The Debye’S Potentials Utilization in the Three-Dimensional Problems of the Radiation and Propagation of the Elastic Waves

Are studied internal and external tasks of radiat ion of a sound by the elastic bodies, excit ing by the harmonic point source, imitating turbulent pulsation of a flow of a liquid. The angular characteristics of radiat ion of a hollow spheroidal shell are calcu lated. The characteristic equations of the axial three-d imensional flexural waves in the hollow cylindrical shell and cylindrical bar are received with the help of Debye’s potentials. The phase velocities of the various forms of these waves for shells and for cylindrical bar are calculated.


Introduction
At study of the three-dimensional characteristics of radiation of a sound by the elastic shells and bars excit ing by the turbulent pulsation of a flo w of a liquid and calculation of the phase velocities of the axial flexural waves in such bodies at such excitation it is necessary to use an artificial way of division variable in Helmholt z vector equation for a vector function A  -to present her through Debye's potentials or "such as Debye's" potentials.

The Radiation of a Sound by the Cylindrical and Spheroidal Shells, Exciting by the Turbulent Pulsation of a Flow of a Liquid
Firstly we will consider a physical model of rad iation of a sound by a cylindrical pipe (an internal task) and spheroidal shell (an external task), which is raised by turbulent pulsations of a liquid flow.
In a monograph [1], devoted to the studying of the hydrodynamic sources of a sound, it is noted that for the range of problems regarding the radiations of a sound effected by turbulent pulsations of a liquid flow, the calculation, based on the concentrated force, caused by this pulsation, gives rise to certain interest.. The similar physical model is used in a present article. Such approach is based on the earlier obtained results in the course of authors' research of the three-dimensional problems of diffraction and radiation of a sound by the elastic bodies of cylindrical and spheroidal forms [2 -5].
Let's turn to an internal task for the consideration of: the harmonic point source The external radius of the cylindrical p ipe a r 0 = , on the outside region to the shell ( ) a r 2 = is the vacuum. In order to except the presence of the source (a peculiarity) on the boundary surface, we use a reciprocity theorem (6) and trade places a source Q and a point of observation P. As a result the formulat ion of the task will be the fo llowing: we will search potential ( )  [5,7]. The potential i Φ of the harmonic point source in a point Q is defined by the series [8 -10]: where: where: κ-the wave nu mber of the lateral wave in the shell's material; A vector function A  is are described by Debye's potentials U and V [12 -16]: where: R  -is a radius-vector of view point. The efficiency of such representation becomes obvious if we take into account, that the functions U and V submit to the Helmholt z scalar equation, divided in circular cy lindrical coordinates: The other representations for a vector function A  in the Cartesian and cylindrical coordinate systems are given in [17 -19], but it is [5] in a spheroidal system.
The potentials Φ, U and V are also expanded in serieses by eigen-functions of the Helmholt z scalar equation [16,5,11]: -are unknown coefficients and functions correspondingly and they are determined by the next boundary conditions at the external and internal surfaces of an elastic shell: 1) a normal co mponent of a displacement vector r U is continuous at an internal shell's boundary; 2) a sound pressure in a fluid is equal to the normal strain in a shell at an internal boundary; 3) a normal strain in a shell at an external boundary is equal to zero; 4) the tangent strains at the shell's boundaries are equal to zero. An analytic form o f the enumerated boundary conditions are a following representation: A substitution of the series (1), (2), (10) - (12) in the boundary conditions (13) - (17) results in an infinite system of the equations to define the unknown coefficients and functions ( ) ; A a n γ , As the trigonometrical functions cos(nφ) and sin(nφ) are opthogonal, an infinite system breaks out into seven equation with fixed index n for finding the seven combinations of the unknown coefficients and functions.
A product ( ) γ A a n for a potential of a diffused wave ( ) , A a n ∆ ∆ γ ′ = (18) where: Δ -is determinant of a system, but ∆′ -is minor; The rest of elements of the rows and columnes of a determinants Δ и ∆′ can be taken out [5]. Except fo r the first column elements Δ and ∆′ -identical.
An influence of a turbulent pulsation at a pro late spheroidal shell is considered as an external problem. At figure 2 is show hollow spheroidal shell, by streamline flow of a liquid. The points А, В, С mark the possible positions of a point source, imitating turbulent pulsation. With help [2 -4], can be calculated the angular characteristics of a radiat ion of a spheroidal shell under an influence of a turbulent pulsation.   A dipole character of a radiation of a turbulent pulsation [1] can foresee an introduction of a second source (with an other sign), disposed at a small fro m a first source, and to calculate in a point of observation total pressure fro m two sources. By an use of a reciprocity theorem a transference of a turbulent pulsation Q together with a flow of a fluid along of an internal surface of a shell (figure 1) is substituted for a transference of a point of observation P parallel to a boundary in a opposite direction.

The Phase Velocities of Three-Dimensional Flexural Waves in Cylindrical Shells and Bars
Further we shall proceed to consideration of phase velocities of one of types of waves existing in cylindrical shells and bars: axial three-dimensional flexural waves. Wave of a similar type were in detail investigated in [5, 15, 20 -28]. The characteristic equation for wave numbers of three-dimensional flexural waves of form m in the isotropic cylindrical shell it turns out by equating to zero of the determinant ∆ of six order [5,15,23,25,27,28 figure 7 the phase velocities of first three forms of flexu ral waves in the steel cylindrical bar, received with the help of "such as Debye's" potentials, are represented. [19,28].
The the phase velocities of axissymmetrical fle xu ral waves (longitudinal and torsional in bars, longitudinal, torsional and flexural in shells) are submitted in [5,24,26,27,28]. As to anisotropic of shells and bars, the jobs are devoted to phase velocities of elastic waves in them [18,29,30].
On known phase velocities the co mponents of a vector of displacement of an elastic body in any point with the help of results of job can be calculated [31].

Conclusions
The decision of a three-d imensional task o f radiation o f a sound by a cylindrical and spheroidal shells, which is raised by turbulent pulsations of a liquid flo w is received. The characteristics of radiation by a spheroidal shell under action of such sources are calculated.
The characteristic equations for wave numbers of the three-dimensional axial flexu ral waves, raised in an cylindrical shell and bar are received. The dispersive curves of phase velocities of the various forms of these waves for steel and aluminiu m shells of various thickness and for steel cylindrical bar are submitted.