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An approximate theory for waves in a slender elastic wedge immersed in liquid
B y Andrey V. Shanin1 & Victor V. Krylov2
1

Department of Acoustics, Faculty of Physics, Moscow M.V.Lomonosov State University, Moscow 119899, Russia 2 Department of Civil and Structural Engineering, The Nottingham Trent University, Burton Street, Nottingham NG1 4BU, UK

The problem is considered of wave propagation along the tip of a slender elastic wedge immersed in liquid. Interaction between wedge mode and acoustic waves is studied. Namely, attenuation of wedge wave occurs if the velocity of wedge mode is greater than the velocity of acoustic waves and wedge mode velocity changes in the opposite case A functional-differential equation is derived for the problem. This equation is solved asymptotically by using Wiener-Hopf method in the case of very light liquid loading.
Keywords: Wiener-Hopf technique, wedge waves, thin plate flexural vibrations

Intro duction
The theory of diffraction and wave propagation in sectorial-shaped region is a challenging branch of mathematical physics. In the early 50-s, a powerful technique was developed to solve 2D stationary diffraction problems for liquid (or gas) angles sub ject to arbitrary boundary conditions (see Maljuzhinetz (1955)). The main idea was to reduce the problem to a set of functional equations. These equations were defined on a complex plane and linked the values of an unknown function at two different points. There were also some important restrictions on the behaviour of the unknown functions on the complex plane. Some methods for solving such functional equations have been developed. The approach enabled one to avoid solving boundary integral equations. A number of papers appeared developing the method of functional equations in order to be able to to apply it to a great variety of problems (for example, see Osipov & Norris (1997) and a large bibliography therein). The main restriction of the method of functional equations is that the problem can involve only one wave velocity. Such important problems as diffraction of electromagnetic waves on a dielectric wedge and diffraction of acoustic waves in a solid wedge involve wave modes with at least two different velocities, and they cannot yet be solved explicitly using functional equations, since although functional equations can be derived for each of these problems, the equations cannot be solved using known methods. There is an important aspect that attracts a great deal of attention for acoustic diffraction in a solid wedge: such a wedge can act as a waveguide for the so-called wedge waves (Lagasse (1973)). The bulk of information about wedge waves has been obtained using numerical calculations, (see e.g., Moss et al. (1973), Hladky-Hennion
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A.Shanin & V.Krylov

et al. (1997)), and no analytical theory is available for waves in a wedge with arbitrary apex angle. However, analytical theories have been developed successfully for two important cases possessing small parameters: symmetrical wave modes in a wedge with the apex angle close to and antisymmetric (flexural) modes in a slender wedge. In the present paper we shall study the latter case. The wedge modes have some important properties. Their energy is localized in the vicinity of the edge, and their velocities can be essentially lower than the velocities of bulk elastic waves in the wedge material. These properties make wedge modes attractive for such applications as nondestructive testing of certain engineering structures (with edges). The theory of wedge waves in a slender wedge is well-developed. The wedge can be treated as a thin plate of variable thickness, and the dynamic equation of thin plates can be either directly solved (as in McKenna et al. (1974)) or a geometrical acoustics approach can be applied using the dispersion curves of flexural modes (as in Krylov (1989) or Mozhaev (1989)). Both methods lead to similar results and are in good agreement with the analytical derivation based on the expansion of an elastic field in a slender wedge as power series in apex angle (see Parker (1992)). Namely, it is known that there is a number of antisymmetric modes roughly estimated as /, where is the apex angle of a wedge. The lowest mode has velocity of order ck , where ck is the velocity of a shear wave in the solid. Higher modes have a complicated structure and their velocities are approximately proportional to the number of a mode. In this paper we study wave propagation in a slender elastic wedge surrounded by liquid or gas. One can expect such effects as attenuation of wedge waves (in the supersonic case) and modification of their velocities. In what follows we pursue the following aims: -- We study wedge wave propagation in immersed wedges. This is an interesting engineering problem of structural acoustics (Hladky-Hennion et al. (1997), Krylov (1994), Krylov (1998)). The results can be used in noise control and in nondestructive testing applications. There are also some ideas to use such waves for modern underwater propulsion techniques (Krylov (1994)). -- We derive a new type of functional equation. This equation is similar to the classical Wiener-Hopf type, but includes a differential operator. As far as we know, there is no known method for solving such equations. It can be expected that such equations emerge in a variety of diffraction problems. Therefore any attempt to solve this equation would be a valuable contribution to the theory of Wiener-Hopf equations. -- We attempt to use functional equations as a starting point for all calculations in this paper. Functional equations proved to be an effective tool in the previous research of one of the authors (Shanin (1996, 1997)). In the present paper, functional equations replace boundary integral equations and enable us to avoid using special functions.

