_{1}

In the review we obtained a strict solution of problem of the sound diffraction by an elastic spheroidal shell, located near the interface of a liquid medium with an elastic solid medium. Calculation of the scattered sound field for ideal bodies (a spheroid and an elliptical cylinder) is performed. These bodies are placed on the interface between the liquid and the ideal medium; it is shown, that the main role is played not by interaction of scatterers (real and imaginary), but by interference of their scattered fields. The spectrum of the scattered impulse signal for the body in an underwater sound channel is calculated. It is shown, that at large distances the dominante role is played by the spectral characteristic of the channel itself. Based on the method of imaginary sources and imaginary scatterers, the solution of the current study is to solve the diffraction problem of sound pulse signals at ideal (soft) prolate spheroid, which is put in the plane waveguide with the hard elastic bottom. In the work, it is proved that with such a formulation of problems eliminated, there exists possibility of using the method of normal waves because pulses are bundies of energy and can therefore only be distributed to the group velocity which is inherent in just the method of imaginary sources. Calculations made in the article showed that imaginary sources with small numbers exert the effect of total internal reflection, as the result of the reflection coefficient V by the hard elastic bottom which is complex and the real part of V is close to 1.0 which corresponds to V absolutely hard bottom. Found sequences of reflected pulses for the elastic hard bottom and the absolutely hard bottom floor confirmed this approach. In the final part of the article, on the basis of the received results, a solution (the method integral equations) is given, which is a much more complex problem of the diffraction at the elastic non-analytical scatterer, put in the plane waveguide with the hard elastic bottom.

In the series of problems on the study of the influence of media interfaces on characteristics of sound scattering by bodies, following variants are usually investigated:

a) An interaction of a scatterer with a single interface between media;

b) The scatterer in the field of interfracting modes of the underwater sound channel;

c) The finding of the total scattered field of the system of real and imaginary sources and scatterers of the plane waveguide.

The interaction of the scatterer with the interface between media is considered in the example of the problem of scattering of sound by an elastic spheroidal body located at the interface between a liquid an elastic medium [_{s}, η_{s}, φ_{s} (s = 1, 2), the first of which (s = 1) we associate with the scatterer, the second (s = 2) with the interface plane. The beginning of the Cartesian coordinate system O_{2} foci of the second spheroidal coordinate system are defined as projections respectively O_{1} and foci of the first coordinate system on the plane of the boundary Z_{2}Y_{2}, so that inter-focus distance 2h_{0} is common for both coordinate systems. The interface plane is two coordinate half-planes ( φ ′ 2 = π / 2 and φ ″ 2 = − π / 2 ) coordinate system. In order to relate this solution to the solution of the diffraction problem on the elastic spheroidal shell, we simplify the formulation of the problem and assume that the wave vector k of the incident wave is in the plane X_{1}Z_{1} (and correspondly X_{2}Z_{2}), φ o s = 0 ∘ (see

Now along with the potential Φ 1 ( 1 ) of the wave, scattered by the shell, will appear the potential Φ 1 ( 2 ) frothe elastic half-space:

Φ 1 ( 2 ) = 2 ∑ m = 0 ∞ ∑ n ≥ m ∞ K m , n S ¯ m , n ( C 1 , η 2 ) R m , n ( 1 ) ( C 1 , ξ 2 ) cos m φ 2 . (1)

The potential Φ 1 ( 2 ) decomposes by radial functions of the first kind, this is due to the fact, that foci of the second coordinate system lie in the plane of the interface, physically this means that the interacting of scatterers are distorted by fields of two plane waves: an incident and reflected by an interface. Expansions of potentials of the shell are accompanied by expansions of potentials of an elastic half-space:

U 2 ( 2 ) = 2 ∑ m = 1 ∞ ∑ n ≥ m ∞ M m , n S ¯ m , n ( C t 2 , η 2 ) R m , n ( 1 ) ( C t 2 , ξ 2 ) sin m φ 2 ; (2)

W 2 ( 2 ) = 2 ∑ m = 0 ∞ ∑ n ≥ m ∞ N m , n S ¯ m , n ( C t 2 , η 2 ) R m , n ( 1 ) ( C t 2 , ξ 2 ) cos m φ 2 , (3)

Φ 2 ( 2 ) = 2 ∑ m = 0 ∞ ∑ n ≥ m ∞ L m , n S ¯ m , n ( C l 2 , η 2 ) R m , n ( 1 ) ( C l 2 , ξ 2 ) cos m φ 2 ; (4)

where C l 2 and C t 2 ―are wave dimensions of longitudinal transverse waves respectively.

