z) J 0 λρ dλ, A x 0 = J Δ x 4π 0 λ ν 0 + α 0 exp ( ν 0 z)+ β 0 exp ( ν 0 z) J 0 λρ dλ, A x g = J Δ x 4π 0 λ ν g + β g exp ( ν g z) J 0 λρ dλ. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaqabeaaqaaaaaaaaaWdbiaadgeapaWaa0baaSqaa8qa caWG4baapaqaa8qadaqadaWdaeaapeGaamyAaaGaayjkaiaawMcaaa aak8aacaaMb8+dbiabg2da9maalaaapaqaa8qacaWGkbGaeuiLdq0d amaaBaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaisdacqaHapaCaa WaiGgGwahabKaAaUWdaeacOb4dbiacObiIWaaapaqaiGgGpeGamGgG g6HiLcqdpaqaiGgGpeGamGgGgUIiYdaakiadObiHXoqypaWaiGgGBa aaleacOb4dbiacOb4GPbaapaqajGgGaOWdbiacObyGLbGaiGgGbIha cGaAagiCaiacObyGGcGaiGgGcIcacWaAaAOeI0IamGgGe27aU9aadG aAaUbaaSqaiGgGpeGaiGgGdMgaa8aabKaAacGcpeGaiGgGdQhacGaA akykaiacOb4GkbWdamacOb4gaaWcbGaAa+qacGaAaIimaaWdaeqcOb iak8qadGaAagWaa8aabGaAa+qacWaAas4UdWMamGgGeg8aYbGaiGgG wIcacGaAaAzkaaGaiGgGdsgacWaAas4UdWMaiGgGcYcaaeaacaWGbb WdamaaDaaaleaapeGaamiEaaWdaeaapeWaaeWaa8aabaWdbiaaicda aiaawIcacaGLPaaaaaGcpaGaaGzaV=qacqGH9aqpdGaxaUaaa8aabG axa+qacGaxaoOsaiadCbyHuoarpaWaiWfGBaaaleacCb4dbiacCb4G 4baapaqajWfGaaGcbGaxa+qacGaxaIinaiadCbiHapaCaaWaiGaGwa habKacaUWdaeacia4dbiaciaiIWaaapaqaiGaGpeGamGaGg6HiLcqd paqaiGaGpeGamGaGgUIiYdaakmacaciz=daadmaapaqaiaiGK9paa8 qadGaGas2=aaqadaWdaeacaciz=daapeWaiaiGK9paaSaaa8aabGaG as2=aaWdbiadaciz=daaeU7aSbWdaeacaciz=daapeGamaiGK9paaq yVd42damacaciz=daaBaaaleacaciz=daapeGaiaiGK9paaGimaaWd aeqcaciz=daaaaaak8qacGaGas2=aaaMi8UamaiGW9paay4kaSIaia iGK9paaGjcVladaciF=daaeg7aH9aadGaGaY3=aaWgaaWcbGaGaY3= aaWdbiacaciF=daaicdaa8aabKaGaY3=aaaaaOWdbiacaciz=daawI cacGaGas2=aaGLPaaacGaGas2=aaaMb8UaiaiG89paaeyzaiacaciF =daabIhacGaGaY3=aaqGWbGaiaiG89paaeiOaiacaciF=daacIcacW aGas2=aaGHsislcWaGas2=aaaH9oGBpaWaiaiGK9paaSbaaSqaiaiG K9paa8qacGaGas2=aaaIWaaapaqajaiGK9paaaGcpeGaiaiGK9paam OEaiacaciz=daacMcacGaGas2=aaaMi8UaiaiGK9paaGPaVladaciq =daagUcaRiacaciz=daayIW7cWaGas2=aaaHYoGypaWaiaiGK9paaS baaSqaiaiGK9paa8qacGaGas2=aaaIWaaapaqajaiGK9paaaGcpeGa iaiGK9paaeyzaiacaciz=daabIhacGaGas2=aaqGWbGaiaiGK9paae iOaiacaciz=daacIcacWaGas2=aaaH9oGBpaWaiaiGK9paaSbaaSqa iaiGK9paa8qacGaGas2=aaaIWaaapaqajaiGK9paaaGcpeGaiaiGK9 paamOEaiacaciz=daacMcaaiacaciz=daawUfacGaGas2=aaGLDbaa cGaGaY0=aaWGkbWdamacacik=daaBaaaleacacik=daapeGaiaiGO8 paaGimaaWdaeqcacik=daaaOGaiaiGO8paaGzaVlacacib=daayIW7 peWaiaiGe8paaeWaa8aabGaGasW=aaWdbiadacib=daaeU7aSjadac ib=daaeg8aYbGaiaiGe8paayjkaiacacib=daawMcaaiacaci++daa dsgacWaGaIV=aaaH7oaBcGaGaIV=aaGGSaaabaGaamyqa8aadaqhaa WcbaWdbiaadIhaa8aabaWdbmaabmaapaqaa8qacaWGNbaacaGLOaGa ayzkaaaaaOWdaiaaygW7peGaeyypa0ZaaSaaa8aabaWdbiaadQeacq qHuoarpaWaaSbaaSqaa8qacaWG4baapaqabaaakeaapeGaaGinaiab ec8aWbaadGaAaAbCaeqcOb4cpaqaiGgGpeGaiGgGicdaa8aabGaAa+ qacWaAaAOhIukan8aabGaAa+qacWaAaA4kIipaaOWaiGgGbmaapaqa iGgGpeWaiGgGlaaapaqaiGgGpeGamGgGeU7aSbWdaeacOb4dbiadOb iH9oGBpaWaiGgGBaaaleacOb4dbiacOb4GNbaapaqajGgGaaaak8qa cWaAaA4kaSIamGgGek7aI9aadGaAaUbaaSqaiGgGpeGaiGgGdEgaa8 aabKaAacaak8qacGaAaAjkaiacObOLPaaacGaAagyzaiacObyG4bGa iGgGbchacGaAagiOaiacObOGOaGamGgGe27aU9aadGaAaUbaaSqaiG gGpeGaiGgGdEgaa8aabKaAacGcpeGaiGgGdQhacGaAakykaiacOb4G kbWdamacOb4gaaWcbGaAa+qacGaAaIimaaWdaeqcObiak8qadGaAag Waa8aabGaAa+qacWaAas4UdWMamGgGeg8aYbGaiGgGwIcacGaAaAzk aaGaiGgGdsgacWaAas4UdWMaiGgGc6caaaaa@027B@

(1)

Подобные выражения имеем и для A z j : MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGbbWdamaaDaaaleaapeGaamOE aaWdaeaapeWaaeWaa8aabaWdbiaadQgaaiaawIcacaGLPaaaaaGcpa GaiGdGcQdaaaa@4440@

A z i = J Δ x 4π x 0 η i exp ( ν i z) J 0 λρ λ dλ, A z 0 = J Δ x 4π x 0 η 0 exp ( ν 0 z)+ γ 0 exp ( ν 0 z) J 0 λρ λ dλ, A z g = J Δ x 4π x 0 γ g exp ( ν g z) J 0 λρ λ dλ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaqabeaaqaaaaaaaaaWdbiaadgeapaWaa0baaSqaa8qa caWG6baapaqaa8qadaqadaWdaeaapeGaamyAaaGaayjkaiaawMcaaa aakiabg2da9iabgkHiTmaalaaapaqaa8qacaWGkbGaeuiLdq0damaa BaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaisdacqaHapaCaaWaaS aaa8aabaWdbiabgkGi2cWdaeaapeGaeyOaIyRaamiEaaaadaGfWbqa bSWdaeaapeGaaGimaaWdaeaapeGaeyOhIukan8aabaWdbiabgUIiYd aakiabeE7aO9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaqGLbGa aeiEaiaabchacaqGGcGaaiikaiabgkHiTiabe27aU9aadaWgaaWcba WdbiaadMgaa8aabeaak8qacaWG6bGaaiykamaalaaapaqaa8qacaWG kbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbmaabmaapaqaa8qacq aH7oaBcqaHbpGCaiaawIcacaGLPaaaa8aabaWdbiabeU7aSbaacaWG KbGaeq4UdWMaaiilaaqaaiaadgeapaWaa0baaSqaa8qacaWG6baapa qaa8qadaqadaWdaeaapeGaaGimaaGaayjkaiaawMcaaaaak8aacaaM i8+dbiadObOH9aqpcaaMb8UaeyOeI0YaaSaaa8aabaWdbiaadQeacq qHuoarpaWaaSbaaSqaa8qacaWG4baapaqabaaakeaapeGaaGinaiab ec8aWbaadGaDaUaaa8aabGaDa+qacWaDaAOaIylapaqaiqhGpeGamq hGgkGi2kac0b4G4baaamacqbOfWbqajafGl8aabGaua+qacGauaIim aaWdaeacqb4dbiadqbOHEisPa0Wdaeacqb4dbiadqbOHRiI8aaGcdG acagWaa8aabGaca+qacWacas4TdG2damacia4gaaWcbGaca+qacGac aIimaaWdaeqciaiak8qacGacagyzaiaciayG4bGaiGaGbchacGacag iOaiaciaOGOaGamaiGW9paayOeI0IamaiGW9paaqyVd42damacaciC =daaBaaaleacaciC=daapeGaiaiGW9paaGimaaWdaeqcaciC=daaaO WdbiacaciC=daadQhacGaGas2=aaGGPaGaiGaGyIW7cWacaA4kaSIa iGaGyIW7cWacas4SdC2damacia4gaaWcbGaca+qacGacaIimaaWdae qciaiak8qacGacagyzaiaciayG4bGaiGaGbchacGacagiOaiaciaOG OaGamGaGe27aU9aadGacaUbaaSqaiGaGpeGaiGaGicdaa8aabKacac GcpeGaiGaGdQhacGacakykaaGaiGaGwIcacGacaAzkaaWaiaiG89pa aSaaa8aabGaGaY3=aaWdbiacaciF=daadQeapaWaiaiGK9paaSbaaS qaiaiGK9paa8qacGaGas2=aaaIWaaapaqajaiGK9paaaGccGaGaY3= aaaMb8+dbmacaciF=daabmaapaqaiaiG89paa8qacWaGaY3=aaaH7o aBcWaGaY3=aaaHbpGCaiacaciF=daawIcacGaGaY3=aaGLPaaaa8aa bGaGaY3=aaWdbiadaciF=daaeU7aSbaacGaGac3=aaWGKbGamaiGW9 paaq4UdWMaiaiGW9paaiilaaqaaiaadgeapaWaa0baaSqaa8qacaWG 6baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaaaaki abg2da9iabgkHiTmaalaaapaqaa8qacaWGkbGaeuiLdq0damaaBaaa leaapeGaamiEaaWdaeqaaaGcbaWdbiaaisdacqaHapaCaaWaaSaaa8 aabaWdbiabgkGi2cWdaeaapeGaeyOaIyRaamiEaaaadaGfWbqabSWd aeaapeGaaGimaaWdaeaapeGaeyOhIukan8aabaWdbiabgUIiYdaaki abeo7aN9aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacaqGLbGaaeiE aiaabchacaqGGcGaaiikaiabe27aU9aadaWgaaWcbaWdbiaadEgaa8 aabeaak8qacaWG6bGaaiykamaalaaapaqaa8qacaWGkbWdamaaBaaa leaapeGaaGimaaWdaeqaaOWdbmaabmaapaqaa8qacqaH7oaBcqaHbp GCaiaawIcacaGLPaaaa8aabaWdbiabeU7aSbaacaWGKbGaeq4UdWMa aiilaaaaaa@465B@

(2)

где:

k j 2 λ 2 =i ϰ j 2 + λ 2 i ν j ,   ϰ j =i k j ,  k j = ω c ε j ε 0 +i σ j ω ε 0 = ω c ε ˜ j , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaqabeaaqaaaaaaaaaWdbmaakaaapaqaa8qacaWGRbWd amaaDaaaleaapeGaamOAaaWdaeaapeGaaGOmaaaakiabgkHiTiabeU 7aS9aadaahaaWcbeqaa8qacaaIYaaaaaqabaGccqGH9aqpcaWGPbWa aOaaa8aabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiy aapeGae8h8dK=damaaDaaaleaapeGaamOAaaWdaeaapeGaaGOmaaaa kiabgUcaRiabeU7aS9aadaahaaWcbeqaa8qacaaIYaaaaaqabaGccq GHHjIUcaWGPbGaeqyVd42damaaBaaaleaapeGaamOAaaWdaeqaaOWd biaacYcacaGGGcGaaiiOaiab=b=a5=aadaWgaaWcbaWdbiaadQgaa8 aabeaak8qacqGH9aqpcqGHsislcaWGPbGaam4Aa8aadaWgaaWcbaWd biaadQgaa8aabeaak8qacaGGSaGaaiiOaaqaaiaadUgapaWaaSbaaS qaa8qacaWGQbaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiabeM8a 3bWdaeaapeGaam4yaaaadaGcaaWdaeaapeWaaSaaa8aabaWdbiabew 7aL9aadaWgaaWcbaWdbiaadQgaa8aabeaaaOqaa8qacqaH1oqzpaWa aSbaaSqaa8qacaaIWaaapaqabaaaaOWdbiabgUcaRiaadMgadaWcaa WdaeaapeGaeq4Wdm3damaaBaaaleaapeGaamOAaaWdaeqaaaGcbaWd biabeM8a3jabew7aL9aadaWgaaWcbaWdbiaaicdaa8aabeaaaaaape qabaGccqGH9aqpdaWcaaWdaeaapeGaeqyYdChapaqaa8qacaWGJbaa amaakaaapaqaa8qacuaH1oqzpaGbaGWdbyaafaWaaSbaaSqaaiaadQ gaaeqaaaqabaGccaGGSaaaaaa@8AB5@

j= g,0,i , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGQbGaeyypa0ZaamWaa8aabaWd biaadEgacaGGSaGaaGimaiaacYcacaWGPbaacaGLBbGaayzxaaGaai ilaaaa@468C@ c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ скорость света.