1. Formulation of the problem
Consider the space (x, y , z ) filled with a homogenous liquid and a thin elastic plate of variable thickness (a slender wedge) occupying the area x > 0, - < z < . The geometry of the problem is shown in Fig. 1. The local thickness of the wedge is
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Slender elastic wedge in liquid

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presumed to be negligible everywhere (the apex angle of the wedge is very small), so we can say that the y -coordinate of the points of the wedge is everywhere equal to zero.

Figure 1. Geometry of the wedge

We are going to use the plate theory equation to decribe the motion of the wedge and its interaction with the fluid. Of course, it is a simplification of the original problem (wave propagation in a solid wedge with fluid loading), but only this assumption enables us to derive some analytical results. We must note, that our approach is not bad, because wedge waves are known to have exponential decay with the distance from the edge, and the waves in liquid have either exponential or algebraic decay, so at the distances from the edge where the local thickness of the wedge is considerable (comparable with wavelength) both elastic and acoustic values are small. We shall consider the wedge of finite thickness only in the Appendix in order to derive the equations of the limiting case 0. Suppose that all variables (displacements of the wedge and pressure in the liquid) have the dependence on the z coordinate and the time of the type e-i( z+t) . Note that the wave travels in the negative direction of the z -axis, when > 0, > 0. We assume in this paper that is fixed and is the parameter to be determined. Instead of the attenuation of the wave we study the damping in time of a spatially sinusoidal wave, which corresponds to with a negative imaginary part. Note that in practice it is usually an imaginary part of which is sought, with fixed. For small attenuation, we can use the relationship I m[ ] = - I m[ ]/c to find the estimation of spatial attenuation of the wedge wave (c is the velocity of undisturbed wedge wave). Below we seek only the solutions antisymmetric in y . The reason for this is following. Only antisymmetric modes exist in a slender wedge in vacuum. Symmetric mode appears only for close to . Besides, antisymmetric modes in slender wedge are much slower than bulk modes in the solid, so for some the velocities of wedge mode and acoustic wave coincide and the interaction between them can be very effective. There are two subsystems under consideration: fluid one and elastic one. For each subsystem we must formulate the equation of motion, boundary and radiation conditions and, besides, we must desribe the interaction between the subsystems.
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A.Shanin & V.Krylov (a ) Fluid subsystem Helmholtz' equation is satisfied in the liquid
2 p + k 0 p = 0 ,

(1.1)

where p is the amplitude of the acoustic pressure (factor eit is assumed), and k0 = /c0 is the wavenumber of sound in the liquid, is (complex) circular frequency, and c0 is the velocity of sound. As the pressure takes the form p(x, y , z ) = p(x, y )e-i z , we have the following equation for the function p(x, y ): 2 2 +2 2 x y where
2 2 = 2 - k 0 .

p - 2 p = 0 ,

(1.2)

(1.3)

There are two different cases: Re[ ] > Re[k0 ] (subsonic case) and Re[ ] < Re[k0 ] (supersonic case). In the supersonic case, the wave travelling along the edge is faster than sound in the liquid, so that excitation of non-decaying waves is possible. This case is more difficult for analysis because formally one should obtain the solution for p that shows exponential increase at y . The subsonic case is more simple, since the solution must decay exponentially with y . We use a simple model for the supersonic case by supposing that the liquid possesses some absorption (i.e., by considering an imaginary part of c0 ). This is enough to exclude the exponential growth of the solution. We formulate the radiation condition for the liquid in the following way: there must be no wave components coming from infinity or having exponential growth at infinity. This condition will be satisfied by choosing an appropriate branch of square root in the Fourier integral (see the next section). Edge conditions must be imposed, describing the behaviour of p in the vicinity of the edge. Acoustic energy must be integrable in this area, and energy flow from the origin must be equal to zero. Taking into account that an antisymmetric solution is sought, we conclude that p(r) = o(1), where r is the distance from the edge. (b ) Elastic subsystem For the decription of antisymmeric motion of the wedge we will use the theory of flexural vibrations of plates with variable thickness. Namely, the thickness of the plate (wedge) grows linarly with the x-coordinate (as in McKenna et al. (1974)). Let the y -component of the dislpacement of the wedge be denoted by w(x, z , t) = w(x)e
-i( z + t)

p = o(1/r),

(1.4)

.