The potential of the incident wave Φ_{0} in two coordinate systems has a form [

Φ 0 = 2 ∑ m = 0 ∞ ∑ n ≥ m ∞ i − n ε m S ¯ m , n ( C 1 , η 0 ) S ¯ m , n ( C 1 , η s ) R m , n ( 1 ) ( C 1 , ξ s ) cos m φ s , ( s = 1 , 2 ) . (5)

The potential of a diffracted field Φ Σ = Φ 0 + Φ 1 ( 1 ) + Φ 2 ( 2 ) obeys simultaneously to boundary conditions on a surface of a shell and on a planar interface between a liquid and an elastic medium. To boundary conditions on a surface of a shell are supplemented by conditions on an inter-face between a liquid-elastic medium:

λ 0 k 2 ( Φ 0 + Φ 1 ( 1 ) + Φ 1 ( 2 ) ) = λ 2 k l 2 2 Φ 2 ( 2 ) + 2 μ 2 u φ φ ( 2 ) | φ = π / 2 ; − π / 2 ; (6)

( h φ ( 2 ) / h ξ ( 2 ) ) ( ∂ / ∂ ξ ) ( u φ ( 2 ) / h φ ( 2 ) ) + ( h ξ ( 2 ) / h φ ( 2 ) ) ( ∂ / ∂ φ ) ( u ξ ( 2 ) / h ξ ( 2 ) ) | φ = π / 2 ; − π / 2 = 0 ; (7)

( h φ ( 2 ) / h ξ ( 2 ) ) ( ∂ / ∂ ξ ) ( u φ ( 2 ) / h φ ( 2 ) ) + ( h ξ ( 2 ) / h φ ( 2 ) ) ( ∂ / ∂ φ ) ( u ξ ( 2 ) / h ξ ( 2 ) ) | φ = π / 2 ; − π / 2 = 0 ; (8)

− ( h φ ) − 1 ( ∂ / ∂ φ ) ( Φ 0 + Φ 1 ( 1 ) + Φ 1 ( 2 ) ) = ( h φ ( 2 ) ) − 1 ( ∂ Φ 2 / ∂ φ ) + ( h ξ ( 2 ) h η ( 2 ) ) − 1 [ ( ∂ / ∂ ξ ) ( h η ( 2 ) ψ η ( 2 ) ) − ( ∂ / ∂ η ) ( h ξ ( 2 ) ψ ξ ( 2 ) ) ] | φ = π / 2 ; − π / 2 , (9)

where λ_{2} and μ_{2}―Lame coefficients of an elastic half-space; k l ( 2 ) ―a wavt number of a longitudinal wave of an elastic half-space.

In our formulation of a problem because of a panty of a solution respect a plane XZ boundary conditions for φ = − π / 2 completely repeat conditions for φ = + π / 2 , that do not provide any additional information in this case. When substituting potential expansions into boundary conditions for a shell and a planar interface when an expansion coefficients is used an addition theorem for wave spheroidal functions [

