Diffraction of Sound Impulses on Isotropic Bodies of Spherical Form (Strict Solution)

Considerable quantity of scientific works is devoted to sound scattering on bodies of spherical form [1 – 4]. The problem of diffraction of sound impulses with the harmonic infill on elastic spherical shell and elastic isotropic sphere is being reviewed in this paper. At the basis of the dynamic theory of elasticity are found form and duration of sound impulses scattered from elastic spherical shell and elastic isotropic sphere (which in general case can be with random radius, but in this paper both of sphere and shell (outer radius) got 2 4,0 R = m). The scattering of frequency-modulated signal (impulse) can also be found in a similar way, using the basis of the dynamic theory of elasticity, but this paper contains computed results only for signal (impulse) with the harmonic infill.


Introduction
Considerable quantity of scientific works is devoted to sound scattering on bodies of spherical form [1 -4]. The problem of diffraction of sound impulses with the harmonic infill on elastic spherical shell and elastic isotropic sphere is being reviewed in this paper. At the basis of the dynamic theory of elasticity are found form and duration of sound impulses scattered from elastic spherical shell and elastic isotropic sphere (which in general case can be with random radius, but in this paper both of sphere and shell (outer radius) got 2 4, 0 R = m). The scattering of frequency-modulated signal (impulse) can also be found in a similar way, using the basis of the dynamic theory of elasticity, but this paper contains computed results only for signal (impulse) with the harmonic infill.

The First Part of the Article Investigates the Diffraction Problem of the Impulse with the Harmonic Infill on Elastic Spherical Shell
Let's assume that monochrome sound wave falls on elastic spherical shell (Fig. 1). To find form and duration of reflected impulses it is necessary to know angular characteristics of the scattering, which can be wrote like [5]: where: m a -unknown expansion coefficients, ( ( )) m P Cos θ -Legendre polynomial, k -wave number. In contrast to well-known solutions [6,7], which are based on theory of the thin shells, in our case we will use strict approach, which based on dynamic theory of elasticity [5,8], to find the numerical values of angular characteristics of the scattering. Threshold value of m take as 10 m = . Unknown expansion coefficients are determined from physical boundary conditions preset at two surfaces of the shell, which are consists in: a) continuity of normal displacements on the border fluid and elastic spherical shell (outer radius); b) equality of normal stress in spherical shell (on its outer border) to diffracted pressure p Σ in flow media; c) absence of tangent stresses on inner and outer borders of the spherical shell. Mathematically, boundary conditions look like (they acquire such form after deduction from the Lame equation with using Helmholtz theorem and generalized Hooke's law, which is invariant to the coordinate system): [ ] Legendre polynomials are orthogonal to each other on angle interval θ from 0  to 180  , i.e.: By substituting expressions (6) -(9) to the boundary conditions (2) -(5) with the use of property of orthogonality of Legendre polynomials (as well as shifted Legendre polynomials and derivatives from them) we will receive for each mode m its system of equations, from which finds unknown expansion coefficients m a . Using the Cramer's rule unknown expansion coefficients will be found like: where ∆ -system determinant, ' ∆ -determinant, in which the column of unknown coefficient replaced by the column with free terms.

The Second Part of the Article Investigates the Diffraction Problem of the Impulse with the Harmonic Infill on Elastic Isotropic Sphere
In case of diffraction on the elastic isotropic sphere the system of equation becomes simpler; cause of there is no more boundary condition for inner border like it was in a case of shell.
So, let's assume that monochrome sound wave falls on elastic isotropic sphere (Fig. 11). To find the numerical values of angular characteristics of the scattering for the elastic isotropic sphere, we'll also use the dynamic theory of elasticity. The radius of sphere is 2 R . To find form and duration of reflected impulses it is necessary to know angular characteristics of the scattering, which can be wrote like: where m a -unknown expansion coefficients, ( ( )) m P Cos θ -Legendre polynomial, k -wave number.   By substituting expressions (18) -(21) to the boundary conditions (15) -(17) with the use of property of orthogonality of Legendre polynomials (as well as shifted Legendre polynomials and derivatives from them) we will receive for each mode m its system of equations, from which finds unknown expansion coefficients m a . As in case with shell, unknown expansion coefficients can be found by using the Cramer's rule.
As an example let's see which system of equation appears for 3 m = : from condition (15): As you can see, the system of equations became simpler then it was in case with the shell, it lost all of the terms containing the Neumann functions. Numerical values appears from integrals, which, for their turn, appears by using property of orthogonality of Legendre polynomials (as well as shifted Legendre polynomials and derivatives from them) and changes along with change of m .