The equation of flexural vibrations of the plate is L[w(x)] = p(x, +0) - p(x, -0),
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(1.5)

Slender elastic wedge in liquid

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where the operator L[w] is equal to (see Timoshenko & Woinovsky-Krieger (1959)) L[w] = where M
xx

2 Mzz 2 Mxz 2 Mxx + -2 + 2 m(x)w, 2 2 x z x z 2w 2w + 2 z2 x

(1.6)

= -D

2w 2w + 2 x2 z

,

M

zz

= -D

,

Mxz = D(1 - ) D = D0 x3 3 ,

2w , x z E , D0 = 12(1 - 2 )

m(x) = s x, E is Young's modulus of the material, is Poisson's ratio, s is the mass density of the solid and is the apex angle of the wedge. The values Mxx and Mzz are the bending moments per unit length of sections of the wedge perpendicular to the x and z axes, respectively, Mxz is the twisting moment per unit length of a section of the wedge perpendicular to the x axis, and m(x) is the mass per unit area. Since the problem is chosen to be antisymmetrical, we can consider the fluid motion only in the region y > 0 and rewrite (1.7) in the form L[w(x)] = 2p(x, 0). (1.7)

We seek for the solutions w(x) decaying at infinity. Beside that, the tip of the wedge is unconstrained, so w must satisfy the boundary conditions at x = 0 (Timoshenko & Woinovsky-Krieger (1959)): Mxx = 0, Mxx Mxz -2 = 0. x z (1.8)

(c ) Interaction between elastic and fluid subsystems Consider the liquid near the plate. It follows from the equations of fluid acoustics that the acceleration of liquid is proportional to the gradient of pressure. It is obvious that the y -component of the displacement is continuous at the plate, so we conclude that w(x) = 1 p l 2 y ,
y =0

(1.9)

where l is the mass density of the liquid. Substituting (1.9) into (1.7), we obtain a higher-order boundary condition for the liquid. A detailed study shows that the following asymptotic series satisfying Meixner's conditions for the fluid and boundary conditions for the solid can be substituted for p near the edge

p
n=1

an rn sin n +

(1.10)

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n

A.Shanin & V.Krylov m r sin mr + cn,m m
+1

b
n=0 m=0

n,m

sin[( + 1)] m

,
=2n+1/2

where is the angle from the negative direction of the x-axis. Note that the second series contains logarithmic terms. Such behaviour is peculiar to the solutions of impedance boundary problems. Anzats (1.10) enables to make an estimation of w ^ near the edge: w(x) = w(0) + xw (0) + o(x). (1.11)

2. Derivation of the functional equation
Function p(x, y ) in the region y > 0 can be represented in the form of Fourier integral 1 p(x, y ) = 2

p(k )e- ^
-

ikx- k2 +2 y

dk

(2.1)

The appropriate choice of the branch of the square root is necessary for satisfying the radiation condition in the liquid. Namely, the real part of the root must be positive. Note, that a special choice of the contour of integration can help to overcome some difficulties in the the supersonic case with real c0 , in which the exponential growth of p can appear. However, here we do not consider this case. Also, we expect the exponential decay of w and p as |x| , so the Fourier integrals here and below are convergent. Taking into account, that p(x, 0) = 0 for x < 0 since the problem is antisymmetric, we find that

p(k ) = ^
0

p(x)e

ik x

dx.

(2.2)

Normal derivative of the field can be obtained by differentiation of (2.1): p(x, y ) 1 =- y 2

k 2 + 2 p(k )e ^
-

-ikx- k2 +2 y

dk .

(2.3)

Note, that p/ y at the plate can be found from (1.9), so the inverse Fourier transform of (2.3) for y = 0 can be written in the form - k 2 + 2 p(k ) = s(k ) + l 2 w(k ), ^ ^ ^ where
0

(2.4)

s( k ) = ^
-

p ik x e dx, y

(2.5)

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Slender elastic wedge in liquid

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w(k ) = ^
0

w(x) e

ik x

dx,

(2.6)

^ Let L be the Fourier image of the operator L/2, i.e. an operator, such that ^^ ^ L[w(k )] = p(k ). The factor 1/2 is chosen for convenience. The Fourier transform yields iD0 ^^ L[w(k )] = - 2
3

(2.7)

(k 4 + 2 2 k 2 + 4 )

d3 w ^ d2 w ^ + 6k (k 2 + 2 ) 2 + 3 dk dk dw ^ , dk

(2.8)

2 6(k 2 + 2 + (1 - )kT /2 )

2 where kT = 2 /c2 is the wavenumber of shear waves in the solid, and c2 = E /[2(1 - T T )s ]. We show in the Appendix how equation (2.8) can be obtained directly from the governing equations for the solid wedge. Note that the operator L is defined only for x > 0. It might seem that the Fourier transform of (1.7) should include terms corresponding to the values of functions at x = 0 (as in formulas concerning Laplace's transform). However, boundary conditions (1.8) and edge conditions, following from (1.11) precisely make these terms zero. The behaviour of unknown functions at infinity does not cause any problems, because under the assumptions made above, all functions have exponential decay both in the supersonic and the subsonic case. Equation (2.4) can be rewritten in the form

s(k ) + l 2 w(k ) + ^ ^

^^ k 2 + 2 L[w(k )] = 0.