R p , q ( 1 ) , ( 3 ) ( C j , ξ j ) S ¯ p , q ( C j , η j ) exp ( i p φ j ) = ∑ n = 0 ∞ ∑ m = − n n R m , n ( 1 ) ( C s , ξ s ) × Q ¯ m , n , p , q ( 1 ) , ( 3 ) ( C j , C s ; l ; θ j s ) S ¯ m , n ( C s , η s ) exp ( i p φ s ) , (10)

where

Q ¯ m , n , p , q ( 1 ) , ( 3 ) = 2 ( − i ) n − q ∑ r = 0 , 1 ∞ ' ∑ t = 0 , 1 ∞ ' d r p q ( C j ) d t m n ( C s ) ∑ σ = | r + p − t − m | r + p + t + m ( − i ) σ × b ˜ σ ( r + p , p , t + m , m ) Z σ ( 1 ) , ( 3 ) ( k l ) P ˜ σ p − m ( θ j s ) exp [ i ( p − m ) φ j s ] ;

θ_{js}―polar angle of a point O_{s}―a beginning of a s-th coordinate system in an i-th system (_{j} and O_{s}; d r p q ( C j ) and d t m n ( C s ) ―сoefficients of expansions f functions S ¯ p , q ( C j , η j ) and S ¯ m , n ( C s , η s ) by functions P ˜ r p ( η j ) and P ˜ t m ( η s ) , which up to a constant factor coincide with normalized adjoint Legendre functions; Z σ ( 1 ) = j σ ( k l ) ―spherical Bessel functions; Z σ ( 3 ) = h σ ( 1 ) ( k l ) ―spherical Hanrtl functions of a first kind; coefficients b ˜ σ ( r + p , p , t + m , m ) are obtained from coefficients b σ ( r + p , p , t + m , m ) [

A strict solution can be obtained for another orientation of a spheroidal shell with respect to a plane bourder, namely, under a condition of perpendicularity of a rotation axis of a shell to a plane interface between media (_{s} (S = 1, 2) of waves scattered by spheroids are chosen in a form of expansions (taking into account an axial symmetry) [

Φ s ( ξ s , η s ) = ∑ n = 0 ∞ B 0 , n s S ¯ 0 , n ( C s , η s ) R 0 , n ( 3 ) ( C s , ξ s ) . (11)

Since spheroids (real and imaginary) are ideally soft, then on their surfaces ( ξ 0 = ξ 01 = ξ 02 ) is satisfied a homogeneous Dirichlet:

Φ 0 + ∑ s = 1 2 Φ s = 0 | ξ = ξ 0 ; s = 1 , 2 . (12)

A potential of a falling plane wave is given by a decomposition:

Φ 0 ( ξ s , η s ) = 2 ∑ n = 0 ∞ i − n S ¯ 0 , n ( C s , η s ) R 0 , n ( 1 ) ( C s , ξ s ) S ¯ 0 , n ( C s , 1 ) , s = 1 , 2. (13)

Unknown coefficients B 0 , n s of expansions (11) are sought from an infinite system of equations―boundary conditions (12) [

B 0 , n s + ∑ q = 0 ∞ B 0 , q t R 0 , n ( 1 ) ( C s , ξ 0 s ) [ R 0 , n ( 3 ) ( C s , ξ 0 s ) ] − 1 Q ¯ 0 n 0 q ( 3 ) ( C t , C s ; l ; θ t s ) = − 2 i − n S ¯ 0 n ( C s , 1 ) R 0 , n ( 1 ) ( C s , ξ 0 s ) [ R 0 , n ( 3 ) ( C s , ξ 0 s ) ] − 1 , s = 1 , 2 ; t = 1 , 2 ; s ≠ t , (14)

where l―a distance between centers of coordinate systems O_{1} and O_{2} (

θ 12 = 0 , θ 21 = π

Q ¯ 0 n 0 q ( 3 ) ( C t , C s ; l ; θ t s ) = 2 i − n + q ∑ r = 0 , 1 ∞ ' ∑ j = 0 , 1 ∞ ' d r 0 q ( C t ) d j 0 n ( C s ) ∑ σ = | r − j | r + j i − σ b σ ( r , 0 , j , 0 ) h 0 ( 1 ) ( k l ) P σ ( cos θ t s ) ;

b σ ( r , 0 , j , 0 ) = ( r j 00 | σ O ) 2 .

