1. Introduction

At study of the three-dimensional characteristics of radiation of a sound by the elastic shells and bars exciting by the turbulent pulsation of a flow 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 Helmholtz vector equation for a vector function - to present her through Debye’s potentials or “such as Debye’s” potentials.

2. 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 radiation 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 of frequency ω, that imitate turbulent pulsation, is found on the interior of the shell , but the point of observation Р with the cylindrical coordinates , , is situated inside of a liquid flow (figure 1):

Figure 1. The isotropic cylindrical shell, exciting from within by a point harmonic source

The external radius of the cylindrical pipe , on the outside region to the shell 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 formulation of the task will be the following: we will search potential in point Q of an internal border, resulting of the dispersion by pipe of a sound from dot source with potential placed in a point P, basing on[5, 7]. The potential of the harmonic point source in a point Q is defined by the series[8 - 10]:

(1)

where: ; - is the sound velocity; γ and - are the axial and radial components of a wave vector correspondingly; (figure 1);

By analogy with in a point Q, a potential of the diffused wave shell can be written as[7, 5, 11]:

(2)

where and - are unknown coefficients and functions of γ correspondingly and they are determined by the boundary conditions.

The cylindrical components of a displacement vector can be written by way of potential Ф and the cylindrical components of a function [12-14]:

(3)

The cylindrical components of the function can be described by way of Debye’s potentials U and V[5, 12 - 16]:

(4)

(5)

(6)

where: κ–the wave number of the lateral wave in the shell’s material;

A vector function is are described by Debye’s potentials U and V[12 - 16]:

(7)

where: - 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 Helmholtz scalar equation, divided in circular cylindrical coordinates:

(8)

(9)

The other representations for a vector function 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 Helmholtz scalar equation[16, 5, 11]:

(10)

(11)

(12)

где: ; - is a longitudinal wave number; - is a longitudinal wave velocity in the shell’s material; - Neiman’s cylindrical; ; ,, и,, - 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 component of a displacement vector 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 of the enumerated boundary conditions are a following representation:

(13)

(14)

(15)

(16)

(17)

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 , , , , , , . 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 for a potential of a diffused wave is calculated by the Cramer rule on a basis of a ratio of the who determinants of the seven degree:

(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 for the first column elements Δ and - identical.

An influence of a turbulent pulsation at a prolate 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 radiation of a spheroidal shell under an influence of a turbulent pulsation.

Figure 2. The isotropic prolate spheroidal shell, exciting on the outside by a point harmonic source

At figures 3 and 4 are shown the modulus of the angular characteristics of a radiation this shell by an excitement in a point В (figure 3) and point А (figure 4). is a wave dimension of a body, - is a half of an interfocus distance (figure 2). A figure 3 corresponds С=8,1, but a figure 4 – С=5,5.

Figure 3. The modulus of an angular characteristic of a radiation hollow spheroidal shell, exciting on the exciting on the outside by a point harmonic source in a point В; С=8,1

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 from a first source, and to calculate in a point of observation total pressure from 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.

Figure 4. The modulus of an angular characteristic of a radiation of a steel of a steel hollow spheroidal shell, outside by a point harmonic source in a point A; С=5,5

3. 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]:

(19)

where : а₁₁ = - (λ + 2μ) J_m¹¹(h¹a) + λ[a^-2m²J_m(h¹a) - a^-1J_m¹(h¹a)+ k² J_m(h¹a)];

а₁₂ = - (λ + 2μ) N_m¹¹(h¹a) + λ[a^-2m²N_m(h¹a) - a^-1N_m¹(h¹a)+ k² N_m(h¹a)];

а ₁₃ = (λ + 2μ){-iχm²[ J_m(χ¹a) - J_m¹(χ¹a)] + iχ J_m¹¹(χ¹a) + iχk²[J_m(χ¹a) + a J_m¹(χ¹a)]} + λ{ iχa^-2 J_m(χ¹a)( m² + k²) - iχa^-1 J_m¹(χ¹a) ( m² + k²) - 2iχ k² J_m(χ¹a) - iχ J_m¹¹(χ¹a)};

а ₁₄ = (λ + 2μ){-iχm²[ N_m(χ¹a) - N_m¹(χ¹a)] + iχ N_m¹¹(χ¹a) + iχk²[N_m(χ¹a) + a N_m¹(χ¹a)]} + λ{ iχa^-2 N_m(χ¹a)( m² + k²) - iχa^-1 N_m¹(χ¹a) ( m² + k²) - - 2iχ k² N_m(χ¹a) - iχ N_m¹¹(χ¹a)};

а ₁₅ = 2(λ + 2μ)( a^-2-1)ikm J_m(χ¹a); а ₁₆ = 2(λ + 2μ)( a^-2-1)ikm N_m(χ¹a);

а₂₁ = - (λ + 2μ) J_m¹¹(h¹b) + λ[b^-2m²J_m(h¹b) - b^-1J_m¹(h¹b)+ k² J_m(h¹b)];

а₂₂ = - (λ + 2μ) N_m¹¹(h¹b) + λ[b^-2m²N_m(h¹b) - b^-1N_m¹(h¹b)+ k² N_m(h¹b)];