(2.9)

Unknown functions w and s have some important properties following from ^ ^ (2.5), (2.6). The function w(k ) must be analytic in the upper half of the complex ^ k -plane and have no singularities on the real positive half-axis. The function s(k ) ^ must be analytic in the lower half of the complex k -plane and have no singularities on the real negative half-axis. Besides, w(k ) and s(k ) have the behavior at infinity ^ ^ w(k ) = w(0) ^ i 1 1 - w (0) 2 + o( 2 ), k k k s(k ) = o(1). ^ (2.10)

Note that equation (2.9) is similar to the classical homogeneous functional Wiener-Hopf equation (see Noble (1958)). However, it contains a differentional operator, so we shall call (2.9) a functional-differentional equation (FDE).

3. Application of the p erturbation technique to the FDE in the case of a very light fluid
Unfortunately, no theory is known for equations like (2.9) with analytic restrictions on unknown functions. Here we shall consider only a particular case containing a small parameter.
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A.Shanin & V.Krylov

Let the liquid surrounding the wedge be very light. By using the perturbation technique, we shall construct the solution of equations (2.9), (2.8) in the form of a series in a small dimensionless parameter. We start from the solution for the none-immersed wedge (zero-order approximation): ^^ L[w0 (k )] = 0. (3.1)

Note that equation (3.1) is a second order equation for w and can be solved ^ using standard methods. Equation (3.1) has two singular points in the finite part of the complex k -plane: k = i and k = -i . The behaviour of the solution at k = i is of great importance because the solution must be analytic in the upper half plane. The general solution of (3.1) is w0 (k ) = C ^ where = 1 2 4 + 6(1 - )(
2 kT - 1). 22 1

k - i k + i

+C

2

k - i k + i

-

+ C3 ,

(3.2)

The solution (3.2) must have no singularities in the upper half plane, particularly at the point k = i . So, we must insist that is an integer and C2 = 0. The constant C3 is chosen such that w0 (k ) 0 as k . ^ Finally, we rewrite the zero-order approximation of the solution in the form w0 = ^ where = kT 3(1 - ) . (2N 2 + 1 - 3 ) (3.4) k - i k + i
N

- 1,

N = 1, 2, 3 . . .

(3.3)

The last equation determines the eigenvalues of the slender wedge as a waveguide. It coincides with the condition from McKenna et al. (1974). It can be shown that the eigenwaves corresponding to (3.2) can be expressed by Laguerre polynomials in the same way as in McKenna et al. (1974). Let us introduce a small parameter of the problem = It is easy to show that l N 2 . s (3.6)
2 l 0 . 32 D0

(3.5)

This parameter is small for a very light liquid (or gas) loading.
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Slender elastic wedge in liquid Introducing the notations s (k ) = ^ s( k ) ^ , D0 3 2

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^^ L[w(k )] ^^ L [w(k )] = , D0 3 we rewrite (2.9) in the form s (k ) + ^ ^^ k 2 + 2 L [w(k )] = -w(k ). ^ (3.7)

Traditional perturbation techniques can be applied to such equation (Nayfeh, 1981). We suppose that the parameter is fixed and look for a correction to 0 , i.e., to the initial value of . Since the problem is regularly perturbed, the variables can be expanded as series in : 0 + 1 + 2 2 + . . . , w(k ) = w0 (k ) + w1 (k ) + 2 w2 (k ) + . . . , ^ ^ ^ ^ 2 s (k ) = s1 (k ) + s2 (k ) + . . . ^ ^ ^ = Each term of the expansions for w and s must obey the same analytic and edge ^ ^ restrictions, as formulated for w and s themselves. ^ ^ 2 ^ ^ The operator L depends on kT ( ), and is the function of . Therefore, L and also must be expanded as a power series in : ^^ L [w(k )] = = ^2 ^ ^1 ^ ^0 ^ L [w(k )] + L [w(k )] + 2 L [w(k )] + . . . 2 0 + 1 + 2 . . .