Так как в процессе вычислений фиксировали ветвь корня таким образом, что Im k j 2 λ 2 >0, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaqGjbacbaqcL9vacaWFTbGcdaGc aaWdaeaapeGaam4Aa8aadaqhaaWcbaWdbiaadQgaa8aabaWdbiaaik daaaGccqGHsislcqaH7oaBpaWaaWbaaSqabeaapeGaaGOmaaaaaeqa aOGaeyOpa4JaaGimaiaacYcaaaa@49E1@ то Re ν j >0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaqGsbGaaeyzaiabe27aU9aadaWg aaWcbaWdbiaadQgaa8aabeaak8qacqGH+aGpcaaIWaGaaiOlaaaa@452E@ Система уравнений для неизвестных αi, α0, β0, βg, η0, ηi, γ0, γg получается в результате использования граничных условий.

Граничные условия при h дают следующую систему уравнений:

α i exp ( ν i h) α 0 exp ( ν 0 h) β 0 exp ( ν 0 h)= λ ν 0 exp ( ν 0 h), η i exp ( ν i h) η 0 exp ( ν 0 h) γ 0 exp ( ν 0 h)=0, α i ν i exp ( ν i h)+ α 0 ν 0 exp ( ν 0 h) β 0 ν 0 exp ( ν 0 h)= =λexp ( ν 0 h), η i ν i k 0 2 exp ( ν i h) η 0 ν 0 k i 2 exp ( ν 0 h)+ γ 0 ν 0 k i 2 exp ( ν 0 h)= =λ k i 2 λ ν 0 + α 0 exp ( ν 0 h)+ β 0 exp ( ν 0 h) α i k 0 2 exp ( ν i h) , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiabeg7aH9aadaWgaaWcbaWd biaadMgaa8aabeaak8qacaqGLbGaaeiEaiaabchacaqGGcGaiqhGcI cacWaxaAOeI0IamWfGe27aU9aadGauaUbaaSqaiafGpeGaiafGdMga a8aabKauacGcpeGaiafGdIgacGarakykaiacCbiMi8UaaGPaVladSa OHsislcGaxaIjcVladCbiHXoqypaWaiWfGBaaaleacCb4dbiacCbiI WaaapaqajWfGaOWdbiacCbyGLbGaiWfGbIhacGaxagiCaiacCbyGGc GaiafGcIcacWaoaAOeI0IamGdGe27aU9aadGaiaUbaaSqaiacGpeGa iacGicdaa8aabKaiacGcpeGaiacGdIgacGacakykaiacGaiMi8UamW cGgkHiTiaaykW7cGaiaIjcVladGaiHYoGypaWaiacGBaaaleacGa4d biacGaiIWaaapaqajacGaOWdbiacuayGLbGaiqbGbIhacGafagiCai acuayGGcGaiGaGcIcacWaGac3=aaaH9oGBpaWaiaiGK9paaSbaaSqa iaiGK9paa8qacGaGas2=aaaIWaaapaqajaiGK9paaaGcpeGaiaiGK9 paamiAaiacacit=daacMcacGaGas2=aaaMi8UaaGPaVladaciz=daa g2da9iacaciz=daaykW7caaMc8+aiaiGK9paaSaaa8aabGaGas2=aa Wdbiadaciz=daaeU7aSbWdaeacaciz=daapeGamaiGK9paaqyVd42d amacaciz=daaBaaaleacaciz=daapeGaiaiGK9paaGimaaWdaeqcac iz=daaaaaak8qacGaGaY0=aaqGLbGaiaiGm9paaeiEaiacacit=daa bchacGaGaY0=aaqGGcGaiaiGa9paaiikaiadacik=daagkHiTiadac ih=daae27aU9aadGaGacX=aaWgaaWcbGaGacX=aaWdbiacacie=daa icdaa8aabKaGacX=aaaak8qacGaGacX=aaWGObGaiaiG47paaiykai acaci1+daacYcaaeaacqaH3oaApaWaaSbaaSqaa8qacaWGPbaapaqa baGcpeGaaeyzaiaabIhacaqGWbGaaeiOaiaacIcacqGHsislcqaH9o GBpaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaamiAaiaacMcacqGH sislcqaH3oaApaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaeyzai aabIhacaqGWbGaaeiOaiaacIcacqGHsislcqaH9oGBpaWaaSbaaSqa a8qacaaIWaaapaqabaGcpeGaamiAaiaacMcacqGHsislcqaHZoWzpa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaeyzaiaabIhacaqGWbGa aeiOaiaacIcacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe GaamiAaiaacMcacqGH9aqpcaaIWaGaaiilaaqaaiabgkHiTiabeg7a H9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqaH9oGBpaWaaSbaaS qaa8qacaWGPbaapaqabaGcpeGaaeyzaiaabIhacaqGWbGaaeiOaiaa cIcacqGHsislcqaH9oGBpaWaaSbaaSqaa8qacaWGPbaapaqabaGcpe GaamiAaiaacMcacqGHRaWkcqaHXoqypaWaaSbaaSqaa8qacaaIWaaa paqabaGcpeGaeqyVd42damaaBaaaleaapeGaaGimaaWdaeqaaOWdbi aabwgacaqG4bGaaeiCaiaabckacaGGOaGaeyOeI0IaeqyVd42damaa BaaaleaapeGaaGimaaWdaeqaaOWdbiaadIgacaGGPaGaeyOeI0Iaeq OSdi2damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabe27aU9aadaWg aaWcbaWdbiaaicdaa8aabeaak8qacaqGLbGaaeiEaiaabchacaqGGc Gaaiikaiabe27aU9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaWG ObGaaiykaiabg2da9aqaaiabg2da9iabgkHiTiabeU7aSjaabwgaca qG4bGaaeiCaiaabckacaGGOaGaeyOeI0IaeqyVd42damaaBaaaleaa peGaaGimaaWdaeqaaOWdbiaadIgacaGGPaGaaiilaaqaaiabeE7aO9 aadGaJaUbaaSqaiWiGpeGaiWiGdMgaa8aabKaJacGcpeGamGgGe27a U9aadGaAaUbaaSqaiGgGpeGaiGgGdMgaa8aabKaAacGcpeGaiGgGdU gapaWaiGgGDaaaleacOb4dbiacObiIWaaapaqaiGgGpeGaiGgGikda aaGccGaAagyzaiacObyG4bGaiGgGbchacGaAagiOaiacObOGOaGamG gGgkHiTiadObiH9oGBpaWaiafGBaaaleacqb4dbiacqb4GPbaapaqa jafGaOWdbiacqb4GObGaiafGcMcacaaMc8UaiGgGyIW7cWaoaAOeI0 IaiGgGyIW7cWauas4TdG2damacqb4gaaWcbGaua+qacGaAaIimaaWd aeqcqbiak8qacWauasyVd42damacqb4gaaWcbGaua+qacGauaIimaa Wdaeqcqbiak8qacGauao4Aa8aadGaua2baaSqaiafGpeGaiafGdMga a8aabGaua+qacGauaIOmaaaakiac4ayGLbGaiGdGbIhacGaoagiCai ac4ayGGcGaiGdGcIcacWaiaAOeI0IamacGe27aU9aadGaiaUbaaSqa iacGpeGaiacGicdaa8aabKaiacGcpeGaiacGdIgacGacakykaiaayk W7cGaraIjcVladaciF=daagUcaRiacObiMi8UamacGeo7aN9aadGac aUbaaSqaiGaGpeGaiGaGicdaa8aabKacacGcpeGamaiGy9paaqyVd4 2damacaciw=daaBaaaleacaciw=daapeGaiaiGy9paaGimaaWdaeqc aciw=daaaOWdbiacaciw=daadUgapaWaiaiGy9paa0baaSqaiaiGy9 paa8qacGaGaI1=aaWGPbaapaqaiaiGy9paa8qacGaGaI1=aaaIYaaa aOGaiaiGa9paaeyzaiacaciq=daabIhacGaGac0=aaqGWbGaiaiGa9 paaeiOaiacaciq=daacIcacWaGac0=aaaH9oGBpaWaiaiGa9paaSba aSqaiaiGa9paa8qacGaGac0=aaaIWaaapaqajaiGa9paaaGcpeGaia iGa9paamiAaiacaciq=daacMcacWaGac0=aaGH9aqpaeaacqGH9aqp caaMi8UamWiGeU7aSjaaykW7caaMb8+aamWaa8aabaWdbiac0b4GRb Wdamac0byhaaWcbGaDa+qacGaDaoyAaaWdaeac0b4dbiac0biIYaaa aOWdaiaaygW7peWaaeWaa8aabaWdbmaabmaapaqaa8qadaWcaaWdae aapeGaeq4UdWgapaqaa8qacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaa paqabaaaaOWdbiaayIW7cWaPaA4kaSIaaGjcVlad8ciHXoqypaWaiW lGBaaaleac8c4dbiac8ciIWaaapaqajWlGaaGcpeGaayjkaiaawMca aiaaygW7cGaJagyzaiacmcyG4bGaiWiGbchacaqGGcGaaiikaiadOb OHsislcWaAasyVd42damacqb4gaaWcbGaua+qacGauaIimaaWdaeqc qbiak8qacGauaoiAaiac4aOGPaGaaGjcVladObOHRaWkcaaMi8UamW iGek7aI9aadGaJaUbaaSqaiWiGpeGaiWiGicdaa8aabKaJacGccaaM b8+dbiaabwgacaqG4bGaaeiCaiaabckacaGGOaGaaGzaVlabe27aU9 aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaWGObGaaiykaaGaayjk aiaawMcaaiaayIW7cqGHsislcaaMi8UamGjGeg7aH9aadaWgaaWcba WdbiaadMgaa8aabeaak8qacaWGRbWdamaaDaaaleaapeGaaGimaaWd aeaapeGaaGOmaaaakiaabwgacaqG4bGaaeiCaiaabckacaGGOaGaey OeI0IaeqyVd42damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaadIga caGGPaaacaGLBbGaayzxaaGaaiilaaaaaa@87E3@

(3)

а при 0:

α 0 + β 0 β g =λ ν 0 ν g ν 0 ν g , α 0 β 0 + β g ν g ν 0 =0, η 0 + γ 0 γ g =0, η 0 ν 0 k g 2 + γ g ν g k 0 2 =λ k 0 2 λ ν g + β g k g 2 λ ν 0 + α 0 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiabeg7aH9aadaWgaaWcbaWd biaaicdaa8aabeaakiaaygW7peGamGjGgUcaRiaaysW7cqaHYoGypa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyOeI0IaeqOSdi2damaa BaaaleaapeGaam4zaaWdaeqaaOWdbiabg2da9iabeU7aSnaalaaapa qaa8qacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyOe I0IaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqaaaGcbaWdbiabe2 7aU9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqaH9oGBpaWaaSba aSqaa8qacaWGNbaapaqabaaaaOWdbiaacYcacaaMf8UaeqySde2dam aaBaaaleaapeGaaGimaaWdaeqaaOGaaGzaV=qacqGHsislcqaHYoGy paWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGamGgGgUcaRiabek7aI9 aadaWgaaWcbaWdbiaadEgaa8aabeaak8qadaWcaaWdaeaapeGaeqyV d42damaaBaaaleaapeGaam4zaaWdaeqaaaGcbaWdbiabe27aU9aada WgaaWcbaWdbiaaicdaa8aabeaaaaGcpeGaeyypa0JaaGimaiaacYca aeaacqaH3oaApaWaaSbaaSqaa8qacGaMaIimaaWdaeqaaOWdbiabgU caRiabeo7aN9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGHsisl cqaHZoWzpaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGaeyypa0JaaG imaiaacYcaaeaacqaH3oaApaWaiGjGBaaaleacyc4dbiacyciIWaaa paqajGjGaOWdbiabe27aU9aadaWgaaWcbaWdbiaaicdaa8aabeaak8 qacaWGRbWdamaaDaaaleaapeGaam4zaaWdaeaapeGaaGOmaaaakiab gUcaRiabeo7aN9aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacqaH9o GBpaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGaam4Aa8aadaqhaaWc baWdbiaaicdaa8aabaWdbiaaikdaaaGccqGH9aqpcqaH7oaBdaWada WdaeaapeGaam4Aa8aadaqhaaWcbaWdbiaaicdaa8aabaWdbiaaikda aaGcdaqadaWdaeaapeWaaSaaa8aabaWdbiabeU7aSbWdaeaapeGaeq yVd42damaaBaaaleaapeGaam4zaaWdaeqaaaaak8qacqGHRaWkcqaH YoGypaWaaSbaaSqaa8qacaWGNbaapaqabaaak8qacaGLOaGaayzkaa GaeyOeI0Iaam4Aa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaaikda aaGcdaqadaWdaeaapeWaaSaaa8aabaWdbiabeU7aSbWdaeaapeGaeq yVd42damaaBaaaleaapeGaaGimaaWdaeqaaaaak8qacqGHRaWkcqaH XoqypaWaaSbaaSqaa8qacaaIWaaapaqabaaak8qacaGLOaGaayzkaa aacaGLBbGaayzxaaGaaiOlaaaaaa@BA94@

(4)

В дальнейшем нам понадобятся коэффициенты α0 и β0, поэтому ниже приведем их явные выражения:

α 0 = λ ν 0 ν g ν 0 × × ν 0 + ν i + ν 0 ν i exp (2 ν 0 h) ν 0 + ν g ν 0 + ν i ν 0 ν g ν 0 ν i exp (2 ν 0 h) , β 0 = 2λ ν 0 ν i exp (2 ν 0 h) ν 0 + ν g ν 0 + ν i ν 0 ν g ν 0 ν i exp (2 ν 0 h) . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiabeg7aH9aadaWgaaWcbaWd biaaicdaa8aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaeq4UdW2aae Waa8aabaWdbiabe27aU9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qa cqGHsislcqaH9oGBpaWaaSbaaSqaa8qacaWGNbaapaqabaaak8qaca GLOaGaayzkaaaapaqaa8qacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaa paqabaaaaOWdbiabgEna0cqaaiabgEna0oaalaaapaqaa8qadaqada WdaeaapeGaeqyVd42damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiab gUcaRiabe27aU9aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawI cacaGLPaaacqGHRaWkdaqadaWdaeaapeGaeqyVd42damaaBaaaleaa peGaaGimaaWdaeqaaOWdbiabgkHiTiabe27aU9aadaWgaaWcbaWdbi aadMgaa8aabeaaaOWdbiaawIcacaGLPaaaciGGLbGaaiiEaiaaccha caGGGcGaaiikaiabgkHiTiaaikdacqaH9oGBpaWaaSbaaSqaa8qaca aIWaaapaqabaGcpeGaamiAaiaacMcaa8aabaWdbmaabmaapaqaa8qa cqaH9oGBpaWaaSbaaSqaa8qacaaIWaaapaqabaGccaaMi8+dbiabgU caRiabe27aU9aadaWgaaWcbaWdbiaadEgaa8aabeaaaOWdbiaawIca caGLPaaadaqadaWdaeaapeGaeqyVd42damaaBaaaleaapeGaaGimaa WdaeqaaOGaaGjcV=qacqGHRaWkcqaH9oGBpaWaaSbaaSqaa8qacaWG Pbaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0YaaeWaa8aabaWdbi abe27aU9aadaWgaaWcbaWdbiaaicdaa8aabeaakiaayIW7peGaeyOe I0IaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqaaaGcpeGaayjkai aawMcaamaabmaapaqaa8qacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaa paqabaGccaaMi8+dbiabgkHiTiabe27aU9aadaWgaaWcbaWdbiaadM gaa8aabeaaaOWdbiaawIcacaGLPaaaciGGLbGaaiiEaiaacchacaGG GcGaaiikaiabgkHiTiaaikdacqaH9oGBpaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeGaamiAaiaacMcaaaGaaiilaaqaaiabek7aI9aadaWg aaWcbaWdbiaaicdaa8aabeaakiaayIW7peGaeyypa0JaaGjcVpaala aapaqaa8qacaaIYaGaeq4UdW2aaeWaa8aabaWdbiabe27aU9aadaWg aaWcbaWdbiaaicdaa8aabeaak8qacqGHsislcqaH9oGBpaWaaSbaaS qaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaciyzaiaacIha caGGWbGaaiiOaiaacIcacqGHsislcaaIYaGaeqyVd42damaaBaaale aapeGaaGimaaWdaeqaaOWdbiaadIgacaGGPaaapaqaa8qadaqadaWd aeaapeGaeqyVd42damaaBaaaleaapeGaaGimaaWdaeqaaOGaaGjcV= qacqGHRaWkcaaMi8UaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqa aaGcpeGaayjkaiaawMcaamaabmaapaqaa8qacqaH9oGBpaWaaSbaaS qaa8qacaaIWaaapaqabaGccaaMi8+dbiabgUcaRiaayIW7cqaH9oGB paWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaaG jcVlabgkHiTmaabmaapaqaa8qacqaH9oGBpaWaaSbaaSqaa8qacaaI WaaapaqabaGccaaMi8+dbiabgkHiTiaayIW7cqaH9oGBpaWaaSbaaS qaa8qacaWGNbaapaqabaaak8qacaGLOaGaayzkaaWaaeWaa8aabaWd biabe27aU9aadaWgaaWcbaWdbiaaicdaa8aabeaakiaayIW7peGaey OeI0IaaGjcVlabe27aU9aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWd biaawIcacaGLPaaaciGGLbGaaiiEaiaacchacaGGGcGaaGjcVlaacI cacqGHsislcaaIYaGaeqyVd42damaaBaaaleaapeGaaGimaaWdaeqa aOWdbiaadIgacaGGPaaaaiaac6caaaaa@00ED@

(5)

Имея результат вычисления для вектора-потенциала, можно определить электромагнитное поле и, соответственно, импеданс. Чтобы не усложнять расчетов, будем рассматривать составляющую импеданса Z yx = E y / H x . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGAbWdamaaBaaaleaapeGaamyE aiaadIhaa8aabeaakiaayIW7peGamWlGg2da9iaayIW7caaMe8Uaam yra8aadaWgaaWcbaWdbiaadMhaa8aabeaakiaayIW7peGaai4laiaa yIW7caWGibWdamaaBaaaleaapeGaamiEaaWdaeqaaOGaaiOlaaaa@5119@ Она имеет более простой вид по сравнению с другими, но в то же время отражает основные закономерности в поведении импеданса. Рассмотрим величину импеданса в нижней среде. Отметим, что в силу непрерывности горизонтальных составляющих полей импеданс на границе при подходе к ней снизу будет равен импедансу при подходе сверху.

Используя связь полей с вектором-потенциалом:

H g =rot  A g , E g =iω μ 0 A g iω μ 0 ϰ g 2 grad div  A g . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaqefeezVjwzGmuyZX2BUbcuY9gic9gBKbacgmaeaaaa aaaaa8qacaWFibWdamaaCaaaleqabaWdbmaabmaapaqaa8qacaWGNb aacaGLOaGaayzkaaaaaOGaeyypa0JaaeOCaiaab+gacaqG0bGaaeiO aiaa=feapaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaadEgaaiaawI cacaGLPaaaaaGccaGGSaGaaGjbVlaa=veapaWaaWbaaSqabeaapeWa aeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGccqGH9aqpcaWGPb GaeqyYdCNaeqiVd02damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaa =feapaWaaWbaaSqabeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcaca GLPaaaaaGccWaAaAOeI0YaaSaaa8aabaWdbiaadMgacqaHjpWDcqaH 8oqBpaWaaSbaaSqaa8qacaaIWaaapaqabaaakeaatuuDJXwAK1uy0H wmaeHbfv3ySLgzG0uy0Hgip5wzaGWba8qacqGFWpq+paWaa0baaSqa a8qacaWGNbaapaqaa8qacaaIYaaaaaaakiaabEgacaqGYbGaaeyyai aabsgacaqGGcGaaeizaiaabMgacaqG2bGaaeiOaiaa=feapaWaaWba aSqabeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGcpa GaaiOlaaaa@84E7@

можно получить

H x g = y A z g , E y g = iω μ 0 ϰ g 2 y A x g x + A z g z . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGibWdamaaDaaaleaapeGaamiE aaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGccq GH9aqpdaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG5baa aiaadgeapaWaa0baaSqaa8qacaWG6baapaqaa8qadaqadaWdaeaape Gaam4zaaGaayjkaiaawMcaaaaakiaacYcacaaMe8UaaGjbVlaadwea paWaa0baaSqaa8qacaWG5baapaqaa8qadaqadaWdaeaapeGaam4zaa GaayjkaiaawMcaaaaakiabg2da9iabgkHiTmaalaaapaqaa8qacaWG PbGaeqyYdCNaeqiVd02damaaBaaaleaapeGaaGimaaWdaeqaaaGcba Wefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiyaapeGae8h8 dK=damaaDaaaleaapeGaam4zaaWdaeaapeGaaGOmaaaaaaGcdaWcaa WdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG5baaamaadmaapaqa a8qadaWcaaWdaeaapeGaeyOaIyRaamyqa8aadaqhaaWcbaWdbiaadI haa8aabaWdbmaabmaapaqaa8qacaWGNbaacaGLOaGaayzkaaaaaaGc paqaa8qacqGHciITcaWG4baaaiabgUcaRmaalaaapaqaa8qacqGHci ITcaWGbbWdamaaDaaaleaapeGaamOEaaWdaeaapeWaaeWaa8aabaWd biaadEgaaiaawIcacaGLPaaaaaaak8aabaWdbiabgkGi2kaadQhaaa aacaGLBbGaayzxaaGaiGjGc6caaaa@85BD@

Подставляя выражения для A x g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGbbWdamaaDaaaleaapeGaamiE aaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaaaaa@42A4@ и A z g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGbbWdamaaDaaaleaapeGaamOE aaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaaaaa@42A6@ из (1) и (2), находим:

E y g = = J Δ x 4π i ω μ 0 ϰ g 2 x y 0 λ ν g +β ν g γ g λ exp ( ν g z) J 0 λρ dλ, H x g = J Δ x 4π x y 0 γ g λ exp ( ν g z) J 0 λρ dλ. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiabg2da9aqaaiabg2da9iaayIW7cqGHsisldGaAaUaaa8aabGaA a+qacGaAaoOsaiadObyHuoarpaWaiGgGBaaaleacOb4dbiacOb4G4b aapaqajGgGaaGcbGaAa+qacGaAaIinaiadObiHapaCaaGaiWfGdMga dGaraUaaa8aabGara+qacWarasyYdCNamqeGeY7aT9aadGaraUbaaS qaiqeGpeGaiqeGicdaa8aabKaracaakeaceb4efv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiyaapeGamqeG=b=a5=aadGara2baaS qaiqeGpeGaiqeGdEgaa8aabGara+qacGaraIOmaaaaaaGcdGaoaUaa a8aabGaoa+qacWaoaAOaIylapaqaiGdGpeGamGdGgkGi2kac4a4G4b aaamacSa4caaWdaeacSa4dbiadSaOHciITa8aabGala+qacWalaAOa IyRaiWcGdMhaaaWaiqbGwahabKafaUWdaeacua4dbiacuaiIWaaapa qaiqbGpeGamqbGg6HiLcqdpaqaiqbGpeGamqbGgUIiYdaakmaciaya daWdaeacia4dbmacaciC=daalaaapaqaiaiGW9paa8qacWaGac3=aa aH7oaBa8aabGaGac3=aaWdbiadaciC=daae27aU9aadGaGac3=aaWg aaWcbGaGac3=aaWdbiacaciC=daadEgaa8aabKaGac3=aaaaaaGcpe GaiGaGykW7cGacaIjcVladaciC=daagUcaRiacSaiMi8UamqbGek7a IjacSaiMi8UamWcGgkHiTiacSaiMi8UamqeGe27aU9aadGacaUbaaS qaiGaGpeGaiGaGdEgaa8aabKacacGccGacaIjcV=qadGacaUaaa8aa bGaca+qacWacas4SdC2damacia4gaaWcbGaca+qacGacao4zaaWdae qciaiaaOqaiGaGpeGamGaGeU7aSbaaaiaciaOLOaGaiGaGwMcaaiac aciC=daabwgacGaGac3=aaqG4bGaiaiGW9paaeiCaiacaciC=daabc kacGaGas2=aaGGOaGamaiGm9paaqyVd42damacacie=daaBaaaleac acie=daapeGaiaiGq8paam4zaaWdaeqcacie=daaaOWdbiacacie=d aadQhacGaGacX=aaGGPaGaiaiGC8paamOsa8aadGaGaIV=aaWgaaWc bGaGaIV=aaWdbiacaci++daaicdaa8aabKaGaIV=aaaakiacaci4+d aayIW7peWaiaiGG7paaeWaa8aabGaGacU=aaWdbiadaci4+daaeU7a Sjadaci4+daaeg8aYbGaiaiGG7paayjkaiacaci4+daawMcaaiacac i1+daadsgacWaGasT=aaaH7oaBcGaGasT=aaGGSaaabaGaamisa8aa daqhaaWcbaWdbiaadIhaa8aabaWdbmaabmaapaqaa8qacaWGNbaaca GLOaGaayzkaaaaaOGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadQea cqqHuoarpaWaaSbaaSqaa8qacaWG4baapaqabaaakeaapeGaaGinai abec8aWbaadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG 4baaamaalaaapaqaa8qacqGHciITa8aabaWdbiabgkGi2kaadMhaaa WaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiabg6HiLcqdpaqaa8qa cqGHRiI8aaGcdaWcaaWdaeaapeGaeq4SdC2damaaBaaaleaapeGaam 4zaaWdaeqaaaGcbaWdbiabeU7aSbaacaqGLbGaaeiEaiaabchacaqG GcGaaiikaiabe27aU9aadaWgaaWcbaWdbiaadEgaa8aabeaak8qaca WG6bGaaiykaiaadQeapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWa aeWaa8aabaWdbiabeU7aSjabeg8aYbGaayjkaiaawMcaaiaadsgacq aH7oaBcaGGUaaaaaa@659A@

Выразив коэффициенты βg, γg через α0, β0, η0 и γ0 с помощью (3) и (4) получим:

 

