Acoustic Waves Part 4 ppt

Research of the Scattering of Non-linearly Interacting Plane Acoustic Waves by an Elongated Spheroid 79 wave Ω=− 12 ω ω , the summation frequency wave 12 ω ω + , and the second harmonic waves 1 2 ω , 2 2 ω . The wave equation (5) is solved by the method of successive approximations. In the first approximation, the solution is represented by the expression (4) for the total acoustic pressure of the primary field )1( p . To determine solution in the second approximation )2( p , the right-hand side of equation (5) should feature four frequency components: second harmonics of the incident waves ( 1 2 ω , 2 2 ω ) and ( 21 ω ω + , Ω = − 12 ω ω ). The expression for the volume density of secondary waves sources at the difference frequency Ω is: ∑∑∑∑ ∞ = ∞ ≥ ∞ = ∞ ≥ − +−+Ω ⎢ ⎢ ⎣ ⎡ +Ω Ω = 0 0201 0 0201 0 4 0 2 2 2 mml mlml mml mlml mlthkDhkBthkBhkB c Q )cos()()(cos)()( ϕπ ρ ε ⎥ ⎥ ⎦ ⎤ Ω+−+Ω+ ∑∑∑∑ ∞ = ∞ ≥ ∞ = ∞ ≥ 0 0201 0 0102 2 mml mlml mml mlml thkDhkDlmthkDhkB cos)()()cos()()( πϕ . (6) To solve the inhomogeneous wave equation (5) with the right-hand side given by equation (6) in the second approximation, we seek the solution in the complex form .)).()(exp( )()( сctiPp ++Ω= −− δ 22 2 1 . (7) Substitution of the expression (7) into the inhomogeneous wave equation (5) gives the inhomogeneous Helmholtz equation: ),,( )()( ϕηξ −−−− −=+∇ qPkP 2222 , (8) where − k is the wave number of the difference frequency Ω, and ⎢ ⎢ ⎣ ⎡ +Ω Ω = ∑∑ ∞ = ∞ ≥ − 0 0201 0 4 0 2 2 mml mlml tihkBhkB c q )exp()()(),,( ρ ε ϕηξ [] ∑∑ ∞ = ∞ ≥ +−+Ω+ 0 0201 2 mml mlml mltihkDhkB )(exp)()( ϕπ [] ∑∑ ∞ = ∞ ≥ +−+Ω+ 0 0102 2 mml mlml lmtihkDhkB )(exp)()( πϕ ⎥ ⎥ ⎦ ⎤ Ω+ ∑∑ ∞ = ∞ ≥0 0201 mml mlml tihkDhkD )exp()()(. The solution to the inhomogeneous Helmholts equation (8) has the form of a volume integral of the product of the Green function with the density of the secondary wave sources [Novikov et al., 1987] [Lyamshev & Sakov, 1992]: Acoustic Waves 80 ∫ −− = V dddhhhrGqP '''''')( ''' )(),,(),,( ϕηξϕηξϕηξ ϕηξ 1 2 , (9) where )( 1 rG is the Green function, 1 r is the distance between the current point of the volume ),,( '''' ϕηξ M and the observation point ),,( ϕηξ M (Fig.4), and ' ξ h , ' η h , ' ϕ h are the scale factors [Corn & Corn, 1968]: 1 2 22 0 − − = ' '' ' ξ ηξ ξ hh , 2 22 0 1 ' '' ' η ηξ η − − = hh , ))(( '' ' 22 0 11 ηξ ϕ −−= hh . In the far field r r < < ' , the Green function is determined by the asymptotic expression ξηηξηηξξ 0 2 2 000111 11 hhhhikrrikrG ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −−−−−≈−= −− ))((exp)exp()( '''' . The integration in equation (9) is performed over the volume V occupied by the second wave sources and bounded in the spheroidal coordinates by the relations S ξξξ ≤≤ ' 0 , 11 ≤≤− ' η , πϕ 20 ≤≤ ' . This volume has the form of a spheroidal layer of the medium, stretching from the spheroid’s surface to the non-linear interaction boundary (Fig.4). An external spheroid with coordinate S ξ appears to be the boundary of this area. Coordinate S ξ is defined by the size of the non-linear interaction area between the initial