We shall find first-order term of the expansion for , i.e. 1 . This term provides the information about the energy losses due to wave radiation into the liquid and the changes of edge mode velocity. For this purpose we substitute the expansions for the unknown functions into (3.7) and obtain the equation at O(): s (k ) + ^1 where 6i0 (1 - ) d ^1 L = -1 2 2 c2 dk T ^ ^0 ^ f (k ) L [w1 (k )] ^^ is also analytic in the same region. According to (2.10), we conclude that L0 [w1 (k )] O(1) at infinity. ^ We seek the functions s (k ) and f (k ) by using the standard procedure of Wiener^1 Hopf method. First decompose into factors the coefficient on the left-hand side of the equation: s (k ) ^ 1 + k - i 0 w0 (k ) ^ ^ - k + i0 f (k ) = - k - i 0 ^1 ^ k + i0 L [w0 (k )]. (3.10) (3.9) ^ ^ k 2 + 2 L [w1 (k )] = -w0 (k ) - 0 0^ ^ k 2 + 2 L [w0 (k )], 0 1^ (3.8)

Note that w1 (k ) is analytic in the upper half-plane of k . Therefore, the function ^

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A.Shanin & V.Krylov

In agreement with the restrictions imposed on s(k ) and w(k ), the factors are chosen ^ ^ so that the term containing s is analytic in the lower half-plane while the term ^1 ^ involving f (k ) is analytic in the upper one. Next the r.-h.s. of (3.10) is similarly split as the sum of two terms, one of which is regular in the upper half plane and another one is regular in the lower half-plane. ^^ Note that the function k + i0 L [w0 (k )] is regular in the upper half plane and 1 w0 (k )/ k - i0 has only a pole of order N at k = -i (i.e. in the lower half-plane). ^ The required decomposition can be obtained by subtracting a rational fraction from the last expression and adding it to the first. Let be w (k ) ^ 1 0 = - + (k + i )-N g (k ) k - i 0 k - i0 where ( k - i ) N = g (k ) = k - i0
N -1 n=0

(3.11)

1 g n!

(n)

(k + i )n + RN (k ),

(3.12)

so that (k + i )-N RN (k ) is analytic in the lower half-plane. Taking into account the conditions (2.10), we conclude that s ( k ) ^ w0 (k ) ^ 1 = - + k - i0 k - i 0
N -1 n=0

g

(n)

(k + i ) n!
N -1 n=0

n-N

,

(3.13)

^ ^(k ) = 1 6i(1 - )0 k + i0 dw0 (k ) - k + i0 f 2 2 c2 dk T After performing some manipulations we get 12(1 - )N 0 (k - i )N ^0 ^ L [w1 (k )] = -1 2 c2 (k + i )N T For example, for N = 1 g
(0) -1 +1

g

(n)

(k + i ) n!

n-N

. (3.14)

-

1 k + i

N -1 0 n=0

g

(n)

(k + i )n n!

-N

. (3.15)

=-

2i -i( + 0 )

,

(3.16)

1 12(1 - )0 ^0 ^ L [w1 (k )] = -1 + 2 c2 (k + i )2 T

2i (0 - ik )( + 0 )(k + i )

.

(3.17)

Equation (3.15) is an inhomogenous linear ordinary differential equation with respect to the function w1 . This equation has a singular point in the upper half^ plane (at k = i ). The solution of such equation generally has a singularity at the singular point, so we must find a condition, which guarantees the existence of a solution regular in the upper half-plane. This condition will enable to find 1 .
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4. Solvability condition for the first-order term equation
Rewrite (3.15) in the form ^ ^0 L [ (k )] = f (k ), (4.1)

^ where f (k ) is the function obtained in the previous section (namely, the r.-h.s. of ^ (3.15)), and stands for w1 . The function f is by construction analytic in the ^ upper k -plane, particularly at the point k = i . We need a condition providing the analiticity of at this point. ^ Let be analytic at k = i . Introduce the notation = k - i . Represent f (k ) and (k ) near the point = 0 as power series in : ^ f (k ) =
n=0

^ fn n ,

(4.2)

(k ) =
n=0

n n ;

(4.3)

Equation (4.1) can be rewritten in the form of a difference equation: K [n ] P0 (n)n + P1 (n)
n-1

+ P2 (n)

n-2

^ = 2i fn-

1

(4.4)

where n = 1, 2, 3 . . . and the coefficients are P 0 ( n) = P 1 ( n) = P 2 ( n) = -4 2 n(n2 - N 2 ) 2i n(n - 1)(2n - 1) n(n - 1)(n - 2),

N is the number of wedge mode (see (3.3)). The term 0 can be chosen arbitrary, all others must be found recursevely by using (4.4). Note, that P0 = 0 for n = N . This means that generally the nth term ^ cannot be calculated. However, if fn has some specific property, namely, if ^ 2i fN
-1

- P1 (N )

N -1

- P2 (N )