E y g = J Δ x 4π i ω μ 0 ϰ g 2 x y 0 λ ν 0 + α 0 + β 0 ν g λ η 0 + γ 0 exp ( ν g z) J 0 αρ dλ, H x g = J Δ x 4π x y 0 η 0 + γ 0 λ exp ( ν g z) J 0 λρ dλ. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiabg2da9iabgkHiTmaalaaapaqaa8qacaWGkbGaeuiLdq0damaa BaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaisdacqaHapaCaaGaam yAamaalaaapaqaa8qacqaHjpWDcqaH8oqBpaWaaSbaaSqaa8qacaaI WaaapaqabaaakeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5 wzaGGba8qacqWFWpq+paWaa0baaSqaa8qacaWGNbaapaqaa8qacaaI Yaaaaaaakmaalaaapaqaa8qacqGHciITa8aabaWdbiabgkGi2kaadI haaaWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaeyOaIyRaamyEaaaa daGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaeyOhIukan8aabaWdbi abgUIiYdaakmaadmaapaqaa8qadaWcaaWdaeaapeGaeq4UdWgapaqa a8qacqaH9oGBpaWaaSbaaSqaa8qacaaIWaaapaqabaaaaOWdbiabgU caRiabeg7aH9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacWaAaA4k aSIamGgGek7aI9aadGaAaUbaaSqaiGgGpeGaiGgGicdaa8aabKaAac GccaaMb8+dbiabgkHiTmaalaaapaqaa8qacqaH9oGBpaWaaSbaaSqa a8qacaWGNbaapaqabaaakeaapeGaeq4UdWgaamaabmaapaqaa8qacq aH3oaApaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4kaSIaeq4S dC2damaaBaaaleaapeGaaGimaaWdaeqaaaGcpeGaayjkaiaawMcaaa Gaay5waiaaw2faaiaabwgacaqG4bGaaeiCaiaabckacaGGOaGaeqyV d42damaaBaaaleaapeGaam4zaaWdaeqaaOWdbiaadQhacaGGPaGaam Osa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qadaqadaWdaeaapeGa eqySdeMaeqyWdihacaGLOaGaayzkaaGaamizaiabeU7aSjaacYcaae aacaWGibWdamaaDaaaleaapeGaamiEaaWdaeaapeWaaeWaa8aabaWd biaadEgaaiaawIcacaGLPaaaaaGccqGH9aqpcqGHsisldaWcaaWdae aapeGaamOsaiabfs5ae9aadaWgaaWcbaWdbiaadIhaa8aabeaaaOqa a8qacaaI0aGaeqiWdahaamaalaaapaqaa8qacqGHciITa8aabaWdbi abgkGi2kaadIhaaaWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaeyOa IyRaamyEaaaadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaeyOhIu kan8aabaWdbiabgUIiYdaakmaalaaapaqaa8qadaqadaWdaeaapeGa eq4TdG2damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgUcaRiabeo 7aN9aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaa a8aabaWdbiabeU7aSbaacaqGLbGaaeiEaiaabchacaqGGcGaaiikai abe27aU9aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacaWG6bGaaiyk aiaadQeapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8aaba WdbiabeU7aSjabeg8aYbGaayjkaiaawMcaaiaadsgacqaH7oaBcaGG Uaaaaaa@DBC9@

(6)

При возбуждении волн низкочастотного диапазона (КНЧ и СНЧ, частоты менее 300 Гц) хорошим приближением при рассмотрении поля является квазистационарное приближение [Терещенко, 2017], в рамках которого полагают ϰ 0 0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=b=a5=aadaWgaaWcbaWdbiaaicdaa8aabe aak8qacqGHsgIRcaaIWaGaaiOlaaaa@4E65@ Воспользуемся условием ϰ 0 =0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=b=a5=aadaWgaaWcbaWdbiaaicdaa8aabe aak8qacqGH9aqpcaaIWaGaaiOlaaaa@4D7E@ Тогда из систем уравнений (3) и (4) получаем следующие соотношения:

η 0 | ϰ 0 0 = 1+ α 0 | ϰ 0 0 ,   γ 0 | ϰ 0 0 = β 0 | ϰ 0 0 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaH3oaApaWaaSbaaSqaa8qacaaI WaaapaqabaGcpeGaaiiFa8aadaWgaaWcbaWefv3ySLgznfgDOfdary qr1ngBPrginfgDObYtUvgaiyaapeGae8h8dK=damaaBaaameaapeGa aGimaaWdaeqaaSWdbiabgkziUkaaicdaa8aabeaak8qacqGH9aqpcq GHsisldaqadaWdaeaapeGaaGymaiabgUcaRiabeg7aH9aadaWgaaWc baWdbiaaicdaa8aabeaak8qacaGG8bWdamaaBaaaleaapeGae8h8dK =damaaBaaameaapeGaaGimaaWdaeqaaSWdbiabgkziUkaaicdaa8aa beaaaOWdbiaawIcacaGLPaaacaGGSaGaaiiOaiaacckacqaHZoWzpa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiiFa8aadaWgaaWcbaWd biab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabeaal8qacqGHsgIRca aIWaaapaqabaGcpeGaeyypa0JaeqOSdi2damaaBaaaleaapeGaaGim aaWdaeqaaOWdbiaacYhapaWaaSbaaSqaa8qacqWFWpq+paWaaSbaaW qaa8qacaaIWaaapaqabaWcpeGaeyOKH4QaaGimaaWdaeqaaOGaiqeG c6caaaa@7C4C@

Подставляя эти выражения в (6) с учетом (5) имеем:

 

E y g | ϰ 0 0 = J Δ x iω μ 0 2π ϰ g 2 x y 0 exp ( ν g z) J 0 λρ dλ+ 0 f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ , H x g | ϰ 0 0 = J Δ x 2π x y 0 1 λ+ ν g exp ( ν g z) J 0 λρ dλ+ 0 ν g ϰ g 2 f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabe aal8qacqGHsgIRcaaIWaaapaqabaGcpeGaeyypa0JaeyOeI0YaaSaa a8aabaWdbiaadQeacqqHuoarpaWaaSbaaSqaa8qacaWG4baapaqaba GcpeGaamyAaiabeM8a3jabeY7aT9aadaWgaaWcbaWdbiaaicdaa8aa beaaaOqaa8qacaaIYaGaeqiWdaNae8h8dK=damaaDaaaleaapeGaam 4zaaWdaeaapeGaaGOmaaaaaaGcdaWcaaWdaeaapeGaeyOaIylapaqa a8qacqGHciITcaWG4baaamaalaaapaqaa8qacqGHciITa8aabaWdbi abgkGi2kaadMhaaaWaamWaa8aabaWdbmaawahabeWcpaqaa8qacaaI Waaapaqaa8qacqGHEisPa0WdaeaapeGaey4kIipaaOGaaeyzaiaabI hacaqGWbGaaeiOaiaacIcacqaH9oGBpaWaaSbaaSqaa8qacaWGNbaa paqabaGcpeGaamOEaiaacMcacaWGkbWdamaaBaaaleaapeGaaGimaa WdaeqaaOWdbmaabmaapaqaa8qacqaH7oaBcqaHbpGCaiaawIcacaGL PaaacaWGKbGaeq4UdWMaey4kaSYaaybCaeqal8aabaWdbiaaicdaa8 aabaWdbiabg6HiLcqdpaqaa8qacqGHRiI8aaGccaWGMbWaaeWaa8aa baWdbiabeU7aSjaacYcacqWFWpq+paWaaSbaaSqaa8qacaWGNbaapa qabaGcpeGaaiilaiab=b=a5=aadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacaGGSaGaamiAaaGaayjkaiaawMcaaiaabwgacaqG4bGaaeiCai aabckacaGGOaGaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqaaOWd biaadQhacaGGPaGaamOsa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8 qadaqadaWdaeaapeGaeq4UdWMaeqyWdihacaGLOaGaayzkaaGaamiz aiabeU7aSbGaay5waiaaw2faaiaacYcaaeaacaWGibWdamaaDaaale aapeGaamiEaaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGL PaaaaaGccaGG8bWdamaaBaaaleaapeGae8h8dK=damaaBaaameaape GaaGimaaWdaeqaaSWdbiabgkziUkaaicdaa8aabeaak8qacqGH9aqp daWcaaWdaeaapeGaamOsaiabfs5ae9aadaWgaaWcbaWdbiaadIhaa8 aabeaaaOqaa8qacaaIYaGaeqiWdahaamaalaaapaqaa8qacqGHciIT a8aabaWdbiabgkGi2kaadIhaaaWaaSaaa8aabaWdbiabgkGi2cWdae aapeGaeyOaIylcbiGaa4xEaaaadaWadaWdaeaapeWaaybCaeqal8aa baWdbiaaicdaa8aabaWdbiabg6HiLcqdpaqaa8qacqGHRiI8aaGcda WcaaWdaeaapeGaaGymaaWdaeaapeGaeq4UdWMaey4kaSIaeqyVd42d amaaBaaaleaapeGaam4zaaWdaeqaaaaak8qacaqGLbGaaeiEaiaabc hacaqGGcGaaiikaiabe27aU9aadaWgaaWcbaWdbiaadEgaa8aabeaa k8qacaWG6bGaaiykaiaadQeapaWaaSbaaSqaa8qacaaIWaaapaqaba GcpeWaaeWaa8aabaWdbiabeU7aSjabeg8aYbGaayjkaiaawMcaaiaa dsgacqaH7oaBcqGHRaWkdaGfWbqabSWdaeaapeGaaGimaaWdaeaape GaeyOhIukan8aabaWdbiabgUIiYdaakmaalaaapaqaa8qacqaH9oGB paWaaSbaaSqaa8qacaWGNbaapaqabaaakeaapeGae8h8dK=damaaDa aaleaapeGaam4zaaWdaeaapeGaaGOmaaaaaaGccaWGMbWaaeWaa8aa baWdbiabeU7aSjaacYcacqWFWpq+paWaaSbaaSqaa8qacaWGNbaapa qabaGcpeGaaiilaiab=b=a5=aadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacaGGSaGaamiAaaGaayjkaiaawMcaaiaabwgacaqG4bGaaeiCai aabckacaGGOaGaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqaaOWd biaadQhacaGGPaGaamOsa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8 qadaqadaWdaeaapeGaeq4UdWMaeqyWdihacaGLOaGaayzkaaGaamiz aiabeU7aSbGaay5waiaaw2faaiaacYcaaaaa@1EE1@

(7)

где

f λ, ϰ g , ϰ i ,h = 2 λ ν g λ ν i exp(2λh) λ+ ν g λ+ ν i λ ν g λ ν i exp(2λh) . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGMbWaaeWaa8aabaWdbiabeU7a SjaacYcatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGGbci ab=b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacaGGSaacgaGa e4h8dK=damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaacYcacaWGOb aacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbiaaikdadaqadaWd aeaapeGaeq4UdWMaeyOeI0IaeqyVd42damaaBaaaleaapeGaam4zaa WdaeqaaaGcpeGaayjkaiaawMcaamaabmaapaqaa8qacqaH7oaBcqGH sislcqaH9oGBpaWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOa GaayzkaaGaciyzaiaacIhacaGGWbGaaiikaiabgkHiTiaaikdacqaH 7oaBcaWGObGaaiykaaWdaeaapeWaaeWaa8aabaWdbiabeU7aSjabgU caRiabe27aU9aadaWgaaWcbaWdbiaadEgaa8aabeaaaOWdbiaawIca caGLPaaadaqadaWdaeaapeGaeq4UdWMaey4kaSIaeqyVd42damaaBa aaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgkHiTmaa bmaapaqaa8qacqaH7oaBcqGHsislcqaH9oGBpaWaaSbaaSqaa8qaca WGNbaapaqabaaak8qacaGLOaGaayzkaaWaaeWaa8aabaWdbiabeU7a SjabgkHiTiabe27aU9aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbi aawIcacaGLPaaaciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0IaaGOm aiabeU7aSjaadIgacaGGPaaaaiaac6caaaa@9745@

Формула (7) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ это представление полей в виде суммы поля в двухслойной среде и дополнения, отражающего влияние ионосферы. При h, т.е. в отсутствие ионосферы, второе слагаемое в квадратных скобках стремится к нулю. Первые слагаемые в квадратных скобках несложно вычислить, используя два интеграла Ватсона [Терещенко, 2017]:

0 exp( ν g z) ν g J 0 λρ λdλ= exp( ϰ g )R R ,   0 exp( ν g z) ν g J 0 λρ dλ= I 0 r + K 0 r , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbmaawahabeWcpaqaa8qacaaI Waaapaqaa8qacqGHEisPa0WdaeaapeGaey4kIipaaOWaaSaaa8aaba WdbiGacwgacaGG4bGaaiiCaiaacIcacqaH9oGBpaWaaSbaaSqaa8qa caWGNbaapaqabaGcpeGaamOEaiaacMcaa8aabaWdbiabe27aU9aada WgaaWcbaWdbiaadEgaa8aabeaaaaGcpeGaamOsa8aadaWgaaWcbaWd biaaicdaa8aabeaak8qadaqadaWdaeaapeGaeq4UdWMaeqyWdihaca GLOaGaayzkaaGaeq4UdWMaamizaiabeU7aSjabg2da9maalaaapaqa a8qaciGGLbGaaiiEaiaacchacaGGOaGaeyOeI0Yefv3ySLgznfgDOf daryqr1ngBPrginfgDObYtUvgaiyaacqWFWpq+paWaaSbaaSqaa8qa caWGNbaapaqabaGcpeGaaiykaiaadkfaa8aabaWdbiaadkfaaaGaai ilaiaacckacaGGGcaabaWaaybCaeqal8aabaWdbiaaicdaa8aabaWd biabg6HiLcqdpaqaa8qacqGHRiI8aaGcdaWcaaWdaeaapeGaciyzai aacIhacaGGWbGaaiikaiabe27aU9aadaWgaaWcbaWdbiaadEgaa8aa beaak8qacaWG6bGaaiykaaWdaeaapeGaeqyVd42damaaBaaaleaape Gaam4zaaWdaeqaaaaak8qacaWGkbWdamaaBaaaleaapeGaaGimaaWd aeqaaOWdbmaabmaapaqaa8qacqaH7oaBcqaHbpGCaiaawIcacaGLPa aacaWGKbGaeq4UdWMaeyypa0Jaamysa8aadaWgaaWcbaWdbiaaicda a8aabeaak8qadaqadaWdaeaapeGaamOCa8aadaWgaaWcbaWdbiabgU caRaWdaeqaaaGcpeGaayjkaiaawMcaaiaadUeapaWaaSbaaSqaa8qa caaIWaaapaqabaGcpeWaaeWaa8aabaWdbiaadkhapaWaaSbaaSqaa8 qacqGHsisla8aabeaaaOWdbiaawIcacaGLPaaacaGGSaaaaaa@9A8E@