For a regularization of a system (14) with respect to unknown coefficients B 0 , n s we introduce new unknown X 0 , n s from a ratio [

B 0 , n s = R 0 , n ( 1 ) ( C s , ξ 0 s ) X 0 , n s . (15)

As a result, an infinite system (14) for unknown B 0 , n s is reduced to another infinite system of relatively new unknown X 0 , n s [

X 0 , n s + ∑ q = 0 ∞ X 0 , q t R 0 , q ( 1 ) ( C t , ξ 0 t ) [ R 0 , n ( 3 ) ( C s , ξ 0 s ) ] − 1 Q ¯ 0 n 0 q ( 3 ) ( C t , C s ; l ; θ t s ) = − 2 i − n S ¯ 0 , n ( C s , 1 ) [ R 0 , n ( 3 ) ( C s , ξ 0 s ) ] − 1 . (16)

Further we find by a trunkaction method a solution of a regular system (16). Initially we calculated angular scattering functions D s ( θ s ) of two interacting spheroids distorting a field of a plane monochromatic wave. A

moduluses of angular characteristics | D s ( θ s ) | of tnteracting spheroids (a curve 1 refers to a first spheroid, a curve 2―to a second spheroid). A curve 3 depicts in another scale | D ( θ ) | of a single soft spheroid in an infinite medium. A scale had to be changed so that a curve 3 did not merge with curves 1 and 2. A curve 4 characterizes a modulus | D Σ ( θ 1 ) | of a total angular characteristic in coordinates of a first spheroid (

| D Σ ( θ 1 ) | = | D 1 ( θ 1 ) + D 2 ( θ 1 ) exp ( i k l cos θ 1 ) | .

Calculations were carried out at: C 1 = C 2 = 10.0 , ξ 01 = ξ 02 = ξ 0 = 1.005 , a semi-focus distance h 01 = h 02 = 1 м , l = 8 h 0 . An analysis of curves prestnted in _{1}, C_{2}, l) an interaction of scatterers turned out to be smail, because of this curves 1, 2, 3 are so close to each other. A main role is played by interference effects (especially in a shadow region), so a curve 4 stands out sharply (again in a shadow region) agatnst a background of other curves. In a second stage (based on a calculation of a scattered field of two spheroids) were calculated angular characteristics D Σ ( θ ) of a soft spheroid ( ξ 0 = 1.005 ; C = 10 ), located at a distance l 0 = 4 h 0 = 4 m from an interface between liquid and an ideal medium. Results of calculations | D Σ ( θ ) | are shown in

A strict solution also has a problem of scattering of sound by a spheroidal half-body placed on an interface between a liquid and an ideal medium. With a mirror image of a scatterer and a source with respect to a plane boundary, we obtain a spheroidal scatterer located in a field of two sources (real and imaginary).

A phase of a wave potential from an imaginary source on a plane interface coincides with a phase of a potential of an incident wave in this plane in a liquid borders on an ideally rigid medium and phases of these waves differ at a boundary by 180˚ if a liquid borders on an ideally soft medium. In

In _{0} of oblate ideal hemispheroids located on interface between media, where n under D ( θ ; φ ) we mean a total angular characteristic (from real and iimaginary sources). Curves of _{0} of hard 5 and soft 6 oblate spheroids ( ξ 0 = 0.1005 ) in an infinite liquid medium. It is easy to see that if a hemispheroid and an ideal semibounded medium and same (hard

or soft, curves 1 and 2) that is, a half body represents only a violation of a shape of a boundary hides this unevenness and σ_{0} grows slowly and at smail wave dimensions close to zero and vise versa, for different materials of a hemispheroid and a semibouded ideal medium cross-sections σ_{0} are much larger (curves 3 and 4) in an entre investigated range of wave dimensions. While acurves 5 and 6 tend asymptotically to a value of geometric acoustics, remaining curves increase indefinitely. Mathematical and physical explanations for this phenomenon were given in comments to