N -2

= 0,

(4.5)

then the term N can be chosen arbitrary and all other terms can be calculated using (4.4). The restriction (4.5) is not simple to check, because N -1 and N -2 depend on ^ ^ all fn for n = 0 . . . N - 2. Below we formulate a simplier relation involving all fn for n = 0 . . . N - 1 in explicit form. The solution of difference equation (4.4) can be represented in the form

n = 2i
m=0

^ ^ fm Gm,n + hn ,

(4.6)

^ where hn is a solution of the homogenous equation and G is Green's function with the property K [Gm,n ] = 1 0 for n = m + 1 otherwise (4.7)

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A.Shanin & V.Krylov

Let be m < N and n < N . It is easy to show directly, that in this case Gm,n = 0 for n m. On the other hand, Gm,n for n > m + 1 is a solution of homogeneous difference equation K [Gm,n ] = 0 with respect to n. Difference equation does not provide a convenient representation of Gm,n . Fortunately, operator K can be factorized: K = -nK + K where K - [n ] K + [n ] 2 (n - N )n - i(n - 1) 2 (n + N )n - i(n - 1)
n-1 n-1 -

= -nK - K + ,

(4.8)

, .

(4.9) (4.10)

+ It is obvious that the sequences n of homogeneous equation K [n ] = (4.9), (4.10) in terms of factorials. and n are either greater or smaller

- ± and n , for which K ± [n ] = 0 are the solutions 0. Such sequences can be found explicitly from ± For example, if n > m, m = 1 and both m + 1 than N , then

+ n

= =

(n - 1)(n - 2) . . . m (n + N )(n - 1 + N ) . . . (m + 1 + N ) (n - 1)(n - 2) . . . m (n - N )(n - 1 - N ) . . . (m + 1 - N )

i 2 i 2

n-m

n-m

- n

+ - We seek Gn,m for m < n in the form of a linear combination of n and n . One can check directly, that Green's function can be written in the form

Gm

-1,n

=

8

2 n2

1 (2i )(n

-m)

(m + 1)(m + 2) . . . (n - 1) - (N - n)(N - n + 1) . . . (N - m)

(m + 1)(m + 2) . . . (n - 1) (-N - n)(-N - n + 1) . . . (-N - m)

(4.11)

for m < n < N . Now we can calculate N -1 , N -2 and apply (4.5). Elementary, but tedious calculations show that no singularity appears for n = N if and only if
N n=1

N! f n!(N - n)!

n-1

(2i )n-1 = 0.

(4.12)

Equation (4.12) is the condition of solvability for the inhomogenous equation (4.1). To conclude this section it would be useful to describe the connection of the results obtained in the Fourier domain with those in x domain. It is known (see McKenna et al. (1974)) that the expressions for eigenfunctions of a slender wedge contain Laguerre polynomials: wn (x) = e
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- x

L1 (2 x). n

(4.13)

Slender elastic wedge in liquid

13

Note that the first term in (3.2) can be reduced to (4.13) by means of the inverse Fourier transform and the known integral representation of Laguerre polynomials. The value of n in (4.13) corresponds to - 1 in (3.2). ^ Moreover, let f (k ) be the Fourier transform of f (x). Then condition (4.12) is equivalent to the relation

f (x)e
0

- x

L1 (2 x) dx = 0, n

(4.14)

which follows from the usual solvability conditions for ODEs.

5. Final results
The condition of solvability (4.12) applied to equation (3.15) provides the first-order correction to the circular frequency in the form 1 = c2 3 2 T 3(1 - )0 N 1 k + i
N -1 m=0

N !(2i )m в m!(m + 1)!(N - m - 1)! g (n) n!(k + i )(N

в

dm dk m

N -1 0 n=0

-n) k=i

(5.1)

The last formula seems rather complicated. Consider the case N = 1, so giving 1 = - 3 2 c2 1 T , 3(1 - )0 + (5.2)

where is the square root of r.-h.s. of (1.3) with the branch chosen to be positive real or negative imaginary. Note that in the subsonic case this formula describes change in the velocity of a wedge mode. In the supersonic case, one can calculate the imaginary part of 1 : I m[1 ] = 3 2 c2 T , 3(1 - )0 2 - 2 (5.3)

which describes the attenuation of a wedge mode, in addition to the change in velocity given by Re[1 ]. Using equations (1.3), (3.5), (3.8) and (5.2), calculations have been carried out of the ratio C = cwat /cvac = 1 + 1 /0 between wedge wave velocities in a brass wedge in water and in the same wedge in vacuum. The following parameters have 3 3 been used: s = 8600 kg/m , l = 1000 kg/m , c0 = 1478 m/s, cl = 4350 m/s, ct = 2127 m/s, and = 0.343. The results are shown in Fig.2 as a function of wedge angle (curve -- C1 ). For comparison, the corresponding ratio following from the geometrical acoustics theory (Krylov (1998)) is displayed in the same picture (curve -- C2 ). One can see that the agreement between the present theory and the geometrical acoustics one is reasonably good. The discrepancy at smaller angles can be explained by the fact that, according to equation (3.9), the parameter is no longer small in this region and the present theory becomes inaccurate.
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14

A.Shanin & V.Krylov

Figure 2. Calculated values of C = cwat /cvac as functions of wedge angle for a brass wedge in water: curve C1 -- present theory, curve C2 -- geometrical acoustics theory (Krylov (1998)).