где

R= ρ 2 + z 2 ,  r + = ϰ g R+z 2 ,  r = ϰ g Rz 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGsbGaeyypa0ZaaOaaa8aabaWd biabeg8aY9aadaahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaamOEa8 aadaahaaWcbeqaa8qacaaIYaaaaaqabaGccaGGSaGaaeiOaiaadkha paWaaSbaaSqaa8qacqGHRaWka8aabeaak8qacqGH9aqpdaWcaaWdae aatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGGba8qacqWF Wpq+paWaaSbaaSqaa8qacaWGNbaapaqabaGcpeWaaeWaa8aabaWdbi aadkfacqGHRaWkcaWG6baacaGLOaGaayzkaaaapaqaa8qacaaIYaaa aiaacYcacaqGGcGaamOCa8aadaWgaaWcbaWdbiabgkHiTaWdaeqaaO Wdbiabg2da9maalaaapaqaaGGbc8qacqGFWpq+paWaaSbaaSqaa8qa caWGNbaapaqabaGcpeWaaeWaa8aabaWdbiaadkfacqGHsislcaWG6b aacaGLOaGaayzkaaaapaqaa8qacaaIYaaaaiaacYcaaaa@6CD4@

I0, K0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ модифицированные функции Бесселя. В результате получим

E y g | ϰ 0 0 = J Δ x iω μ 0 2π ϰ g 2 x y z I 0 r + K 0 r + 0 f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGfbWdamaaDaaaleaapeGaamyE aaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGcca GG8bWdamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgi p5wzaGGba8qacqWFWpq+paWaaSbaaWqaa8qacaaIWaaapaqabaWcpe GaeyOKH4QaaGimaaWdaeqaaOWdbiabg2da9iabgkHiTmaalaaapaqa a8qacaWGkbGaeuiLdq0damaaBaaaleaapeGaamiEaaWdaeqaaOWdbi aadMgacqaHjpWDcqaH8oqBpaWaaSbaaSqaa8qacaaIWaaapaqabaaa keaapeGaaGOmaiabec8aWjab=b=a5=aadaqhaaWcbaWdbiaadEgaa8 aabaWdbiaaikdaaaaaaOWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGa eyOaIyRaamiEaaaadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHci ITcaWG5baaamaadmaapaqaa8qadaWcaaWdaeaapeGaeyOaIylapaqa a8qacqGHciITcaWG6baaaiaadMeapaWaaSbaaSqaa8qacaaIWaaapa qabaGcpeWaaeWaa8aabaWdbiaadkhapaWaaSbaaSqaa8qacqGHRaWk a8aabeaaaOWdbiaawIcacaGLPaaacaWGlbWdamaaBaaaleaapeGaaG imaaWdaeqaaOWdbmaabmaapaqaa8qacaWGYbWdamaaBaaaleaapeGa eyOeI0capaqabaaak8qacaGLOaGaayzkaaGaey4kaSYaaybCaeqal8 aabaWdbiaaicdaa8aabaWdbiabg6HiLcqdpaqaa8qacqGHRiI8aaGc caWGMbWaaeWaa8aabaWdbiabeU7aSjaacYcacqWFWpq+paWaaSbaaS qaa8qacaWGNbaapaqabaGcpeGaaiilaiab=b=a5=aadaWgaaWcbaWd biaadMgaa8aabeaak8qacaGGSaacbiGaa4hAaaGaayjkaiaawMcaai aabwgacaqG4bGaaeiCaiaabckacaGGOaGaeqyVd42damaaBaaaleaa peGaam4zaaWdaeqaaOWdbiaadQhacaGGPaGaamOsa8aadaWgaaWcba Wdbiaaicdaa8aabeaak8qadaqadaWdaeaapeGaeq4UdWMaeqyWdiha caGLOaGaayzkaaGaamizaiabeU7aSbGaay5waiaaw2faaiaacYcaaa a@A952@

 

H x g | ϰ 0 0 = J Δ x 2π ϰ g x y 1 ϰ g z exp( ϰ g R) R 1 ϰ g 2 z 2 I 0 r + K 0 r 0 ν g ϰ g f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadIeapaWaa0baaSqaa8qa caWG4baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabe aal8qacqGHsgIRcaaIWaaapaqabaGcpeGaeyypa0JaeyOeI0YaaSaa a8aabaWdbiaadQeacqqHuoarpaWaaSbaaSqaa8qacaWG4baapaqaba aakeaapeGaaGOmaiabec8aWHGbciab+b=a5=aadaWgaaWcbaWdbiaa dEgaa8aabeaaaaGcpeWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaey OaIyRaamiEaaaadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciIT caWG5baaamaadeaapaqaa8qadaWcaaWdaeaapeGaaGymaaWdaeaape Gae8h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaaaak8qadaWcaaWd aeaapeGaeyOaIylapaqaa8qacqGHciITcaWG6baaamaalaaapaqaa8 qaciGGLbGaaiiEaiaacchacaGGOaGae8h8dK=damaaBaaaleaapeGa am4zaaWdaeqaaOWdbiaadkfacaGGPaaapaqaa8qacaWGsbaaaiabgk HiTmaalaaapaqaa8qacaaIXaaapaqaa8qacqWFWpq+paWaaSbaaSqa a8qacaWGNbaapaqabaaaaOWdbmaalaaapaqaa8qacqGHciITpaWaaW baaSqabeaapeGaaGOmaaaaaOWdaeaapeGaeyOaIyRaamOEa8aadaah aaWcbeqaa8qacaaIYaaaaaaakiaadMeapaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeWaaeWaa8aabaWdbiaadkhapaWaaSbaaSqaa8qacqGH RaWka8aabeaaaOWdbiaawIcacaGLPaaacaWGlbWdamaaBaaaleaape GaaGimaaWdaeqaaOWdbmaabmaapaqaa8qacaWGYbWdamaaBaaaleaa peGaeyOeI0capaqabaaak8qacaGLOaGaayzkaaaacaGLBbaacqGHsi slaeaacqGHsisldaWacaWdaeaapeWaaybCaeqal8aabaWdbiaaicda a8aabaWdbiabg6HiLcqdpaqaa8qacqGHRiI8aaGcdaWcaaWdaeaape GaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqaaaGcbaWdbiab=b=a 5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaaGcpeGaamOzamaabmaapa qaa8qacqaH7oaBcaGGSaGae8h8dK=damaaBaaaleaapeGaam4zaaWd aeqaaOWdbiaacYcacqWFWpq+paWaaSbaaSqaa8qacaWGPbaapaqaba GcpeGaaiilaiaadIgaaiaawIcacaGLPaaacaqGLbGaaeiEaiaabcha caqGGcGaaiikaiabe27aU9aadaWgaaWcbaWdbiacCb4GNbaapaqaba GcpeGaamOEaiaacMcacaWGkbWdamaaBaaaleaapeGaaGimaaWdaeqa aOWdbmaabmaapaqaa8qacqaH7oaBcqaHbpGCaiaawIcacaGLPaaaca WGKbGaeq4UdWgacaGLDbaacaGGUaaaaaa@C704@

(8)

После того, как получили выражения для полей горизонтального заземленного вибратора (8), можно перейти к рассмотрению импеданса.

ПОВЕРХНОСТНЫЙ ИМПЕДАНС ПОЛЯ ЗАЗЕМЛЕННОЙ ГОРИЗОНТАЛЬНОЙ АНТЕННЫ

Прежде чем преобразовать формулы (8), рассмотрим падение плоской монохроматической волны на границу раздела двух сред [Бреховских, 1957]. Ее импеданс Zg равен:

 

Z g = E τ H τ = μ 0 ε ˜ ε 0 = μ 0 ε 0 1 1+i σ g / ω ε 0 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGAbWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGfbWdamaaBaaale aapeGaeqiXdqhapaqabaaakeaapeGaamisa8aadaWgaaWcbaWdbiab es8a0bWdaeqaaaaak8qacqGH9aqpdaGcaaWdaeaapeWaaSaaa8aaba WdbiabeY7aT9aadaWgaaWcbaWdbiaaicdaa8aabeaaaOqaa8qacuaH 1oqzpaGbaGWdbyaafaGaeqyTdu2damaaBaaaleaapeGaaGimaaWdae qaaaaaa8qabeaakiabg2da9maakaaapaqaa8qadaWcaaWdaeaapeGa eqiVd02damaaBaaaleaapeGaaGimaaWdaeqaaaGcbaWdbiabew7aL9 aadaWgaaWcbaWdbiaaicdaa8aabeaaaaGcpeWaaSaaa8aabaWdbiaa igdaa8aabaWdbiaaigdacqGHRaWkcaWGPbGaeq4Wdm3damaaBaaale aapeGaam4zaaWdaeqaaOWdbiaac+cadaqadaWdaeaapeGaeqyYdCNa eqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaaGcpeGaayjkaiaawM caaaaaaSqabaGccaGGSaaaaa@65F2@

где Eτ и Hτ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ тангенциальные составляющие электрического и магнитного полей, соответственно. При σ/ ω ε 0 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaHdpWCcaGGVaWaaeWaa8aabaWd biabeM8a3jabew7aL9aadaWgaaWcbaWdbiaaicdaa8aabeaaaOWdbi aawIcacaGLPaaacqWIRjYpcaaIXaaaaa@48BA@ импеданс будет равен

 

 

Z g = iω μ 0 σ g . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGAbWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabg2da9maakaaapaqaa8qacqGHsisldaWcaaWdae aapeGaamyAaiabeM8a3jabeY7aT9aadaWgaaWcbaWdbiaaicdaa8aa beaaaOqaa8qacqaHdpWCpaWaaSbaaSqaa8qacaWGNbaapaqabaaaaa WdbeqaaOGaaiOlaaaa@4BE6@

(9)

Рассмотрим поведение поля и, соответственно, импеданса при z, стремящемся к нулю, т.е. на поверхности Земли. Из (8), учитывая (9), получаем при z0:

 

E y g | ϰ 0 0,z0 = J Δ x 2π ϰ g Z g x y × × 1 ρ + 0 f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ | z0 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aak8aacaaMb8+dbiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqe guuDJXwAKbstHrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbi aaicdaa8aabeaal8qacqGHsgIRcaaIWaGaaiilaiaayIW7caWG6bGa eyOKH4QaaGimaaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGkb GaeuiLdq0damaaBaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaikda cqaHapaCcqWFWpq+paWaaSbaaSqaa8qacaWGNbaapaqabaaaaOWdbi aadQfapaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeWaaSaaa8aabaWd biabgkGi2cWdaeaapeGaeyOaIyRaamiEaaaadaWcaaWdaeaapeGaey OaIylapaqaa8qacqGHciITcaWG5baaaiabgEna0cqaaiabgEna0oaa dmaapaqaa8qadaWcaaWdaeaapeGaaGymaaWdaeaapeGaeqyWdihaai abgUcaRmaawahabeWcpaqaa8qacaaIWaaapaqaa8qacqGHEisPa0Wd aeaapeGaey4kIipaaOGaamOzamaabmaapaqaa8qacqaH7oaBcaGGSa Gae8h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaOWdbiaacYcacqWF Wpq+paWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaaiilaiaadIgaai aawIcacaGLPaaacaqGLbGaaeiEaiaabchacaqGGcGaaiikaiabe27a U9aadaWgaaWcbaWdbiaadEgaa8aabeaak8qacaWG6bGaaiykaiaadQ eapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8aabaWdbiab eU7aSjabeg8aYbGaayjkaiaawMcaaiaadsgacqaH7oaBcaGG8bWdam aaBaaaleaapeGaamOEaiabgkziUkaaicdaa8aabeaaaOWdbiaawUfa caGLDbaacaGGSaaaaaa@A7E2@

H x g | ϰ 0 0,z0 = J Δ x 2π ϰ g x y × × ϰ g 2 I 0 ρ ϰ g 2 K 0 ρ ϰ g 2 + I 1 ρ ϰ g 2 K 1 ρ ϰ g 2 + MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadIeapaWaa0baaSqaa8qa caWG4baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabe aal8qacqGHsgIRcaaIWaGaaiilaiaayIW7caWG6bGaeyOKH4QaaGim aaWdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGkbGaeuiLdq0dam aaBaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaikdacqaHapaCcqWF Wpq+paWaaSbaaSqaa8qacaWGNbaapaqabaaaaOWdbmaalaaapaqaa8 qacqGHciITa8aabaWdbiabgkGi2kaadIhaaaWaaSaaa8aabaWdbiab gkGi2cWdaeaapeGaeyOaIyRaamyEaaaacqGHxdaTaeaacqGHxdaTda WabaWdaeaapeWaaSaaa8aabaWdbiab=b=a5=aadaWgaaWcbaWdbiaa dEgaa8aabeaaaOqaa8qacaaIYaaaamaadmaapaqaa8qacaWGjbWdam aaBaaaleaapeGaaGimaaWdaeqaaOWdbmaabmaapaqaa8qadaWcaaWd aeaapeGaeqyWdiNae8h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaa GcbaWdbiaaikdaaaaacaGLOaGaayzkaaGaam4sa8aadaWgaaWcbaWd biaaicdaa8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabeg 8aYjab=b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaOqaa8qacaaI YaaaaaGaayjkaiaawMcaaiabgUcaRiaadMeapaWaaSbaaSqaa8qaca aIXaaapaqabaGcpeWaaeWaa8aabaWdbmaalaaapaqaa8qacqaHbpGC cqWFWpq+paWaaSbaaSqaa8qacaWGNbaapaqabaaakeaapeGaaGOmaa aaaiaawIcacaGLPaaacaWGlbWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaeqyWdiNae8h8dK=dam aaBaaaleaapeGaam4zaaWdaeqaaaGcbaWdbiaaikdaaaaacaGLOaGa ayzkaaaacaGLBbGaayzxaaaacaGLBbaacqGHRaWkaaaa@A39E@