high-frequency waves. This size is inversely proportional to the coefficient of viscous sound attention associated with the corresponding pumping frequency. Beyond this area, the initial waves are assumed to attenuate linearly. After the integration with respect to coordinates ' ϕ and ' η (considering the high-frequency approximation), equation (9) takes the form =+++= −−−− − ),,(),,(),,(),,(),,( )()()()( )( ϕηξϕηξϕηξϕηξϕηξ 2 4 2 3 2 2 2 1 2 PPPPP ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −= ∫∫ − − − − SS d hk TdhkT hk C ξ ξ ξ ξ ξ ξ ηξ ξηξξ η 00 0 0 0 1 ' ' ' ''' )sin( )sin( , (10) where ξρ ξεπ 0 4 0 0 22 0 8 c hikh C )exp( − − −Ω = , [] ⎢ ⎢ ⎣ ⎡ +−+= ∑∑∑∑ ∞ = ∞ = ∞ ≥ ∞ ≥ 00 02010201 2 mmml mlml ml mlml mlihkDhkBhkBhkBT )(exp)()()()( ϕπ [] ⎥ ⎥ ⎦ ⎤ +−+ ∑∑∑∑ ∞ = ∞ ≥ ∞ = ∞ ≥ 0 0201 0 0102 2 mml mlml mml mlml hkDhkDlmihkDhkB )()()(exp)()( πϕ (from here on, the time factor )exp( ti Ω is omitted). Research of the Scattering of Non-linearly Interacting Plane Acoustic Waves by an Elongated Spheroid 81 The expression (10) for the total acoustic pressure of the difference-frequency wave ),,( )( ϕηξ 2 − P consists of four spatial components. The first component ),,( )( ϕηξ 2 1 − P corresponds to the part of the acoustic pressure of the difference-frequency wave, that is formed in the spheroidal layer of the non-linear interaction area by the incident high- frequency plane waves 1 ω and 2 ω . The second component ),,( )( ϕηξ 2 2 − P describes the interaction of the incident plane wave of frequency 1 ω with the scattered spheroidal wave of frequency 2 ω . The third component ),,( )( ϕηξ 2 3 − P corresponds to the interaction of the scattered plane wave of frequency 2 ω with the scattered spheroidal wave of 1 ω . The fourth component ),,( )( ϕηξ 2 4 − P characterises the interaction of two scattered spheroidal waves with frequencies 1 ω and 2 ω . 4. Results To obtain the final expression of the total acoustic pressure of the difference-frequency wave ),,( )( ϕηξ 2 − P , consider the first spatial component ),,( )( ϕηξ 2 1 − P from equation (10), which characterises the non-linear interaction between incident plane waves of highfrequency: ⎢ ⎢ ⎢ ⎣ ⎡ −= ∫ ∑∑ ∞ = ∞ ≥ − − − − S mml mlml dhkhkBhkB hk C P ξ ξ ξηξξ η ϕηξ 0 0 00201 0 2 1 ''' )( )sin()()(),,( ⎥ ⎥ ⎥ ⎦ ⎤ − ∫ ∑∑ ∞ = ∞ ≥ − S mml mlml d hk hkBhkB ξ ξ ξ ξ ηξ 0 0 0 0201 ' ' ' )sin( )()( . (11) It should be noted that this is the only component that gives no information about the scatterer. The boundaries of the integration layer are directly defined by the elongated spheroid shape. Using representation of the plane wave in the spheroidal coordinate system and substituting )( 0 hkB nml , the expression (11) takes the form [] [] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ −−−= ∫∫ − −−− − − − SS d hk hikdhkhik hk C P ξ ξ ξ ξ ξ ξ ηξ ηξξηξξηξ η ϕηξ 00 0 000 0 2 1 ' ' ' ''''' )( )sin( exp)sin(exp),,( . (12) After the final integration with respect to the coordinate ' ξ , the expression for the first component (12) has the form )()()()()( ),,( 2 14 2 13 2 12 2 11 2 1 −−−−− +++= PPPPP ϕηξ , (13) where [][][] 000000 0 2 0 2 2 1211 2 ξηηξξηηξ ηηη )(exp)(exp )( )( , ∓∓ ∓ ∓ hikhik hk C P SS −− − − −− −≈ , [][][] 00000 2 1413 2 ξηηξηη )(Ei)(Ei )( , ∓∓∓ hikhik i C P S −− − −− −+−−≈ , and ∫ = dx x ax ax )exp( )Ei( is the integral exponential function. Acoustic Waves 82 From the expression (13) for the first component ),,( )( ϕηξ 2 1 − P of the total acoustic pressure of the difference-frequency wave, it follows that the scattering diagram of this component is determined by the function )( ηη ± 0 1 . This function depends on the coordinate 0 η or, the polar coordinate system, equivalent to the angle of incidence 0 θ of the highfrequency plane waves. The scattering diagram of the first component ),,( )( ϕηξ 2 1 − P are shown in Fig.5 for angle of incidence of the high-frequency plane waves 0 0 30= θ )( 5 0 = − hk . Fig. 5. Scattering diagram of the spatial component ),,( )( ϕηξ 2 1 − P of the total acoustic pressure produced by the difference-frequency wave by a rigid elongated spheroid for: 2 f =1000 kHz, 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 θ = 0 30 , ≈ 021 hk , 40, 0 h =0,01 м, 0 ξ =1,005 (relations axis - 1:10), ξ =7. In the direction of the angle of incidence (with respect to the z-axis), the scattering diagrams have major maximums. Increase of the amplitude of the spheroidal wave produced by the scatterer leads to additional maximums in lateral directions (irrespective of the angle of incidence). This result is connected with the increase of the function η 1 . Increasing the extent of the interaction region (the coordinate S ξ ) results in the narrowing of the scattering lobes; this scenario corresponds to increasing the size of the re-radiating volume around the scatterer. The elongated spheroid has radial dimension 0051 0 ,= ξ with the semi-axes correlation 1:10. Acoustic pressure of the difference frequency wave has been calculated in the far field of the scattering spheroid, i.e. in the Fraunhofer region. Therefore, the scattering field can be considered as being shaped by. Shadowing of the secondary waves sources by the scatterer itself can occur in the Rayleigh region. Here it is necessary to take into account wave dimensions of the scatterer as well as the distance to the point of observation ),,( ϕ η ξ M . In the cases presented in this contribution, the point of Research of the Scattering of Non-linearly Interacting Plane Acoustic Waves by an Elongated Spheroid 83 observation was at radial distances 7 = ξ and 15, which exceeded the length of the elongated spheroid by an order magnitude. Now consider the second ),,( )( ϕηξ 2 2 − P and third ),,( )( ϕηξ 2 3 − P components from the equation (10) for the total acoustic pressure of the difference-frequency wave, these components characterise the non-linear interaction of the incident plane waves with the scattered spheroidal ones waves: [] ⎢ ⎢ ⎢ ⎣ ⎡ −−= ∫ ∑∑ ∞ = ∞ ≥ − − − − S mml mlml dhkmlihkDhkB hk C P ξ ξ ξηξξϕπ η ϕηξ 0 0 '' 0 ' 0201 0 )2( 2 )sin()2(exp)()(),,( [] ⎥ ⎥ ⎥ ⎦ ⎤ −− ∫ ∑∑ ∞ = ∞ ≥ − S mml mlml d hk mlihkDhkB ξ ξ ξ ξ ηξ ϕπ 0 0 ' ' ' 0 0201 )sin( )2(exp)()( . (14) Values of )( 0 hkB nml and )( 0 hkD nml are substituted into equation (14) and the plane wave expansion is used. For the axially symmetrical scattering problem (perfect spheroid), the high-frequency asymptotic forms the angular spheroidal 1 st - order function ),( η 0 hkS nml and the radial spheroidal 3 rd - order function ),( ' )( ξ 0 3 hkR n ml [Kleshchyov & Klyukin, 1987], [Abramovitz & Stegun, 1971]: [] ' ' ' )( exp ),( ' ξ ξ ξ ξ 0 0 1 0 3 0 hik hk i hkR n n l hk n ml n −− ∞→ ≈ . Then equation (11) takes the form [] ⎢ ⎢ ⎢ ⎣ ⎡ −−− − ≈ ∫ − − − − S dhkhkhki hkk hkAiC P ξ ξ ξηξξη ηη ϕηξ 0 000102 2 02 02 2 2 12 2 ''' )( )sin()(exp )( )( ),,( [] ⎥ ⎥ ⎥ ⎦ ⎤ −−− ∫ − S d hk hkhki ξ ξ ξ ξ ηξ ξη 0 2 0 00102 ' ' ' ' )sin( )(exp . (15) After the final integration [Prudnikov et al., 1983], the expression for the 2 nd component of the total acoustic pressure of the difference-frequency wave takes the form )()()()()( ),,( 2 24 2 23 2 22 2 21 2 2 −−−−− +++= PPPPP ϕηξ , (16) where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − −− ≈ − − −− 2 022 0 2 02 02 2 2221 112 u iuiu hkk hkAiC P S )exp()exp( ))(( )( )( , ξξ ηηη ∓ , [] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −−−−− − −− ≈ − − −− )Ei()Ei( )exp()exp( ))(( )( )( , 0222 0 022 0 2 02 02 2 2423 112 ξξ ξ ξ ξ ξ ηηη iuiuu iuiu hkk hkAC P S S S ∓ , )( η η 0001022 hkhkhku − −= ∓ . Acoustic Waves 84 The expression for the 3 rd component ),,( )( ϕηξ 2 3− P is similar to the expression (15). An analysis of equation (15) shows that the behaviour of scattering diagrams for the components ),,( )( ϕηξ 2 2− P and ),,( )( ϕηξ 2 3− P is determined mainly by the function ))(( ηηη −− 111 0 , where the dependence on the angle of incident 0 θ (that is 0 η ) is not clear. The scattering diagram of these components are shown in Fig.6, for 0 0 30= θ )( 5 0 = − hk . These diagrams have maximums in the backward and side directions ( 0 0 and ) 0 90± . The increase of the wave size of the spheroidal scatterer leads to additional maximums, which depend on the angle of incident of the high-frequency plane waves. Fig. 6. Scattering diagram of the spatial components ),,( )( ϕηξ 2 2− P , ),,( )( ϕηξ 2 3− P by a rigid elongated spheroid for: 2 f = 1000 kHz, 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 θ = 0 30 , 0 ξ =1.005, ξ =7. Now, we consider the fourth component ),,( )( ϕηξ 2 4− P of the total acoustic pressure of the difference-frequency wave. This component characterises the non-linear interaction of the scattered spheroidal waves with frequencies 1 ω and 2 ω : ⎢ ⎢ ⎢ ⎣ ⎡ −= ∫ ∑∑ ∞ = ∞ ≥ − − − − S mml mlml dhkhkDhkD hk C P ξ ξ ξηξξ η ϕηξ 0 0 00201 0 2 4 ''' )( )sin()()(),,( ⎥ ⎥ ⎥ ⎦ ⎤ − ∫ ∑∑ ∞ = ∞ ≥ − S mml mlml d hk hkDhkD ξ ξ ξ ξ ηξ 0 0 0 0201 ' ' ' )sin( )()( . (17) After some algebraic manipulations, equation (17) takes the form Research of the Scattering of Non-linearly Interacting Plane Acoustic Waves by an Elongated Spheroid 85 )()()()()( ),,( 2 44 2 43 2 42 2 41 2 4 −−−−− +++= PPPPP ϕηξ , (18) where [][] )Ei()Ei( ))(( )()( )( , 0444 0 2 012 0202 2 4241 112 ξξ ηηη iuiuu hkkik hkAhkAC P S −−−− −− ≈ − − −− ∓ , [] ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −−−+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − −− ≈ − − −− )Ei()Ei( )exp()exp( ))(( )()( )( , 044 2 4 0 044 4 0 2 012 0202 2 4443 114 ξξ ξ ξ ξ ξ ηηη iuiuu iuiu iu hkkik hkAhkAC P S S S ∓ , )( η 004 hkhku −− = ∓ . The scattering diagram of the fourth component ),,( )( ϕηξ 2 4− P are shown in Fig.7, for 0 0 30= θ )( 5 0 = − hk . Their configuration is primarily determined by the function ))(( η η η −− 111 0 of equation (18). As indicated above, this function has a maximum in the backward direction and slightly depends on the angle of incidence. Increasing of the spheroidal scatterer wave size results increases lateral scattering. Fig. 7. Scattering diagram of the spatial component ),,( )( ϕηξ 2 4− P by a rigid elongated spheroid for: 2 f = 1000 kHz, 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 θ = 0 30 , 0 ξ =1.005, ξ =7. Fig.8 presents the scattering diagram of the total acoustic pressure in the difference- frequency wave ),,( )( ϕηξ 2 − P according to the asymptotic expressions for spatial components. In this case, the angle of incidence is 0 0 30= θ )( 5 0 = − hk , and the coordinate 7= ξ . Fig.9 shows wave scattering diagrams of difference frequency ),,( )( ϕηξ 2 − P on rigid elongated spheroid 0 ξ =1,005 with different incidence angle values of inflation incident waves 0 θ = 0 0 ; 0 90 . Acoustic Waves 86 Fig. 8. Scattering diagram of the total acoustic pressure the difference-frequency wave ),,( )( ϕηξ 2 − P by a rigid elongated spheroid for: 2 f = 1000 kHz, 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 θ = 0 30 , 0 ξ =1,005, ξ =7. Fig. 9. Scattering diagrams of the total acoustic pressure the difference-frequency wave ),,( )2( ϕηξ − P by a rigid elongated spheroid for: 2 f = 1000 kHz, 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 ξ =1,005, ξ =7, 0 θ = 0 0 ; 0 90 . Research of the Scattering of Non-linearly Interacting Plane Acoustic Waves by an Elongated Spheroid 87 With incidence angle 0 θ = 0 0 diagrams have got the basic maximums back, with the increase of spheroid wave dimension, the modest lateral scattering appears. With incidence angle 0 θ = 0 60 diagrams are of the similar form 0 θ = 0 30 , with conformable maximums in decrease direction, in mirrorlike, as well as back. With incidence angle 0 θ = 0 90 diagrams have got the basic maximums back and lateral directions. With the wave dimension growth, modest intermediate levels can be observed. It follows from Fig.9 that angle value change 0 θ leads generally to the change of maximums position in the line of incidence and reflex angle. It is emphasized that the figures illustrate the dependence of acoustic pressure ),,( )( ϕηξ 2 − P on the polar angle η θ arccos = but not on the angle of asymptote of the hyperbola η . This presentation is conventionally employed for the scattering diagrams in spheroidal coordinates [Cpence & Ganger, 1951], [Kleshchyov & Sheiba, 1970]. The diagrams are presented in the xoz plane (Fig.4). Polar angle θ varies in the range 0 0 to 0 360 ; the value