Let us turn to passive characteristics (scattering indicatrix) of hemispheroids located on an interface between a liquid and an ideal medium. We imagine for this that on a fairly large distance from a boundary along a line LM a combined system (source-receiver) moves, while we are interested in a reflected signal to a point finding a combined system. A movement of a system is so slow that a Doppler effect can be ignored. Two orientations of half-bodies are possible which admit are depicted in

ideally sofy medium (a curve 2, ξ 0 = 0.1005 ; С = 10 ). A orientation of hemibodies corresponds to

We will pass from stationary (harmonic) irradiation to nonstationary radiation in a form of sound pulses with a rectangular envelope and monochromatic

filling. As before, we will consider interfaces of media of three types: a liquid―an elastic bottom; a liquid―an ideally soft medium (Dirichlet condition); a liquid―an ideally hard medium (Neumann condition).

If a scatterer (oblate hard spheroid) is placed at an interface between media (a liquid―an elastic bottom) at an observation point, come first an impulse of a mirror reflection from a scatterer.

We orient a hard prolate hemispheroid in such a way that it’s a major semiaxis will be in a plane of an interface between media. We calculate mirror-reflected pulses Ψ Σ ( t ′ ) at an angle of incidence θ 1 = 60 ∘ for two variants: 1) a hard prolate hemispheroid on a boundary with a hard medium; 2) a hard prolate

hemispheroid on a boundary with a soft medium, In

We see in

In

A picture of a reflection of sound by a spheroidal body located at a boundary of a liquid―an elastic bottom, we supplement it with angular scattering characteristics R ( θ , φ ) for a stationary sound signal with a fixed frequency. In

A study of an effect of two boundaries on a field of a spheroidal scatterer will begin withan ideal spheroid placed in an underwater sound channel with non-reflecting boundaries [_{0} of such a waveguide we place a point source of a rulsed signal and at a horizontal distance r_{ }from it and at a depth z_{2}―a spheroidal scatterer (

S 2 ( ω ) = ρ 0 2 ( r 1 ) − 1 ∑ m = 1 M ∑ n = 1 N m P m P n , m D n m ( ω ) exp [ − i ( κ m r 1 + κ n m r 1 − π / 2 ) ] , (17)

where: P m = p m ( ρ 0 ) − 1 φ m ( z 2 ) φ m ( z 0 ) ; p_{m}―a mode excitation coefficient m; φ m ( z 2 ) ―a intrinsic wave-guide function determined by boundary conditions;

ρ_{0}―density at source and receiver depth; P n m = ( 1 / ρ 2 ) φ n ( z 2 ) φ n ( z 0 ) ; ρ_{2}― density at a depth of a center of a scatterer; D n m ( ω ) ―space-transfer function of body scattering for a m-th mode of a source and a n-th mode of a scatterer; κ_{m} and κ_{nm}―horizontal components of wave numbers of modes of an incident and scattered waves res-pectively; M―a largest admissible source mode; N―a largest admissible scatterer mode for m-th mode of a source.

A dependence of a velocity of sound on a coordinate z for a symmetric waveguide (see

c ( z ) = ( p | z | + q ) 1 / 2 .

We will find a spectrum S 2 ( ω ) at a combined point of a source and receiver for a ideal soft scatterer in a forn of a prolate spheroid with a coordinate of an outher surface ξ 0 = 1.005 . We place a source (receiver) on an axis of a symmetric waveguide ( z 2 = z 0 = 0 ), a major axis of a spheroid being perpendicular to an axis Z. An interfocus spheroid distance 2h_{0} is assumed equal to 9.7 m. A source generates a pulse signal with a duration τ 0 = 0.05 s at a frequency f 0 = 400 Hz ( С = 8.0 ). Space-transfer function D n m ( ω ) is determined by a frequency and angular (in a plane XOZ) characteristics of a spheroid scattering. With a chosen velocity profile of a sound (see

even with a maximum wave size ( C = 10.0 ) used in calculations D ( 90 ∘ ; φ ) = | D ( 90 ∘ ; φ ) | exp [ i ψ ( 90 ∘ ; φ ) ] (an angular characteristic) is practically non-directional within angles ϕ = 0 ÷ 16 ∘ (see