+ 0 ν g ϰ g f λ, ϰ g , ϰ i ,h exp ( ν g z) J 0 λρ dλ | z0 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqGHRaWkdaWacaWdaeaapeWaaybC aeqal8aabaWdbiaaicdaa8aabaWdbiabg6HiLcqdpaqaa8qacqGHRi I8aaGcdaWcaaWdaeaapeGaeqyVd42damaaBaaaleaapeGaam4zaaWd aeqaaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiy aapeGae8h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaaaak8qacaWG MbWaaeWaa8aabaWdbiabeU7aSjaacYcacqWFWpq+paWaaSbaaSqaa8 qacaWGNbaapaqabaGcpeGaaiilaiab=b=a5=aadaWgaaWcbaWdbiaa dMgaa8aabeaak8qacaGGSaGaamiAaaGaayjkaiaawMcaaiaabwgaca qG4bGaaeiCaiaabckacaGGOaGaeqyVd42damaaBaaaleaapeGaam4z aaWdaeqaaOWdbiaadQhacaGGPaGaamOsa8aadaWgaaWcbaWdbiaaic daa8aabeaak8qadaqadaWdaeaapeGaeq4UdWMaeqyWdihacaGLOaGa ayzkaaGaamizaiabeU7aSjaacYhapaWaaSbaaSqaa8qacaWG6bGaey OKH4QaaGimaaWdaeqaaaGcpeGaayzxaaGaaiOlaaaa@7D59@

Если воспользоваться асимптотическими разложениями модифицированных функций Бесселя при ρ ϰ g /2 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qadaabdaWdaeaapeGaeqyWdi3efv3y SLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiyaacqWFWpq+paWaaS baaSqaa8qacaWGNbaapaqabaGcpeGaai4laiaaikdaaiaawEa7caGL iWoacqWIRjYpcaaIXaaaaa@53C6@ [Градштейн, 1962], то несложно заметить, что внеинтегральный член в H x g | ϰ 0 0,z0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaacbiaeaaaaaaaaa8qacaWFibWdamaaDaaaleaapeGa amiEaaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaa GccaGG8bWdamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGGba8qacqGFWpq+paWaaSbaaWqaa8qacaaIWaaapaqaba WcpeGaeyOKH4QaaGimaiaacYcacaWG6bGaeyOKH4QaaGimaaWdaeqa aaaa@583E@ стремится к 1/ρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaaIXaGaaGzaVlaac+cacaaMb8Ua eqyWdihaaa@4424@ и совпадет с аналогичным в E y g | ϰ 0 0,z0 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGfbWdamaaDaaaleaapeGaamyE aaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGcca GG8bWdamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgi p5wzaGGba8qacqWFWpq+paWaaSbaaWqaa8qacaaIWaaapaqabaWcpe GaeyOKH4QaaGimaiaacYcacaaMi8UaamOEaiabgkziUkaaicdaa8aa beaakiacebOGUaaaaa@5B4F@ Следующий шаг преобразования формул для полей MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ дифференцирование по y. Учтем, что

y = ρ y ρ = y ρ ρ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaapeGaeyOaIylapaqa a8qacqGHciITcaWG5baaaiabg2da9maalaaapaqaa8qacqGHciITcq aHbpGCa8aabaWdbiabgkGi2kaadMhaaaWaaSaaa8aabaWdbiabgkGi 2cWdaeaapeGaeyOaIyRaeqyWdihaaiabg2da9maalaaapaqaa8qaca WG5baapaqaa8qacqaHbpGCaaWaaSaaa8aabaWdbiabgkGi2cWdaeaa peGaeyOaIyRaeqyWdihaaiaacYcaaaa@574D@

и после дифференцирования по ρ получим:

 

E y g | ϰ 0 0,z0 = J Δ x 2π ϰ g Z g x y ρ × × 1 ρ 2 + 0 f λ, ϰ g , ϰ i ,h exp ( ν g z) J 1 λρ dλ | z0 , H x g | ϰ 0 0,z0 = J Δ x 2π ϰ g x y ρ ϰ g ρ I 1 ρ ϰ g 2 K 1 ρ ϰ g 2 + + 0 ν g ϰ g f λ, ϰ g , ϰ i ,h exp ( ν g z) J 1 λρ dλ | z0 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabe aal8qacqGHsgIRcaaIWaGaaiilaiaadQhacqGHsgIRcaaIWaaapaqa baGcpeGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadQeacqqHuoarpa WaaSbaaSqaa8qacaWG4baapaqabaaakeaapeGaaGOmaiabec8aWjab =b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaaGcpeGaamOwa8aada WgaaWcbaWdbiaadEgaa8aabeaak8qadaWcaaWdaeaapeGaeyOaIyla paqaa8qacqGHciITcaWG4baaamaalaaapaqaa8qacaWG5baapaqaa8 qacqaHbpGCaaGaey41aqlabaGaey41aq7aamWaa8aabaWdbmaalaaa paqaa8qacaaIXaaapaqaa8qacqaHbpGCpaWaaWbaaSqabeaapeGaaG OmaaaaaaGccqGHRaWkdaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGa eyOhIukan8aabaWdbiabgUIiYdaakiaadAgadaqadaWdaeaapeGaeq 4UdWMaaiilaiab=b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaak8qa caGGSaGae8h8dK=damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaacY cacaWGObaacaGLOaGaayzkaaGaaeyzaiaabIhacaqGWbGaaeiOaiaa cIcacqaH9oGBpaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGaamOEai aacMcacaWGkbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbmaabmaa paqaa8qacqaH7oaBcqaHbpGCaiaawIcacaGLPaaacaWGKbGaeq4UdW MaaiiFa8aadaWgaaWcbaWdbiaadQhacqGHsgIRcaaIWaaapaqabaaa k8qacaGLBbGaayzxaaGaaiilaaqaaiaadIeapaWaa0baaSqaa8qaca WG4baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaaaa kiaacYhapaWaaSbaaSqaa8qacqWFWpq+paWaaSbaaWqaa8qacaaIWa aapaqabaWcpeGaeyOKH4QaaGimaiaacYcacaWG6bGaeyOKH4QaaGim aaWdaeqaaOWdbiabg2da9iabgkHiTmaalaaapaqaa8qacaWGkbGaeu iLdq0damaaBaaaleaapeGaamiEaaWdaeqaaaGcbaWdbiaaikdacqaH apaCcqWFWpq+paWaaSbaaSqaa8qacaWGNbaapaqabaaaaOWdbmaala aapaqaa8qacqGHciITa8aabaWdbiabgkGi2kaadIhaaaWaaSaaa8aa baWdbiaadMhaa8aabaWdbiabeg8aYbaadaWabaWdaeaapeWaaSaaa8 aabaWdbiab=b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaOqaa8qa cqaHbpGCaaGaamysa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qada qadaWdaeaapeWaaSaaa8aabaWdbiabeg8aYjab=b=a5=aadaWgaaWc baWdbiaadEgaa8aabeaaaOqaa8qacaaIYaaaaaGaayjkaiaawMcaai aadUeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeWaaeWaa8aabaWd bmaalaaapaqaa8qacqaHbpGCcqWFWpq+paWaaSbaaSqaa8qacaWGNb aapaqabaaakeaapeGaaGOmaaaaaiaawIcacaGLPaaaaiaawUfaaiab gUcaRaqaaiabgUcaRmaadiaapaqaa8qadaGfWbqabSWdaeaapeGaaG imaaWdaeaapeGaeyOhIukan8aabaWdbiabgUIiYdaakmaalaaapaqa a8qacqaH9oGBpaWaaSbaaSqaa8qacaWGNbaapaqabaaakeaapeGae8 h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaaaak8qacaWGMbWaaeWa a8aabaWdbiabeU7aSjaacYcacqWFWpq+paWaaSbaaSqaa8qacaWGNb aapaqabaGcpeGaaiilaiab=b=a5=aadaWgaaWcbaWdbiaadMgaa8aa beaak8qacaGGSaGaamiAaaGaayjkaiaawMcaaiaabwgacaqG4bGaae iCaiaabckacaqGOaGaeqyVd42damaaBaaaleaapeGaam4zaaWdaeqa aOWdbiaadQhacaGGPaGaamOsa8aadaWgaaWcbaWdbiaaigdaa8aabe aak8qadaqadaWdaeaapeGaeq4UdWMaeqyWdihacaGLOaGaayzkaaGa amizaiabeU7aSjaacYhapaWaaSbaaSqaa8qacaWG6bGaeyOKH4QaaG imaaWdaeqaaaGcpeGaayzxaaGaaiOlaaaaaa@1BA5@

(10)

От переменной интегрирования λ в формуле (10) перейдем к безразмерной переменной ζ=λρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaH2oGEcqGH9aqpcqaH7oaBcqaH bpGCaaa@4419@ и введем новые обозначения:

 

 

ρ ϰ j = 1i ρ πf μ 0 σ j 1i D j , χ j =ρ ν j = ζ 2 + D j 2 , j= g,i . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiabeg8aYnrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacgaGae8h8dK=damaaBaaaleaape GaamOAaaWdaeqaaOWdbiabg2da9maabmaapaqaa8qacaaIXaGaeyOe I0IaamyAaaGaayjkaiaawMcaaiabeg8aYnaakaaapaqaa8qacqaHap aCcaWGMbGaeqiVd02damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiab eo8aZ9aadaWgaaWcbaWdbiaadQgaa8aabeaaa8qabeaakiabggMi6o aabmaapaqaa8qacaaIXaGaeyOeI0IaamyAaaGaayjkaiaawMcaaiaa dseapaWaaSbaaSqaa8qacaWGQbaapaqabaGcpeGaaiilaaqaaiabeE 8aJ9aadaWgaaWcbaWdbiaadQgaa8aabeaak8qacqGH9aqpcqaHbpGC cqaH9oGBpaWaaSbaaSqaa8qacaWGQbaapaqabaGcpeGaeyypa0ZaaO aaa8aabaWdbiabeA7a69aadaahaaWcbeqaa8qacaaIYaaaaOGaey4k aSIaamira8aadaqhaaWcbaWdbiaadQgaa8aabaWdbiaaikdaaaaabe aakiaacYcacaqGGcGaamOAaiabg2da9maadmaapaqaa8qacaWGNbGa aiilaiaadMgaaiaawUfacaGLDbaacaGGUaaaaaa@7FB0@

(11)

Нетрудно заметить, что Dj MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ это отношение расстояния от антенны до точки наблюдения к толщине скин-слоя среды. В новых переменных получаем следующее выражение для поля:

 

 

E y g | ϰ 0 0,z0 = J Δ x 2π ϰ g Z g x y ρ 3 F E , H x g | ϰ 0 0,z0 = J Δ x 2π ϰ g x y ρ 3 F H , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadweapaWaa0baaSqaa8qa caWG5baapaqaa8qadaqadaWdaeaapeGaam4zaaGaayjkaiaawMcaaa aakiaacYhapaWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbst HrhAG8KBLbacgaWdbiab=b=a5=aadaWgaaadbaWdbiaaicdaa8aabe aal8qacqGHsgIRcaaIWaGaaiilaiaadQhacqGHsgIRcaaIWaaapaqa baGcpeGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadQeacqqHuoarpa WaaSbaaSqaa8qacaWG4baapaqabaaakeaapeGaaGOmaiabec8aWjab =b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaaGcpeGaamOwa8aada WgaaWcbaWdbiaadEgaa8aabeaak8qadaWcaaWdaeaapeGaeyOaIyla paqaa8qacqGHciITcaWG4baaamaalaaapaqaa8qacaWG5baapaqaa8 qacqaHbpGCpaWaaWbaaSqabeaapeGaaG4maaaaaaGccaWGgbWdamaa BaaaleaapeGaamyraaWdaeqaaOWdbiaacYcaaeaacaWGibWdamaaDa aaleaapeGaamiEaaWdaeaapeWaaeWaa8aabaWdbiaadEgaaiaawIca caGLPaaaaaGccaGG8bWdamaaBaaaleaapeGae8h8dK=damaaBaaame aapeGaaGimaaWdaeqaaSWdbiabgkziUkaaicdacaGGSaGaamOEaiab gkziUkaaicdaa8aabeaak8qacqGH9aqpcqGHsisldaWcaaWdaeaape GaamOsaiabfs5ae9aadaWgaaWcbaWdbiaadIhaa8aabeaaaOqaa8qa caaIYaGaeqiWdaNae8h8dK=damaaBaaaleaapeGaam4zaaWdaeqaaa aak8qadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWG4baa amaalaaapaqaa8qacaWG5baapaqaa8qacqaHbpGCpaWaaWbaaSqabe aapeGaaG4maaaaaaGccaWGgbWdamaaBaaaleaapeGaamisaaWdaeqa aOWdbiaacYcaaaaa@9981@

(12)

где:

F E =1+ 0 Φ ζ exp (2ζ h/ρ)dζ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGgbWdamaaBaaaleaapeGaamyr aaWdaeqaaOWdbiabg2da9iaaigdacqGHRaWkdaGfWbqabSWdaeaape GaaGimaaWdaeaapeGaeyOhIukan8aabaWdbiabgUIiYdaakiabfA6a gnaabmaapaqaa8qacqaH2oGEaiaawIcacaGLPaaacaqGLbGaaeiEai aabchacaqGGcGaaiikaiabgkHiTiaaikdacqaH2oGEcaGGGcGaamiA aiaac+cacqaHbpGCcaGGPaGaamizaiabeA7a6jaacYcaaaa@5D02@

 

F H = 1i D g I 1 D g 1+i K 1 D g 1+i + + 0 1+i ζ 2 2 D g 2 Φ ζ exp (2ζh/ρ)dζ, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qa caWGibaapaqabaGcpeGaeyypa0ZaaeWaa8aabaWdbiaaigdacqGHsi slcaWGPbaacaGLOaGaayzkaaGaamira8aadaWgaaWcbaWdbiaadEga a8aabeaak8qacaWGjbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbm aabmaapaqaa8qadaWcaaWdaeaapeGaamira8aadaWgaaWcbaWdbiaa dEgaa8aabeaaaOqaa8qacaaIXaGaey4kaSIaamyAaaaaaiaawIcaca GLPaaacaWGlbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbmaabmaa paqaa8qadaWcaaWdaeaapeGaamira8aadaWgaaWcbaWdbiaadEgaa8 aabeaaaOqaa8qacaaIXaGaey4kaSIaamyAaaaaaiaawIcacaGLPaaa cqGHRaWkaeaacqGHRaWkdaGfWbqabSWdaeaapeGaaGimaaWdaeaape GaeyOhIukan8aabaWdbiabgUIiYdaakmaakaaapaqaa8qacaaIXaGa ey4kaSIaamyAamaalaaapaqaa8qacqaH2oGEpaWaaWbaaSqabeaape GaaGOmaaaaaOWdaeaapeGaaGOmaiaadseapaWaa0baaSqaa8qacaWG Nbaapaqaa8qacaaIYaaaaaaaaeqaaOGaaGPaVlabfA6agnaabmaapa qaa8qacqaH2oGEaiaawIcacaGLPaaacaqGLbGaaeiEaiaabchacaqG GcGaaiikaiabgkHiTiaaikdacqaH2oGEcaWGObGaai4laiabeg8aYj aacMcacaWGKbGaeqOTdONaaiilaaaaaa@7E8D@

(13)

 

Φ ζ = 2ζ ζ χ g ζ χ i ζ+ χ g ζ+ χ i ζ χ g ζ χ i exp(2ζh/ρ) J 1 ζ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqqHMoGrdaqadaWdaeaapeGaeqOT dOhacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbiaaikdacqaH2o GEdaqadaWdaeaapeGaeqOTdONaeyOeI0Iaeq4Xdm2damaaBaaaleaa peGaam4zaaWdaeqaaaGcpeGaayjkaiaawMcaamaabmaapaqaa8qacq aH2oGEcqGHsislcqaHhpWypaWaaSbaaSqaa8qacaWGPbaapaqabaaa k8qacaGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaeqOTdONaaG jcVlabgUcaRiaayIW7cqaHhpWypaWaaSbaaSqaa8qacaWGNbaapaqa baaak8qacaGLOaGaayzkaaWaaeWaa8aabaWdbiabeA7a6jaayIW7cq GHRaWkcaaMi8Uaeq4Xdm2damaaBaaaleaapeGaamyAaaWdaeqaaaGc peGaayjkaiaawMcaaiaayIW7caaMc8UamGgGgkHiTiaayIW7daqada WdaeaapeGaeqOTdONaaGjcVlabgkHiTiaayIW7cqaHhpWypaWaaSba aSqaa8qacaWGNbaapaqabaaak8qacaGLOaGaayzkaaWaaeWaa8aaba WdbiabeA7a6jaayIW7cqGHsislcaaMi8Uaeq4Xdm2damaaBaaaleaa peGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaaiGacwgacaGG4bGaai iCaiaacIcacqGHsislcaaIYaGaeqOTdONaamiAaiaayIW7caGGVaGa aGjcVlabeg8aYjaacMcaaaGaamOsa8aadaWgaaWcbaWdbiaaigdaa8 aabeaak8qadGauagWaa8aabGaua+qacWauasOTdOhacGauaAjkaiac qbOLPaaacGauakOlaaaa@9F47@

 

Формулы (12) и (13) описывают поле горизонтального заземленного диполя. Поле линейной антенны определяется суммой полей, излучаемых источниками, относящимися к антенне. В частности, если обозначим H g ρ,z MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=Tqii9aadaahaaWcbeqaa8qadGaJygWaa8 aabGaJy+qacGaJyo4zaaGaiWiMwIcacGaJyAzkaaaaaOWdaiaayIW7 peWaaeWaa8aabaWdbiabeg8aYjaacYcacaWG6baacaGLOaGaayzkaa aaaa@57F3@ магнитное поле, возбуждаемое линейной заземленной антенной в нижнем полупространстве, то H ρ,z=0 = H g ρ,z0 , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=Tqiinaabmaapaqaa8qacqaHbpGCcaGGSa GaamOEaiabg2da9iaaicdaaiaawIcacaGLPaaacqGH9aqpcqGHris5 ruqqK9MyLbYqHnhBV5giqj3BGi0BSrgaiCWacaGFibWdamaaCaaale qabaWdbmacmIzadaWdaeacmI5dbiacmI5GNbaacGaJyAjkaiacmIPL PaaaaaGcdaqadaWdaeaapeGaeqyWdiNaaiilaiaadQhacqGHsgIRca aIWaaacaGLOaGaayzkaaGaaiilaaaa@6B58@ то есть равно сумме полей источников, находящихся в точке η на антенне (рис. 1). Устремляя Δx к нулю, можно суммирование заменить на интегрирование по η:

J Δ x Jdη, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGkbGaeyiLdq0aaSbaaSqaaiaa dIhaaeqaaOGaeyOKH4QaamOsaiaadsgacqaH3oaAcaGGSaaaaa@474C@

после чего получим:

 

H ρ,z=0 = L L H x g ρ η ,z0 | J Δ x J dη, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=Tqiinaabmaapaqaa8qacqaHbpGCcaGGSa GaamOEaiabg2da9iaaicdaaiaawIcacaGLPaaacqGH9aqpdaGfWbqa bSWdaeaapeGaeyOeI0IaamitaaWdaeaapeGaamitaaqdpaqaa8qacq GHRiI8aaGccaWGibWdamaaDaaaleaapeGaamiEaaWdaeaapeWaaeWa a8aabaWdbiaadEgaaiaawIcacaGLPaaaaaGcdaqadaWdaeaapeGaeq yWdi3damaaBaaaleaapeGaeq4TdGgapaqabaGcpeGaaiilaiaadQha cqGHsgIRcaaIWaaacaGLOaGaayzkaaGaaiiFa8aadaWgaaWcbaWdbi acyc4GkbGaeyiLdq0damaaBaaameaapeGaamiEaaWdaeqaaSWdbiab gkziUkaadQeaa8aabeaak8qacaWGKbGaeq4TdGMaiqkGcYcaaaa@71C4@

где ρ η = (xη) 2 + y 2 . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaHbpGCpaWaaSbaaSqaa8qacqaH 3oaAa8aabeaak8qacqGH9aqpdaGcaaWdaeaapeGaaiikaiaadIhacq GHsislcqaH3oaAcaGGPaWdamaaCaaaleqabaWdbiaaikdaaaGccqGH RaWkcaWG5bWdamaaCaaaleqabaWdbiaaikdaaaaabeaakiaac6caaa a@4C9C@

При интегрировании необходимо подставить под интеграл выражение для поля диполя, находящегося не в центре координат, а в точке x=η,y=0. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWG4bGaeyypa0Jaeq4TdGMaaiil aiaadMhacqGH9aqpcaaIWaGaaiOlaaaa@45B1@ Произведем замену в выражении (12) для магнитного поля xxη,ρ ρ η MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWG4bGaeyOKH4QaamiEaiabgkHi TiabeE7aOjaacYcacaaMe8UaeqyWdiNaeyOKH4QaeqyWdi3damaaBa aaleaapeGaeq4TdGgapaqabaaaaa@4E11@ и /x/η. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqGHciITcaGGVaGaeyOaIyRaamiE aiabgkziUkabgkHiTiabgkGi2kaac+cacqGHciITcqaH3oaAcaGGUa aaaa@4B15@ Выполняя интегрирование по η, получим:

 

H ρ,0 = J 2π ϰ g y ρ η 3 F H | η=L η=L . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=Tqiinaabmaapaqaa8qacqaHbpGCcaGGSa GaaGimaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacaWGkbaa paqaa8qacaaIYaGaeqiWdaNae8h8dK=damaaBaaaleaapeGaam4zaa Wdaeqaaaaak8qadaWcaaWdaeaapeGaamyEaaWdaeaapeGaeqyWdi3d amaaDaaaleaapeGaeq4TdGgapaqaa8qacaaIZaaaaaaakiaadAeapa WaaSbaaSqaa8qacaWGibaapaqabaGcpeGaaiiFa8aadaqhaaWcbaWd biabeE7aOjabg2da9iabgkHiTiaadYeaa8aabaWdbiabeE7aOjabg2 da9iaadYeaaaGccaGGUaaaaa@67B5@

Аналогичные преобразования можно выполнить и для электрического поля. Опуская их, можно написать для электрического поля линейной заземленной антенны E y ρ,0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=btif9aadaWgaaWcbaWdbiaadMhaa8aabe aak8qadaqadaWdaeaapeGaeqyWdiNaaiilaiaaicdaaiaawIcacaGL Paaaaaa@4ECC@ следующее выражение:

 

E y ρ,0 = J 2π ϰ g Z g y ρ η 3 F E | η=L η=L . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvga iyaaqaaaaaaaaaWdbiab=btif9aadaWgaaWcbaWdbiaadMhaa8aabe aak8qadaqadaWdaeaapeGaeqyWdiNaaiilaiaaicdaaiaawIcacaGL PaaacqGH9aqpdaWcaaWdaeaapeGaamOsaaWdaeaapeGaaGOmaiabec 8aWjab=b=a5=aadaWgaaWcbaWdbiaadEgaa8aabeaaaaGcpeGaamOw a8aadaWgaaWcbaWdbiaadEgaa8aabeaak8qadaWcaaWdaeaapeGaam yEaaWdaeaapeGaeqyWdi3damaaDaaaleaapeGaeq4TdGgapaqaa8qa caaIZaaaaaaakiaadAeapaWaaSbaaSqaa8qacaWGfbaapaqabaGcpe GaaiiFa8aadaqhaaWcbaWdbiabeE7aOjabg2da9iabgkHiTiaadYea a8aabaWdbiabeE7aOjabg2da9iaadYeaaaGccaGGUaaaaa@6B85@

Таким образом, поверхностный импеданс зависит от импеданса плоской волны и отношения функций FE и FH и только при FE = FH равен Zg. Функция FH имеет более сложный вид по сравнению с FE, и ее поведение меняется в зависимости от Dg. Рассмотрим поведение FH при D g 1. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiablUMi=iaaigdacaGGUaaaaa@42D5@ Воспользовавшись асимптотическими представлениями для модифицированных функций Бесселя [Градштейн, 1962], получим:

 

I 1 D g 1+i K 1 D g 1+i 1+i 2 D g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGjbWdamaaBaaaleaapeGaaGym aaWdaeqaaOWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaamira8aada WgaaWcbaWdbiaadEgaa8aabeaaaOqaa8qacaaIXaGaey4kaSIaamyA aaaaaiaawIcacaGLPaaacaWGlbWdamaaBaaaleaapeGaaGymaaWdae qaaOWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaamira8aadaWgaaWc baWdbiaadEgaa8aabeaaaOqaa8qacaaIXaGaey4kaSIaamyAaaaaai aawIcacaGLPaaacqGH8iIFdaWcaaWdaeaapeGaaGymaiabgUcaRiaa dMgaa8aabaWdbiaaikdacaWGebWdamaaBaaaleaapeGaam4zaaWdae qaaaaaaaa@5600@

и, соответственно,

 

F H 1+ 0 1+i ζ 2 D g 2 Φ ζ exp (2ζh/ρ)dζ. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGgbWdamaaBaaaleaapeGaamis aaWdaeqaaOWdbiabloKi7iaaigdacqGHRaWkdaGfWbqabSWdaeaape GaaGimaaWdaeaapeGaeyOhIukan8aabaWdbiabgUIiYdaakmaakaaa paqaa8qacaaIXaGaey4kaSIaamyAamaalaaapaqaa8qacqaH2oGEpa WaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaapeGaamira8aadaqhaaWc baWdbiaadEgaa8aabaWdbiaaikdaaaaaaaqabaGccWaJawOPdy0aae Waa8aabaWdbiabeA7a6bGaayjkaiaawMcaaiaabwgacaqG4bGaaeiC aiaabckacaGGOaGaeyOeI0IaaGOmaiabeA7a6jaadIgacaGGVaGaeq yWdiNaaiykaiaadsgacqaH2oGEcGaMakOlaaaa@66FF@  (14)