of the angle 0 0= θ corresponds to the position of x axis, and the value 0 90= θ corresponds to z axis. The arrow here shows the direction of the initial plane wave incidence. The axisymmetry of the diagrams with respect to x axis has been taken into account and two diagrams with positive and negative directions of the angle 0 180±= θ have been combined. Fig.10 shows a spatial simulation of the scattering diagram of the total acoustic pressure ),,( )2( ϕηξ − P for 0 0 30= θ ( 5 0 = − hk , 7 = ξ , an arrow indicates the direction of the initial wave incidence). It is a surface of revolution, and the rotation axis is the larger axis of the elongated spheroid, that is the x- axis. Fig. 10. Spatial model of scattering diagram of the total acoustic pressure the difference- frequency wave ),,( )2( ϕηξ − P by a rigid elongated spheroid for: 1 f =880 kHz, − F =120 kHz, 0 hk − =5, 0 θ = 0 30 , ξ =7. Acoustic Waves 88 5. Discussion Although investigation of the linear scattering of acoustic waves by the elongated spheroid has been considered previously, results of the scattering of the nonlinearly interacting acoustic wave were not reported. In most previous publications, the problem is investigated when the angles of incidence of acoustic waves are 0 0= θ and 0 90 [Kleshchyov & Sheiba, 1970], [Tetyuchin & Fedoryuk, 1989]. In article [Kleshchyov & Sheiba, 1970] the calculated diagrams of plane acoustic wave scattering by a similar size spheroid ( 0051 0 ,= ξ , 10 0 =kh ) at angle of incidence 0 30= θ are presented. Also in this work the scattering diagram has maximums symmetrical to the angle of incidence (mirror lobes) with respect to z axis [Burke, 1966], [Boiko, 1983]. At angle of incidence 0 0= θ forward scattering dominates. The basic maximum is aligned with 0 140 . When the angle of incidence is 0 90= θ (lateral incidence), there are only two maximums – forward and backward. An analysis of the acoustic pressure distribution of the difference-frequency wave scattered field shows that the scattering diagrams have maximums in a backward direction. In direction to the angle of incidence, in lateral and transverse directions, plane waves have maximums. Incident high-frequency plane waves form the scattering field in backward and forward directions, and scattered spheroidal waves form the scattering field in transverse direction. An increase in the wave size of the spheroidal scatterer changes maximum levels, and an increase in the size of the interacting area around the elongated spheroidal scatterer leads to narrowing of these maximums. It is important to note that in this work we considered the case when the scattered field is generated by the secondary wave sources located in the volume around the spheroid. In the case of the linear scattering, these sources are located on the surface of the spheroid. The mirror maximums 0 30 and 0 150 appear as a result of the asymptotics of the first spatial sum ),,( )( ϕηξ 2 1− P as confirmed in [2]. Therefore, the plotted scattering diagrams are in conformity with the results of 0 90 [Burke, 1966], [Kleshchyov & Sheiba, 1970], [Boiko, 1983], [Tetyuchin & Fedoryuk, 1989]. As for the numerical evaluation of the acoustic pressure, it is necessary to note the following. In view of the complexity of mathematical calculations, the obtained asymptotics allow for qualitative evaluation of the spatial distribution of the acoustic pressure in the scattered field. It would be more adequate to compare the results with experimental data. Unfortunately, experiments in non-linear conditions have not been carried out. For the sake of better understanding of contribution of the separated sums into the cumulative acoustic field, results were presented for two values of the wave dimension and the angle of incidence. It should be noted, that description of wave processes in spheroidal coordinates have several peculiarities. For example, comparing the acoustic pressure distribution at the distance from the scatterer, the results given in [Abbasov & Zagrai, 1994], [Abbasov & Zagrai, 1998], [Abbasov, 2007] can be taken. Spheroidal coordinates in a far field transform into spherical ones )( 0 0 →h and ),,(),,( )()( ϕθϕηξ rPP 22 −− → . The results of this research are in agreement with results of prior studies of the scattering process described in spherical coordinates. [...]... Eq (8) leads to 4n homogeneous linear equations for Xl l = (1- 4n), as follows (1) ⎡ H 1,G ⎢ (1) ⎢ H 2 ,G ⎢ (1) ⎢ H 3,G ⎢ ⎢ H (1) ⎣ 4 ,G (2 ) H 1,G … (2 ) H 2 ,G (2 ) H 3,G (2 ) H 4 ,G … ( 4n H 1,G ) ⎤ ⎡ X 1 ⎤ ⎥⎢ ⎥ X ( n H 24G ) ⎥ ⎢ 2 ⎥ , ⎥⎢ ⎥ = HX = 0, ( n ⎥ H 34G ) ⎥ ⎢ , ⎥⎢ ⎥ ( n H 44 G ) ⎥ ⎢ X 4 n ⎥ , ⎦⎣ ⎦ (9) 94 Acoustic Waves where H is a 4n × 4n matrix with components (l) 3( 1( ( 44 H 1,G = CG −G'... equations to determine both ( AG , AG , AG ) and k z 1 2 ⎛ c 11 ( k x + G )( k x + G ' ) + c 44 k z − ρω 2 ⎞ ⎛ AG' ⎞ 0 c 12 ( kx + G ) + c 44 ( k x + G ' )k z ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ (5) ⎜ ⎟⎜ 2 ⎟ 2 0 0 c 44 ( k x + G )( k x + G ' ) + c 44 k z − ρω 2 ⎜ AG' ⎟ = 0, ⎜ ⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ 2 ⎜ c 12 ( kx + G ' ) + c 44 ( k x + G )k z 0 c 44 ( kx + G )( k x + G ' ) + c 11 k z − ρω 2 ⎟ ⎜ A 3 ' ⎟ ⎝ ⎠⎝ G ⎠ Supposing that the materials... G,G′ , ( l = m = 1 − 3 ) are: 11 11 44 M G ,G′ = C G − G′( kx + G′ )( kx + Gx ) + C G − G′G′ Gz x z 12 M G ,G ′ = 0 13 12 44 M G ,G′ = C G-G′G′ ( kx + Gx ) + C G-G′( kx + G′ )Gz z x 21 M G ,G ′ = 0 22 44 44 ′ M G ,G′ = C G-G′( kx + Gx )( kx + Gx ) + C G-G′G′ Gz z 23 M G ,G ′ = 0 31 44 12 M G ,G′ = C G-G′G′ ( kx + Gx ) + C G-G′( kx + G′ )Gz z x 32 M G ,G ′ = 0 (36b) 33 44 11 ′ M G ,G′ = C G-G′( kx + Gx... namely, P and SV waves It is relatively simple to discuss the SH wave so that we focus our attentions to P and SV waves, and the equation of motion for Lamb waves becomes 1 ⎞ ⎛ AG ' ⎞ ⎟ ⎟⎜ ⎟ = 0, ⎟⎜ ' 2 2 ⎟⎜ 3 ⎟ ⎟⎜ A ' ⎟ c 44 ( k x + G )( k x + G ) + c 11 k z − ρω ⎠ ⎝ G ⎠ 2 ⎛ c 11 ( k x + G )( k x + G ' ) + c 44 k z − ρω 2 ⎜ ⎜ ⎜ ⎜ c ( k + G ' ) + c ( k + G )k z 44 x ⎝ 12 x c 12 ( k x + G ) + c 44 ( k x +... the problem of the scattering of non-linearly interacting waves by an elongated spheroid 7 References