1. Introduction

The method of the integral equations is using in problems of the sound diffraction on the ideal and elastic bodies non – analytical form for the calculation of the angular characteristics of the sound scattering on the equal footing with the different: the method of the boundary elements and the method of the T – matrixes.

The main target of the given research is appearing the determination of the angular distributions of the scattered pressure, that is important for the practical hydrolocation.

2. The Ideal Scatterer of the Non – analytical Form

Figure 1. The non – analytical smooth scatterer in the form of the cylinder with the semi – spheres

We discern the ideal non – analytical scatterer in the form of the terminal cylinder with the semi – spheres on its ends (see fig. 1).

The pressure in the scattered wave in the point observation is equally[ 1, 2 ]:

(1)

where Q – the point of the surface of the scatterer.

For the ideal soft scatterer ( the homogeneous Dirichlet condition) the scattered pressure is equally:

for the ideal hard scatterer ( the homogeneous Neimann’s condition):

Then (1) for the Dirichlet condition at the surface is accepting the appearance:

(2)

for the Neimann’s condition:

(3)

We are combining the point of the observation P with the point of the surface scatterer . Тhen (2) is transform in the Fredholm equation of the first kind for the function ( with the account of the Dirichlet condition):

(4)

We are differentiate the left and right parts of the equation (3) with and are use the boundary Neimann’s condition, then we are get integro – differential equation for the funktion :

(5)

This problem can solve with the help of the Fredholm equation of the second kind. For the Dirichlet condition we are get the equation for the function , where the sign «+» is mean the outside domain to S . For the Dirichlet condition th pressure p_Σ is continuous by the crossing through the boundary S: and , while the function (since inside – the vacuum) , but outside - .

The function we are find from the solution of the non – homogeneous Fredholm equation of the second kind[ 1, 2 ]:

(6)

The integral to the left of (6) must understand in the sense of the main meaning.

With the help Ψ we are find and the scattered pressure in the any point of the medium :

(7)

For the Neimann’s condition we are bring the function - the solution of the Fredholm equation of the second kind[ 1, 2 ]:

The scattered pressure in the point we are express through the function Ф:

(8)

The scattered pressure can find either with the help of the integral (2) (for the Fredholm equation of the first kind), or with the help of the equation (7) (for the Fredholm equation of the second kind).

The surface S is consist of S₂ and of the surfaces S₁ и S₃ (see fig. 1).

For the calculation of the integrals (2) and (7) on surface we are choose the grid of the nodal points. We are mark the nodal point through , where the index m is point out the room on the element, while the index n – on guide. The integrals (2) and (7) we are substitute the numerical quadratures

(9)

(10)

where

and - the values of the functions Ф₁, Ф₂, and Ф in the nodal points Q_mn; A_n, D_m – the weight coefficients; М – the quantity of the nodal points on the element, N – the quantity of the nodal points on the guide.

The unknown functions and we are find from the solution of the Fredholm equation of the first kind (4) and of the Fredholm equation of the second kind (6).

Transformed in (4) to the numerical quadratures, we are get the algebraice system of the equations of the order relatively F^mn – of the meanings in nodal points:

(11)

If we are taking advantage of the symmetry of the problem relatively 0f the plane XOZ, that from the system of the equations we transform to the system MN/2 (N – the even number), but by - to the system MN/4 (M and N - the even numbers). Easily see from fig. 1, that by means of the function dependn’t on the angle , that is for everybody n = 1,2,…,N, and we get the system M – order. In correspondence with the integral of Kirchhoff the special point isolates in the equations (4). In the equation (6) the special point appearsn’t isolated, therefore the integral in the left part of the equation is considering in the meaning of the main value. For the calculation of this integral was using the following method. In the expression under the integral sign is appropriating the term of the simple analytical form, containing the singularity, which is integrating analytically. The staying part of the integral has the isolated special point and the corresponding integral is calculating by the help of the numerical method. The integral is showing in the kind of the three integrals:

(12)

The integral has not of the special point by

The integral has the special point of the lesser kind and also is integrating by the numerical method.

We cross to the calculation of the integral and consider firstly the integral:

(13)

where – the radius – vector of the point, which belongn’t to the surface S.

From the Green’s formula is getting:

where V – the volume, limited on the surface S.

We know, that:

where - delta function, satisfying to the relation:

Therefore the integral I in (13) is equal:

if the point is finding inside of the surface S and I=0, if is finding outside of the surface S. In the our version the point is finding on the surface S and the integral in the sense of the main meaning is equal:

(14)

Substituted (14) and (12) in the integral Fredholm equation of the second kind (6), we get:

(15)

If we introduce the designation (fig. 1):

that the equation (15) has the form:

(16)

For the chosen soft scatterer in the form of the cylinder with the semi – spheres were found the values by the following parameters: a=1m, l=1m, ka=1, θ₀ = .