6. Conclusions
It has been demonstrated that the problem of flexural wave propagation along the tip of an immersed wedge can be formulated in terms of functional-differential equation (FDE) which is similar to the classical Wiener-Hopf equation. The first order perturbation solution of this equation has been derived for the case of light liquid loading. The comparison of the perturbation solution applied for a brass wedge in water with the corresponding geometrical acoustics calculations shows that the agreement is reasonably good for relatively large values of wedge apex angle. The increased discrepancy at smaller wedge angles can be explained by the fact that liquid loading in this case is no longer light.

Acknowledgements
The work was supported by the Russian Foundation for Basic Research (RFBR), by the program "Universities of Russia", and by the European INTAS Programme.

App endix A.
Here we derive a homogenous equation for a thin elastic wedge in vacuum starting from the functional equation. The contents of this section is slightly aside from the rest of the paper, so that some notations below are different from those above.
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Slender elastic wedge in liquid (a ) Formulation of the problem Let us describ ui (x, y , z , t) = ui corresponding to The equation

15

e stationary oscillations of the wedge using the displacement vector (x, y )e-i( z+t) , where the index i runs over the numbers 1, 2, 3 coordinates x, y , z respectively (see Fig. 1). of motion of the solid is
ij,j

+ s 2 ui = 0,

(A 1)

where

ij

is the stress tensor. For an isotropic solid ij = E E i,j uk,k + (ui,j + uj,i ). (1 - 2 )(1 + ) 2(1 + ) (A 2)

Here we use the notation implying summation over the repeated indexes, i,j is the Kronecker's delta. Symbol ", i" denotes differentiation. Boundary conditions corresponding to the free boundary are ij nj = 0, (A 3)

where nj is the vector of interior normal to the boundary. We seek an antisymmetric solution with respect to the plane x cos(/2) - y sin(/2) = 0, that decreases in x direction. We are interested in waveguide modes decreasing exponentially, but the results obtained can be generalized to some other cases. (b ) Derivation of functional equations The procedure described below is similar to that performed by Shanin (1997) for a wedge with the apex angle close to . Let us apply the second Green's formula to the problem: [ui ij nj - ui ij nj ]ds = 0, (A 4)

where ui is an arbitrary solution of (A 1), ij is corresponding stress tensor. Integration is performed over the surface including a large fragment of the wedge. Let ui be a plane wave travelling from the edge, with the z -component of its wave vector. The integration in (A 4) can be performed along lines that are pro jections of the sides onto the plane z = 0. There are three bulk modes in a solid. Each of them can replace ui : 2 ikT A1 = - i s 2 - 2 ik T A2 = - i s 2 k 2 kL - k 2 -
2

ei(

kx+y

2 kL -k2 - 2 + z )

,

(A 5)

2 kT - k 2 - k 0

2

e

i(kx+y

2 kT -k2 - 2 + z )

,

(A 6)

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16 2 ikT A3 = - i s 2 where

A.Shanin & V.Krylov k 2 kT - k 2 - 2 2 - kT
s

2

ei(

k x+ y

2 kT -k2 - 2 + z )

,

(A 7)

(1 + )(1 - 2 ) , E (1 - ) 2(1 + ) 2 . k T = 2 s E To provide the convergency of the integral in (A 4), we chose k = k + ik , where k is positive and k is small and positive. having positive imaginary part. The branches of square roots are chosen to have positive imaginary parts. Let ui (x) be the values of ui on the line y = 0. The values on the other side of the wedge can be calculated since the solutions are known to be antisymmetric. Note that on the sides of the wedge, the terms Aj are exponentials. Introduce the i notation
2 kL = 2

ui (k ) = ^
0

ui (x)e

ikx

dx,

(A 8)

Ї k ~ k

= =

k cos + k cos +

2 kL - k 2 - 2 sin , 2 kT

(A 9)

- k - sin .

2

2

Note that k is the pro jection of wavenumber of ui to the side of the angle coinciding Ї ~ with x - axis; k and k are pro jections of a wavevector to the other side (see Fig. 3).

