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Kinematics of Wave Fields in a Sphere

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Abstract

A stable analytical solution is obtained for wave fields of seismic waves in a planetary-sized sphere using the new asymptotic behavior of the Bessel functions. This allows one to obtain wave fields with high detail. The new asymptotic has a clear physical meaning in terms of inhomogeneous waves in a sphere. An analytical solution to the problem for the propagation of waves in a sphere is obtained using the potentials of longitudinal and transverse waves. The classical representation of the total field in terms of potentials is known. In this work, it is taken in a form that allows one to immediately reduce the vector equations of motion (in the spectral region) to the classical Bessel equations. This makes it easier to find an analytical solution. As a result of analytical calculations for a homogeneous globe of the Earth, the following effect was revealed. At small distances (in degrees), the time of the first arrival of P waves for the vector equation of the theory of elasticity and the scalar wave equation differs. Moreover, in both cases, the same speed of longitudinal waves is taken. The P wave for the scalar and vector equations propagates along the line segment connecting the points of excitation and reception (minimum propagation path). However, the first arrival of the P wave for the vector equation comes earlier than the first arrival of the P wave for the scalar equation. That is, there is a difference in the apparent kinematics of the first arrivals. This phenomenon for the sphere must be taken into account in seismological studies. As the distance increases, the first arrival for the vector equation coincides in time with the first arrival for the scalar equation.

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Funding

This work was carried out according to State Task of the Schmidt Institute of Physics of the Earth, Russian Academy of Sciences (project no. 0144-2019-0011) and the Institute of Computational Mathematics and Mathematical Geophysics of the Siberian Branch of the Russian Academy of Sciences (project no. 0251-2021-0004).

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Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    A. G. Fatyanov

  2. Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, 123242, Moscow, Russia

    V. Yu. Burmin

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  1. A. G. Fatyanov
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  2. V. Yu. Burmin
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Correspondence to V. Yu. Burmin.

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Fatyanov, A.G., Burmin, V.Y. Kinematics of Wave Fields in a Sphere. Izv. Atmos. Ocean. Phys. 57, 739–745 (2021). https://doi.org/10.1134/S0001433821070021

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  • DOI: https://doi.org/10.1134/S0001433821070021

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