а ₂₃ = (λ + 2μ){-iχm²[ J_m(χ¹b) - J_m¹(χ¹b)] + iχ J_m¹¹(χ¹b) + iχk²[J_m(χ¹b) + b J_m¹(χ¹b)]} + λ{ iχb^-2 J_m(χ¹b)( m² + k²) - iχb^-1 J_m¹(χ¹b) ( m² + k²) - 2iχ k² J_m(χ¹b) - iχ J_m¹¹(χ¹b)};

а ₂₄ = (λ + 2μ){-iχm²[ N_m(χ¹b) - N_m¹(χ¹b)] + iχ N_m¹¹(χ¹b) + iχk²[N_m(χ¹b) + b N_m¹(χ¹b)]} + λ{ iχb^-2 N_m(χ¹b)( m² + k²) - iχb^-1 N_m¹(χ¹b) ( m² + k²) - 2iχ k² N_m(χ¹b) - iχ N_m¹¹(χ¹b)};

а ₂₅ = 2(λ + 2μ)( b^-2-1)ikm J_m(χ¹b); а ₂₆ = 2(λ + 2μ)( b^-2-1)ikm N_m(χ¹b);

а ₃₁ = 2ma^-1[J_m¹(h¹a) - a^-1J_m(h¹a)];а ₃₂ = 2ma^-1[N_m¹(h¹a) - a^-1N_m(h¹a)];

а ₃₃ = iχm{ J_m(χ¹a)[ a^-2 (2- m²) - k²] - J_m¹¹(χ¹a) - a^-1 J_m¹(χ¹a) };

а ₃₄ = iχm{ N_m(χ¹a)[ a^-2 (2- m²) - k²] - N_m¹¹(χ¹a) - a^-1 N_m¹(χ¹a) };

a₃₅ = ikJ_m¹(χ¹a)aχ² ; a₃₆ = ikN_m¹(χ¹a)aχ² ;

а ₄₁ = 2mb^-1[J_m¹(h¹b) - b^-1J_m(h¹b)]; а ₄₂ = 2mb^-1[N_m¹(h¹b) - b^-1N_m(h¹b)];

а ₄₃ = iχm{ J_m(χ¹b)[ b^-2 (2- m²) - k²] - J_m¹¹(χ¹b) - b^-1 J_m¹(χ¹b) };

а ₄₄ = iχm{ N_m(χ¹b)[ b^-2 (2- m²) - k²] - N_m¹¹(χ¹b) - b^-1 N_m¹(χ¹b) };

a₄₅ = ikJ_m¹(χ¹b)bχ² ; a₄₆ = ikN_m¹(χ¹b)bχ² ;

а ₅₁ = -2ik J_m¹(h¹a); а ₅₂ = -2ik N_m¹(h¹a);

а ₅₃ = - kχ[ J_m(χ¹a)(m² + a^-1k²) + 5 J_m¹(χ¹a) + aJ_m¹¹(χ¹a)];

а ₅₄ = - kχ[ N_m(χ¹a)(m² + a^-1k²) + 5 N_m¹(χ¹a) + aN_m¹¹(χ¹a)];

а ₅₅ = χ²m[a^-1J_m(χ¹a) - J_m¹(χ¹a)]; а ₅₆ = χ²m[a^-1N_m(χ¹a) - N_m¹(χ¹a)];

а ₆₁ = -2ik J_m¹(h¹b); а ₆₂ = -2ik N_m¹(h¹b);

а ₆₃ = - kχ[J_m(χ¹b)(m² + b^-1k²) + 5 J_m¹(χ¹b) + bJ_m¹¹(χ¹b)];

а ₆₄ = - kχ[ N_m(χ¹b)(m² + b^-1k²) + 5 N_m¹(χ¹b) + bN_m¹¹(χ¹b)];

а ₆₅ = χ²m[b^-1J_m(χ¹b) - J_m¹(χ¹b)]; а ₆₆ = χ²m[b^-1N_m(χ¹b) - N_m¹(χ¹b)];

In the figures 5 and 6 the results of calculations of phase velocities of three-dimensional flexural waves for steel and aluminium shells are represented. The following designations one can find : the curves 1, 2 corresponds m=1 (zero mode); the curves 3, 4- m = 2; the curves 5, 6 - m = 3; the curves 7, 8- m = 4; The external radius of shells and was accepted equal 1,0 m. С₁ – the velocity of longitudinal wave ; С_R - the velocity of Relay‘s wave; ; internal radius b = 0,99 м corresponds to a thin shell, b = 0,8 м – thick shell.

The characteristic equation for wave numbers κ of three-dimensional flexural waves in the isotropic cylindrical bar of radius а =1,0, received with the help of Debye’s potentials, looks like[22, 26]:

(20)

In the figure 7 the phase velocities of first three forms of flexural 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 flexural 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 components of a vector of displacement of an elastic body in any point with the help of results of job can be calculated[31].

Figure 5. The phase velocities of three-dimensional flexural waves in steel cylindrical shells

Figure 6. The phase velocities of three- dimensional flexural waves in aluminium cylindrical shells

Figure 7. The phase velocities of first three forms of flexural waves in the steel cylindrical bar

4. Conclusions

The decision of a three-dimensional task of radiation of a sound by a cylindrical and spheroidal shells, which is raised by turbulent pulsations of a liquid flow 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 flexural 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 aluminium shells of various thickness and for steel cylindrical bar are submitted.