Так как D g 1, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiablUMi=iaaigdacaGGSaaaaa@42D3@ то выражение (14) будет отличаться от выражения (12) для FE лишь при больших ζ D g . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaH2oGEcqWIRjYpcaWGebWdamaa BaaaleaapeGaam4zaaWdaeqaaOGaaiOlaaaa@43C7@ Выражение, стоящее под знаком интеграла, представляет собой произведение медленно меняющейся и экспоненциальной функции. В зависимости от показателя экспоненты будет изменяться вклад интегрального члена в функцию FH. Чтобы оценить вклад в интеграл области с ζ D g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaH2oGEcqWIRjYpcaWGebWdamaa BaaaleaapeGaam4zaaWdaeqaaaaa@430B@ рассмотрим экспоненту, входящую в интеграл exp (2ζh/ρ). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaqGLbGaaeiEaiaabchacaqGGcGa aiikaiabgkHiTiaaikdacqaH2oGEcaWGObGaai4laiabeg8aYjaacM cacaGGUaaaaa@4AAC@ Учитывая, что ζ D g , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaH2oGEcqWIRjYpcaWGebWdamaa BaaaleaapeGaam4zaaWdaeqaaOGaaiilaaaa@43C5@ то exp (2ζh/ρ)<exp (2 D g h/ρ). MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbiqaaWZgqaaaaaaaaaWdbiaabwgacaqG4bGaaeiCaiaa bckacaGGOaGaeyOeI0IaaGOmaiabeA7a6jaadIgacaGGVaGaeqyWdi NaaiykaiabgYda8iaabwgacaqG4bGaaeiCaiaabckacaGGOaGaeyOe I0IaaGOmaiaadseapaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGaam iAaiaac+cacqaHbpGCcaGGPaGaaiOlaaaa@588E@ Из (11) следует, что D g /ρ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiaac+cacqaHbpGCaaa@427E@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ это величина, обратная толщине скин-слоя. Следовательно, при отношении удвоенной высоты волновода к толщине скин-слоя много больше единицы имеем экспоненциально малую величину и, соответственно, вклад в интеграл, которым можно пренебречь. Таким образом, F E F H MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGgbWdamaaBaaaleaapeGaamyr aaWdaeqaaOWdbiabgIKi7kaadAeapaWaaSbaaSqaa8qacaWGibaapa qabaaaaa@438E@ при D g 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiablUMi=iaaigdaaaa@4223@ и удвоенной высоте волновода, превышающей длину скин-слоя, и, соответственно, при выполнении этих условий поверхностный импеданс будет равен импедансу плоской волны.

Перейдем теперь к случаю, когда D g <1. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabgYda8iaaigdacaGGUaaaaa@427C@ Здесь даже в отсутствии ионосферы поверхностный импеданс не совпадает с импедансом плоской волны. Рассмотрим Hx, имеющую более сложную структуру, чем Ey. Из формулы (13) при D g <1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabgYda8iaaigdaaaa@41CA@ следует, что:

 

F H = 1i 2 D g × × 1+ 2 1i D g 0 1+i ζ 2 D g 2 Φ ζ exp (2ζh/ρ)dζ . MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGceaGabeaaqaaaaaaaaaWdbiaadAeapaWaaSbaaSqaa8qa caWGibaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaaigdacqGHsi slcaWGPbaapaqaa8qacaaIYaaaaiaadseapaWaaSbaaSqaa8qacaWG NbaapaqabaGccqGHxdaTaeaacqGHxdaTpeWaamWaa8aabaWdbiaaig dacqGHRaWkdaWcaaWdaeaapeGaaGOmaaWdaeaapeWaaeWaa8aabaWd biaaigdacqGHsislcaWGPbaacaGLOaGaayzkaaGaamira8aadaWgaa WcbaWdbiaadEgaa8aabeaaaaGcpeWaaybCaeqal8aabaWdbiaaicda a8aabaWdbiabg6HiLcqdpaqaa8qacqGHRiI8aaGcdaGcaaWdaeaape GaaGymaiabgUcaRiaadMgadaWcaaWdaeaapeGaeqOTdO3damaaCaaa leqabaWdbiaaikdaaaaak8aabaWdbiaadseapaWaa0baaSqaa8qaca WGNbaapaqaa8qacaaIYaaaaaaaaeqaaOGaaGPaVlabfA6agnaabmaa paqaa8qacqaH2oGEaiaawIcacaGLPaaacaqGLbGaaeiEaiaabchaca qGGcGaaiikaiabgkHiTiaaikdacqaH2oGEcaWGObGaai4laiabeg8a YjaacMcacaWGKbGaeqOTdOhacaGLBbGaayzxaaGaaiOlaaaaaa@79AC@

Наличие множителя, превышающего единицу, перед интегралом по ζ, показывает, что относительные изменения в магнитном поле, обусловленные влиянием ионосферы, могут быть более значительными, по сравнению с изменениями в FE. На рис. 2 приведены результаты расчетов функции F H / D g MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qadaabdaWdaeaapeGaamOra8aadaWg aaWcbaWdbiaadIeaa8aabeaaaOWdbiaawEa7caGLiWoacaGGVaGaam ira8aadaWgaaWcbaWdbiaadEgaa8aabeaaaaa@45F1@ как функции параметра (частоты) для ряда значений электродинамических параметров волновода и отношения высоты волновода к расстоянию от источника. При этом проводимость Земли, используемая в расчетах σ g =5 10 5 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaHdpWCpaWaaSbaaSqaa8qacaWG NbaapaqabaGcpeGaeyypa0JaaGynaiabgwSixlaaigdacaaIWaWdam aaCaaaleqabaWdbiabgkHiTiaaiwdaaaaaaa@4881@ См/м, соответствует проводимости Земли для Кольского полуострова. Из графиков следует, что для КНЧ и более низких частот, соответствующих D g <1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabgYda8iaaigdaaaa@41CA@ имеется заметное влияние ионосферы на величину магнитного поля. В область D g >1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabg6da+iaaigdaaaa@41CE@ при рассматриваемых отношениях высоты к расстоянию до антенны зависимость от проводимости ионосферы отсутствует, что совпадает с классическими представлениями о влиянии ионосферы лишь на расстояниях, превышающих двойную высоту ионосферного волновода.

 

Рис. 2. Зависимость FHDg как функции параметра Dg для ряда значений электродинамических параметров волновода и отношения высоты волновода к расстоянию от источника: (a) – h/p = 0.8, (б) – h/p = 1.1.

 

Как уже указано выше, подобное явление наблюдается и в экспериментальных исследованиях [Терещенко, 2007]. Однако малая мощность используемого генератора низкочастотного электромагнитного поля не позволила получить хорошее отношение сигнала к шуму и сделать точные количественные оценки.

Хорошие экспериментальные данные были получены при проведении на Кольском полуострове измерений во время эксперимента FENICS-2014 в августе MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ сентябре 2014 г. [Колобов, 2015], где использовался более мощный передатчик, чем ранее применявшийся, а также более протяженная антенна. Это позволило пренебречь влиянием на принимаемый сигнал как внешних, так и внутренних шумов. В процессе измерений сигнал превышал шум на два порядка и измерялся с точностью многократно превышающей суточные вариации. На рис. 3 представлены результаты измерения магнитного поля Hx 23 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ 29 августа 2014 г. в обсерватории Ловозеро, находящейся на расстоянии около 100 км от излучающей антенны. Геомагнитные условия были спокойными в первой половине эксперимента, а 27 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ 29 августа наблюдалось повышение магнитной активности, и соответственно изменения в ионосфере. Для сопоставления рис. 2 и рис. 3 оценим величину Dg для эксперимента, проведенного на Кольском полуострове. Подставив в (17), определяющую Dg, параметры, соответствующие эксперименту: 80 км MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ высоту волновода, ρ 100 км MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ расстояние от одного из концов антенны и σ g =5 10 5 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacqaHdpWCpaWaaSbaaSqaa8qacaWG NbaapaqabaGccaaMi8+dbiabg2da9iaaiwdacWaMaAyXICTaaGPaVl aaykW7caaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislcaaI1aaa aaaa@4E47@ См/м, получим, что 1 Гц соответствует D g 1.4. MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaeXafv3ySLgzGmvETj2BSbqefm0B 1jxALjhiov2DaebbnrfifHhDYfgasaacHOaM0xg9vrFfpeea0xh9v8 qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpe pae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGacaGaamaabeqaaq aabaqbaaGcbaaeaaaaaaaaa8qacaWGebWdamaaBaaaleaapeGaam4z aaWdaeqaaOWdbiabgIKi7kaaigdacaGGUaGaaGinaiaac6caaaa@4499@

 

Рис. 3. Зависимость амплитуды магнитного поля от частоты при силе тока в передающей антенне 1 А в сеансах 23–29 августа 2014 г. (обс. Ловозеро) – компоненты Hx. Обозначения кривых: 1 – измерения 23–27.08.2014 г.; 2 – 28.08.2014 г.; 3 – 29.08.2014 г.

 

Поэтому сравнивая рис. 3 с теоретическим поведением функции FH (рис. 2), видим, что они подобны. Таким образом, при низкой проводимости подстилающей поверхности наблюдается влияние ионосферы на магнитное поле и соответственно поверхностный импеданс КНЧ и более низком диапазоне.

ВЫВОДЫ

Полученное в работе решение задачи о поле заземленной линейной антенны в трехслойной среде позволяет оценить как поведение электромагнитного поля, так и поверхностного импеданса на Земле в зависимости от электродинамических параметров и высоты плоского волновода Земля MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=nbiaaa@3A35@ ионосфера. Показано, что при отношении расстояния от точки наблюдения до излучающей антенны, превышающего величину скин-слоя Земли, необходимым условием для совпадения поверхностного импеданса электромагнитной волны и импеданса плоской волны является малость скин-слоя по сравнению с высотой волновода.

Теоретически показано и экспериментально подтверждено влияние ионосферы на расстояниях, меньших скин-слоя, при низкой проводимости подстилающей поверхности на магнитное поле и поверхностный импеданс волн КНЧ и более низкого диапазона.

Работа выполнена при поддержке Российского фонда фундаментальных исследований (проект MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqef4uz3r3BUb acfaqcLbwaqaaaaaaaaaWdbiaa=zriaaa@3A39@ 19-05-00823)

E. D. Tereshchenko

Polar Geophysical Institute

Author for correspondence.
Email: tereshchenko@gmail.com

Russian Federation, 26a, Akademgorodok street, Apatity,Murmansk region, 184209

P. E. Tereshchenko

Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Waves Propagation of the Russian Academy of Sciences; Polar Geophysical Institute

Email: tereshchenko@gmail.com

Russian Federation, IZMIRAN, Kaluzhskoye hwy, 4, Troitsk, Moscow, 108840 ;26a, Akademgorodok street, Apatity,Murmansk region, 184209 

  1. Бреховских Л.М. Волны в слоистых средах. М.: изд-во АН СССР. 1957. 502 с.
  2. Велихов Е.П., Кононов Ю.М., Шорин В.И. и др. Способ электромагнитного зондирования земной коры с использованием нормированных источников поля. Патент РФ 2093863 от 20.10.1997 г.
  3. Вешев А.В. Электропрофилирование на постоянном и переменном токе. 2 е изд., перераб. и дополн. Л.: Недра. 1980. 301 с.
  4. Градштейн И.С., Рыжик И.М. Таблицы интегралов, сумм, рядов и произведений. М.: Физматгиз. 1962. 1100 с.
  5. Жданов М.С. Геофизическая магнитная теория и ме¬тоды. М.: Научный Мир. 2012. 680 с.
  6. Тихонов А.Н. Об определении электрических характеристик глубоких слоев земной коры // Докл. АН СССР. 1950. Т. 73. № 2. С. 295–297.
  7. Ковтун А.А., Успенский Н.И. Геоэлектрика: поля естественных источников. Учебное пособие. Санкт-Петербург: изд-во СПбГУ. 2009. 171 с.
  8. Колобов В.В., Баранник М.Б., Жамалетдинов А.А. Опыт применения КНЧ-генератора Энергия-2 для электромагнитных зондирований в ходе международного эксперимента FENICS-2014 // Вестник КНЦ. 2/2015 (28). Вып. 10. С. 52–64.
  9. Терещенко Е.Д., Григорьев В.Ф., Сидоренко А.Е., Миличенко А.Н., Молоков А.В., Собчаков Л.А., Васильев А.В. О возможности квазивертикального радиозондирования ионосферы в крайне низкочастотном диапазоне // Письма в ЖТФ. 2007. Т. 85. Вып. 8. С. 471–473.
  10. Терещенко Е.Д., Терещенко П.Е. Электромагнитное поле горизонтальной линейной заводненной антенны // ЖТФ. 2017. Т. 87. Вып. 3. С. 453–457.
  11. Cagniard L. Basic theory of the magneto-telluric method of geophysical prospecting // Geo-phys. 1953. V. 18. P. 605–535.
  12. Wait J.R. Electromagnetic waves in stratified media. N.Y.: Elmsford. Pergamon press. 1970. 620 p.

Supplementary files

Supplementary Files Action
1. Fig. 1. Geometry of the problem. View (15KB) Indexing metadata
2. Fig. 2. Dependence of FHDg as a function of the parameter Dg for a number of electrodynamic parameters of the waveguide and the ratio of the height of the waveguide to the distance from the source: (a) - h / p = 0.8, (b) - h / p = 1.1. View (52KB) Indexing metadata
3. Fig. 3. The dependence of the amplitude of the magnetic field on the frequency at the current strength in the transmitting antenna 1 A in sessions 23–29 August 2014 (observ. Lovozero) - components Hx. The designations of the curves: 1 - measurements 23–27.08.2014; 2 - 08.28.2014; 3 - 29.08.2014 View (35KB) Indexing metadata

Views

Abstract - 37

PDF (Russian) - 40

PlumX

Refbacks

  • There are currently no refbacks.

Copyright (c) 2019 Russian Academy of Sciences