S 2 ( ω ) = D ( ω ) ρ 0 2 ( r 1 ) − 1 ∑ m = 1 M ∑ n = 1 M P m P n m exp [ − i ( κ m r + κ n m r − π / 2 ) ] . (18)

We turn to a familiar problem of the diffraction of pulses on spheroial bodies I the plane waveguide [

It is known to [

As a result of simple calculations with the help of [^{st} and 4^{th} pulsews of ^{nd}, 3^{rd} and 5^{th} pulses in ^{nd}, 3^{rd} and 5^{th} pulses (see

A similar pattern is observed for anisotropic bottom, such as silicon, in wich quasi-longitudinal wave velocity of about 8300 m/s and quasi-transverse wave velocity of about 5700 m/s, with the secjnd quasi-transverse wave do not occur because of the problem statement [

Based on the obtained solution, consider a more general problem of the diffraction of the pulsed sound signal on an elastic scatterer as a finite cylindrical shell, supplemented with two hemispherical shells (

The first stage will solve the problem of the diffraction of a harmonic wave on a such shell. The density of the material of the shell is ρ_{1}, the Lame’s coefficients―λ and μ. The shell was filled in the internal liquid medium with the density ρ_{2} and the sound velocity C_{3} and it was placad in the external liquid medium with the density ρ_{0} and the sound velocity C_{0}. At the shell falls the plane harmonic wave with pressure p_{i} under the angle Θ_{0} and with the wave vector k .

As was shown in [

u ( r ) = ∬ S { t ( r ′ ) G ( r ′ ; r ) − u ( r ′ ) [ n ^ ′ Σ ( r ′ ; r ) ] } d S ( r ′ ) , r ∈ V , (19)

where t ( r ′ ) = n ^ ′ T ( r ′ ) is the stress vector; n ^ ′ ≡ n ^ ′ ( r ′ ) = n ′ ( r ′ ) is the single vector of the external along the relation to S normal; T ( r ′ ) is the stress tensor of the isotropic material; G ( r ′ ; r ) is the displacement Green’s tensor; Σ ( r ′ ; r ) is the stress Green’s tensor; if r concerns to the point of the surface S, in the left part of the Equation (19) will stand u ( r ′ ) / 2 .

The displacement vector u ( r ) , the stress tensor T ( r ) , the displacement Green’s tensor (

The displacement vector u ( r ) , the stress tensor T ( r ) , the displacement Green’s tensor G ( r ′ ; r ) and the stress Green’s tensor Σ ( r ′ ; r ) were connected between themselves by the following correlations [

T ( r ) = λ I ∇ u ( r ) + μ ( ∇ u + u ∇ ) , (20)

where I = I L + I T ; I L = ( ∇ ∇ ) / ∇ 2 ; I L ⋅ I T = 0 ; I T = − [ ∇ ( ∇ I ) ] / ∇ 2 , I L and I T are the longitudinal and transverse single tensors for the Hamilton’s operator ∇ ;

∑ ( r ′ ; r ) = λ I ∇ G ( r ′ ; r ) + μ [ ∇ G ( r ′ ; r ) + G ( r ′ ; r ) ∇ ] ; (21)

G ( r ′ ; r ) = ( 1 / 4 π ρ t ω 2 ) { k 2 I g ( k 2 | r ′ − r | ) + ∇ ′ [ g ( k 1 | r ′ − r | ) − g ( k 2 | r ′ − r | ) ] ∇ } , (22)

where k_{1} and k_{2} are the wave numbers of the longitudinal and transwerse waves in the material of the shell; g ( k 2 | r ′ − r | ) = exp ( i k 2 | r ′ − r | ) / 4 π | r ′ − r | is the Green’s function.

