Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1788910.31857/S0869-5652489184-88Research ArticleOccurrence of “precursors” of PKP-waves in the layered radial-symmetric EarthFatyanovA. G.burmin@ifz.ruBurminV. Yu.burmin@ifz.ruInstitute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of the Russian Academy of SciencesInstitute of the Earth Physics of the Russian Academy of Sciences10112019489184882511201925112019Copyright © 2019, Russian academy of sciences2019<p>It is generally accepted that <em>PKP</em>‑waves precursors, which are observed on a real data ahead of <em>PKP</em>‑waves, are explained by scattering on small-scale inhomogeneities in the lower mantle. In this paper, a stable analytical solution (without interference) was obtained for the wave field of longitudinal waves in a layered (discrete) ball of planetary size. The calculations of the total wave field, rays and travel-time curves of longitudinal waves for the spherical model of the Earth AK135 with a carrier frequency of 1 hertz are presented. The analytical solution showed that at angles smaller than 145 degrees ahead of the <em>PKP</em>‑waves, low-amplitude waves appear, with a higher frequency of about 1,3 hertz. Indeed, these high-frequency oscillations have the form characteristic for waves scattered at a certain object. The ray pattern and the travel-time graph show that these high-frequency oscillations are due to exclusively to the spherical geometry of the Earth. This could be explained by the interference of refracted and reflected longitudinal waves in the bottom of a discrete outer core. This field propagates even further towards smaller angles due to the interference of diffraction waves.</p>stable analytical solutioninhomogeneous spherediscrete Earth Model AK135wave fieldrefracted seismic wavestravel-timespherical geometry of the Earthhigh-frequency “precursors”устойчивое аналитическое решениенеоднородный шардискретная модель Земли АК135волновое полеотражённые и преломлённые волныгодографсферическая симметрия ЗемлиинтерференцияPKP волнывысокочастотные “предвестники”[Wen L., Helmberger D. V. // Science. 1998. V. 279. Iss. 5357. P. 1701-1703.][Hedlin M. A.H., Shearer P. M., Earle P. S. // Nature. 1997. V. 387 (6629). P. 145-150.][Gutenberg B. // Eos Trans. AGU. 1957. V. 38. P. 750-753.][Bullen K. E., Burke-Gaffney T.N. // Geophys J. Int. 1958. V. 1. P. 9-17.][Bolt B. A. // Nature. 1962. V. 196. P. 122-124.][Sacks I. S., Saa G. // Year Book Carnegie Inst. Washington. 1969. V. 69. P. 419-426.][Бурмин В. Ю. // Физика Земли. 2004. № 6. С. 24-41.][Бурмин В. Ю., Бойко А. Н. // ДАН. 2017. Т. 472. № 2. С. 197-200.][Kennett B. L.N., Engdahl E. R., Buland R. // Geophys. J. Int. 1995. № 122. P. 108-124.][Тихонов А. Н., Самарский А. А. Уравнения математической физики. М.: Наука, 1977. 736 с.][Fatyanov A. G., Burmin V.Yu. // Planetary and Space Science. 2018. V. 153. P. 100-106.][Фатьянов А. Г. // ДАН. 1990. Т. 310. № 2. С. 323-327.][Korneev V. A., Johnson L. R. // Geophys. J. Int. 1993. № 115. P. 230-250.][Фатьянов А. Г. // Матем. заметки СВФУ. 2016. Т. 23. № 3. С. 91-103.][Seiji T., Ando Kazuto, Takayuki M., Daniel P., Komatitsch D., Tromp Jeroen. // Int. J. High Performance Computing Appl. 2016. V. 30 (4). P. 411-422.]