ИСТИНА

The important aspect in the study of the structure of the interiors s of planets is the question of the presence and state of cores inside them. While for the Earth this task was solved long ago, the question of whether the core of the Moon is in a liquid or solid state up to the present is debatable up to present. If the core of the Moon is liquid, then the velocity of longitudinal waves in it should be lower than in the surrounding mantle. If the core is solid, then most likely, the velocity of longitudinal waves in it is higher than in the mantle. Numerical calculations of the wave field allow us to identify the criteria for drawing conclusions about the state of the lunar core. In this report we consid the problem of constructing a stable analytic solution for wave fields in a layered sphere of arbitrary size. After the Fourier- Legendre transformations, the statement of the problem reduces to the consideration of a two-parameter family of boundary-value problems for ordinary differential equations. The solution of the latter problem in each spherical layer is in the form of a linear combination of Bessel functions [1]. The unknown coefficients are determined from known conjugation conditions on the boundary of spherical layers. As a result, a matrix system of linear equations is obtained for their determination. For a small number of layers, its solution can be obtained in explicit form. Since Bessel functions of different types tend to zero and infinity rapidly, uncertainty arises in the solution. And the more the radius of the sphere in relative values (wavelengths), the faster it arises. In this situation, computer calculations become unstable. To construct a stable solution, it is proposed to use the classic asymptotic of Bessel functions [2]. In the article [3] it is shown that the classical asymptotic behavior of Bessel functions gives an error in the solution. To construct the solution, we use the new asymptotes of cylindrical functions obtained in the article [3]. This gives a stable analytical solution for wave fields in an inhomogeneous sphere of arbitrary size.