The second integral equation presents the Kirchhoff integral for the diffracted pressure p Σ ( P 1 ) in the external medium:

C ( P 1 ) p Σ ( P 1 ) = − ∬ S a { p Σ ( Q ) ( ∂ / ∂ n ′ ) [ exp ( i k r 0 / r 0 ) ] − [ exp ( i k r 0 / r 0 ) ] ρ 0 ω 2 ( u n ′ ) } d S a + 4 π p i ( P 1 ) , (23)

where p Σ ( P 1 ) = p i ( P 1 ) + p s ( P 1 ) ; p s ( P 1 ) is the scattered pressure in the point P_{1}; C(P_{1}) is the numerical coefficient, equal 2π, if P 1 ∈ S a and 4π, if P_{1} 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 p 2 ( M 1 ) in the internal liquid medium in the point M_{1} is got the third integral equation:

C ( M 1 ) p 2 ( M 1 ) = ∬ S b { p 2 ( Q ′ ) ( ∂ / ∂ n ′ ) [ exp ( i k r 3 ) / r 3 ] − [ exp ( i k r 3 ) / r 3 ] ρ 0 ω 2 ( u n ′ ) } d S b , (24)

where Q ′ is the point of the internal surface of the shell;

C ( M 1 ) = { 4 π , if M 1 out S b ; 2 π , if M 1 ∈ S b ;

S_{b} is the internal surface of the shell.

To the integral Equations (19), (23) and (24) 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:

τ i | S a = 0 ; τ i | S b = 0 ; i = 1 , 2 ; (25)

2) the normal stress σ n ′ at the external surface of the shell is equally the diffracted pressure p_{Σ}, but at the internal surface is equally the pressure p_{2}

σ n ′ | S a = p Σ ; σ n ′ | S b = p 2 ; (26)

In the conformity with the conditions (7) and (8) the stress vector t ( r ′ ) in the Equation (1) is equal:

t ( r ′ ) = p Σ n ′ | S a ; t ( r ′ ) = p 2 n ′ | S b ; (27)

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

u n ′ = ( 1 / ρ 0 ω 2 ) ( ∂ p Σ / ∂ n ′ ) | S a ; u n ′ = ( 1 / ρ 2 ω 2 ) ( ∂ p 2 / ∂ n ′ ) | S b . } (28)

The substitution of integral Equations (19), (23) and (24) in the boundary conditions gives the system of equations in terms of unknown functions p_{Σ}, p_{2} and the components of the displacement vector u 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 [

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 [

Thus calculated reflection characteristics of the harmonic signal with frequency ν can determine the spectral reflectance function S S ( 2 π ν ) and it can help be applying a Fourier transform we obtain a temporary function of the reflected pulse Ψ S ( t ′ ) [

Ψ S ( t ′ ) = 1 π Re ∫ 0 ∞ S S ( 2 π ν ) e + i 2 π ν t d ( 2 π ν ) (29)

Similarly using spectral reflectance characteristics of elastic bodies of spheroidal form [

In the first part of the review we investigate an interaction of a scatterer and an interface between media; it is shown that a main role in this is played by interference effects. The second part of the review is devoted to a study of a spektrum of a scattered field of an ideal prolate spheroid placed in an underwater sound channel with non-reflecting boundaries. The third part of the review demonstrates the effect of a bottom structure on a series of pulses, reflected from a spheroidal body located in a plane waveguide.

As a result of the research we can draw three conclusions:

1) During studying propagation and diffraction of pulse signals in a plane waveguide, one needs to use the method of imaginary sources as pulses like bundles of energy spread to any directions (including and along the axis of the waveguide) with the group velocity which does not exceed the sound velocity, namely the group velocity based the method of imaginary sources;

2) Replacing the hard elastic bottom on the absolutely hard bottom is acceptable to those sources (real and imaginary) from which waves in the fall to the hard elastic bottom try total internal reflection.

3) We have adopted the model of image sources and image scatterer is gute acceptable (due to internal reflection), at least, for first five calculated reflected pulses in a plane wave-guide with hard elastic bottom.

Kleshchev, A. (2017) Scattering of Stationary and Non- Stationary Sound by Ideal and Elastic Scatterers, Placed near Interface of Media, in Underwater Sound Channel and in Plane Waveguide. Open Access Library Journal, 4: e4141. https://doi.org/10.4236/oalib.1104141