At fig. 2-a and 2-b are present for the chosen parameters by θ₀ = ( fig. 2-a) and θ₀ = ( fig. 2-b) (the curve 1 corresponds to method of the T – matrixes, but the curve 2 – to the method of the integral equations).

3. The elastic Scatterer of the Non – analytical Form

The scattered field for the elastic bodies of the non – analytical form was determined either with the help of the method of the T – matrixes[3] or on the basis of the method of the boundary elements[4].

We are going to spread the method of the integral equations, used in[3] and[2] for the ideal non – analytical scatterers, on the elastic shell of the non – analytical form[5].

Figure 2. 2-a. The modulus of the angular distribution of the scattered pressure. Fig. 2-b. The modulus of the angular distribution of the scattered pressure

In the quality of the such scatterer we are going to consider the terminal isotropic elastic cylindrical shell with the semi – spheres on its ends (see fig. 3). The density of the material of the shell is ρ₁, the Lame’s coefficients - λ and μ. The shell was filled in the internal liquid medium with the density ρ₂ and the sound velocity C₃ and it was placad in the external liquid medium with the density ρ₀ and the sound velocity C₀. At the shell falls the plane harmonic wave with pressure p_i under the angle Θ₀ and with the wave vector .

Figure 3. The elastic shell in the form of the terminal cylinder with the semi – spheres

As was shown in [ 2, 4, 6, 3 ], the initial equation is integral equation , having the sense of the generalized Huygen’s principle, for the displacement vector of the elastic shell:

(12)

where is the stress vector; is the single vector of the external along the relation to S normal; is the stress tensor of the isotropic material; is the displacement Green’s tensor; is the stress Green’s tensor; if concerns to the point of the surface S, in the left part of the equation (12) will stand .

The displacement vector , the stress tensor, the displacement Green’s tensor and the stress Green’s tensor were connected between them selves by the following correlations[ 2, 7, 6, 3 ]:

(13)

where ; ; ;, and are the longitudinal and transverse single tensors for the Hamilton’s operator ;

(14)

(15)

where k₁ and k₂ are the wave numbers of the longitudinal and transwerse waves in the material of the shell; is the Green’s function.

The second integral equation presents the Kirchhoff integral for the diffracted pressure in the external medium[ 2, 7 ]:

(16)

where ; is the scattered pressure in the point P₁; C(P₁) is the numerical coefficient, equal 2π, if and 4π, if P₁ out S_a; S_a is the external surface of the shell; Q is the point of the external surface of the shell.

For the pressure in the internal liquid medium in the point M₁ is got the third integral equation:

(17)

where is the point of the inte;rnal surface of the shell;

S_b is the internal surface of the shell.

To the integral equations (12), (16) and (17) are added the boundary conditions on the external (S_a) and internal (S_b) surfaces of the shell:

1) – at the both surfaces of the shell the tangent stresses are equally null:

(18)

2) – the normal stress at the external surface of the shell is equally the diffracted pressure p_Σ, but at the internal surface is equally the pressure p₂

(19)

In the conformity with the conditions (18) and (19) the stress vector in the equation (12) is equal:

(20)

3) – the continuity of the normal component of the displacement at the both boundaries of the shell:

(21)

The substitution of the integral equations (17), (12) and (18) in the boundary conditions (18) – (20) gives the system of equations in terms of unknown functions p_Σ, p₂ and the components of the displacement vector at the both surfaces of the shell. To obtain numerical solution of this system the integral equations are replaced the quadrature formulas and the grid of the nodal points is chosen at both surfaces of the shell as well as it has be done for the ideal non – analytical scatterers[2, 3].

For choosing boundary conditions we will have the integrals of the two types: the integrals with the isolated special point and the integrals which are considered of the sence of the principal meaning. The method of the calculation of the second types was described in[2].

4. Conclusions

The method of integral equations is appearing effective by the study of the sound diffrac-tion on the ideal and elastic scatterers of the non – analytical forms. With the help of the this method and the integral equations Fredholm of the first and second kinds are calculating the angular distributions of the scattered pressure for the ideal non – analytical forms. At the basis of theory of the dynamics of the elastic bodies is solving the problem of the sound diffraction on the elastic scatterers of the non – analytical forms.

ACKNOWLEDGMENTS

This work was supported as part of research under State Contract no P242 of April 21, 2010, within the Federal Target Program “Scientific and scientific – pedagogical personnel of innovative Russia for the years 2009 – 2013”.