Abbasov, I.B (2007) Scattering nonlinear interacting acoustic waves: sphere, cylinder and a spheroid Fizmatlit, Moscow, 160p Abbasov, I.B., Zagrai, N.P (19 94) Scattering of interacting plane waves by a sphere Acoust Phys Vol 40 , No 4, P 47 3 -47 9 Abbasov, I.B., Zagrai, N.P (1998) The investigation of... into Fourier series, one obtains 93 Acoustic Waves in Phononic Crystal Plates u( x , z , t ) = ∑ e jkx x − jωt ( e jGx AG e jkz z ), (4) G 1 2 3 where kx is a Bloch wave vector and ω is the circular frequency, AG = ( AG , AG , AG ) is the amplitude vector of the partial waves, and k z is the wave number of the partial waves along the z direction Substituting Eqs (2)- (4) into Eq (1), one obtains homogenous... Acoust Phys.( Akust Zh.) Vol 16, No.2, P 2 64- 268 Kleshchyov, A.A (1992) Hydroacoustic scatterers Sudostroenie St Peterburg 248 p Kleshchyov, A.A (20 04) Physical model of sound scattering by jamb of fishes who is at border of section of Akust Zh Vol 50, No 4, P 512-515 Kleshchyov, A.A., Clyukin, I.I (1987) The foundation of hydroacoustic Sudostroenie, Leningrad 224p Kleshchyov, A.A., Rostovtsev, D.M (1986)... ' )k z (6) If one truncates the expansions of Eqs (2) and (3) by choosing n RLVs, one will obtain 4n ( ( eigenvalues kzl ) , (l = 1 − 4n) For the Lamb waves, all of the 4n eigenvalues kzl ) must be included Accordingly, displacement vector of the Lamb waves can be taken of the form (l) ⎞ (l) ⎞ ⎛ 4n ⎛ 4n ( u( x , z , t ) = ∑ ' e i( kx + G )x − iωt ⎜ ∑ AG e iK z z ⎟ = ∑ ' ei ( kx + G )x − iωt ⎜ ∑ Xl... from an pulsted sphere Soviet Physics Acoustics, Vol 38, No 1, P 51-57 Novikov, B.K., Rudenko, O.V., Timoshenko, V.I (1987) Nonlinear underwater acoustic Acoustical Society of America, New York, 2 64 p Prudnikov, A.P., Brychkov, Yu A., Marichhev, O.I (1983) Integrals and rows Nauka Moscow 752p Skudrzyk, E (1971) The foundations of acoustics Springer, New York, 542 p Stanton, T.K (1989) Simple approximate... the band gaps of plate-mode waves in 1D piezoelectric composite plates with substrates 92 Acoustic Waves The chapter is structured as follows: we firstly introduce the theory and modeling used in this chapter in Section 2 In Section 3, we focus on the band gaps of lower-order Lamb waves in 1D composite thin plates without/with substrate In Section 4, we study the lamb waves in 1D quasiperiodic composite . Plane Acoustic Waves by an Elongated Spheroid 85 )()()()()( ),,( 2 44 2 43 2 42 2 41 2 4 −−−−− +++= PPPPP ϕηξ , (18) where [][] )Ei()Ei( ))(( )()( )( , 044 4 0 2 012 0202 2 42 41 112 ξξ ηηη iuiuu hkkik hkAhkAC P S −−−− −− ≈ − − −− ∓. [] ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −−−+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − −− ≈ − − −− )Ei()Ei( )exp()exp( ))(( )()( )( , 044 2 4 0 044 4 0 2 012 0202 2 44 43 1 14 ξξ ξ ξ ξ ξ ηηη iuiuu iuiu iu hkkik hkAhkAC P S S S ∓ , )( η 0 04 hkhku −− = ∓ . The scattering diagram of the fourth component ),,( )( ϕηξ 2 4 P. 123 (,,) GGG A AA and k z . ' ' ' 1 '22 ' 11 44 12 44 '22 2 44 44 ' '22 3 12 44 44 11 ()( ) 0 ()( ) 0()()0 0, ()() 0 ()() xx z x xz G xx z G xxz xxz G A ck