Ї ~ Figure 3. To the definition of k and k

After calculating ij and performing the integration A and A3 respectively: Ї^ Ї Ї^ Ї l1i (k )ui (k ) + l1i (k )ui (k ) - 2l12 (k )u2 (k ) ^ ~)ui (k ) - 2l22 (k )u2 (k ) ~ ~^ ~ l2i (k )ui (k ) + l2i (k ^ ^ ~^ ~ ~^ ~ l3i (k )ui (k ) + l3i (k )ui (k ) - 2l32 (k )u2 (k ) ^
2

in (A 4) we obtain for A1 , = 0, = 0, = 0, (A 10) (A 11) (A 12)

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Slender elastic wedge in liquid where
2 2k kL - k 2 - 2 ; - k 2 + 2k 2 + 2 ; lij (k ) = T 2 2 kT - k 2 - 2 ;

17

2 2 kT - 2(k 2 + 2 ); 2 kL - k 2 - 2 2 - k2 - 2 ; 2k kT k 2 2 2 2 (kT - k 2 - 2 ); (2 2 - kT ) kT - k 2 -

2

Equations (A 10)(A 12) are the functional equations for antisymmetrical cillations of an arbitrary elastic wedge. They are exact. These equations must completed by the restrictions concerning the behaviour of functions ui (k ) in ^ complex plane. As follows from (A 8), the function ui (k ) must be analytic in ^ upper half plane and must decrease at infinity no slower than 1/k . (c ) The limit of thin plates We study the solution of equations (A 10)(A 12) supposing that kT =, where = kL /kT = 1 - [2(1 - )]-1 and 2 . After introducing the notations = k , Ї Їk = , ~ ~k = kL = ,

osbe the the

(A 13)

(A 14)

(A 15)

equations (A 10)(A 12) can be rewritten in the form: 2 Ї - 2 - 1u1 ( ) + 2 - 2 - 1u1 + ^ ^Ї (A 16)

Ї [ - 2( 2 + 1)]u2 ( ) - [ - 2( 2 + 1)]u2 ( ) + ^ ^Ї ^ 2 - 2 - 1u3 ( ) + 2 Ї - 2 - 1u3 ( ) = 0, ^Ї

~ (- + 2 2 + 1)u1 ( ) + (- + 2 2 + 1)u1 ( )+ ^ ^~ 2 ~ - 2 - 1u2 ( ) - 2 ^ ~ - 2 - 1u2 ( ) + ^~ (A 17)

~^ ~ u3 ( ) + u3 ( ) = 0, ^ 2 ~ - 2 - 1u1 ( ) + 2 ^ ~ - 2 - 1u1 ( )+ ^~ (A 18)

~ 2( - 2 - 1)u2 ( ) - 2( - 2 - 1)u2 ( ) + ^ ^~ (2 - ) - 2 - 1u3 ( ) + (2 - ) ^ ~ - 2 - 1u3 ( ) = 0. ^~

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18

A.Shanin & V.Krylov

Suppose that the solution can be written in the form of the following asymptotic series: u1 ( ) = u1 ( ) + 3 u3 ( ) + O[5 ] ^ ^1 ^1 22 u2 ( ) = w( ) + u2 ( ) + O[4 ] ^ ^ ^ 1 u3 ( ) = u3 ( ) + 3 u3 ( ) + O[5 ] ^ ^ ^3 (A 19) (A 20) (A 21)

We expand equations (A 16)(A 18) as power series in using (A 14). Note that equations (A 16) and (A 18) coincide at O(1) and O( ). Therefore, equation (A 18) must be replaced by the difference of (A 16) and (A 18) divided by . At the first order, the system is equivalent to the conditions u1 ^1 u1 ^3 = = - w ( ) + ^ - 1 dw( ) ^ . 2 d dw( ) ^ 2 d , (A 22) (A 23)

At the second order, the equations are satisfied identically. The third order yields:
2 24( u3 + u3 + 2 ) + 12(1 + 2 ) ^1 ^3

du2 ^2 + 8 w + ^ d (A 24)

(7 - 6

2

+ 19 2 )

dw ^ d2 w ^ + 9 (1 + 2 ) 2 + d d d3 w ^ = 0, 3 d du2 ^2 ]+ d

(1 + 2 2 + 4 )

^2 ^3 12[(1 + 2 2 )u3 + u3 + (1 + 2 2 )u2 + (1 + 2 ) ^1 dw ^ + d

4(1 + 2 2 )w + (16 - 6 ^ 3(1 + 4 2 + 3 4 )

2

+ 19 2 )

(A 25)

d2 w ^ d3 w ^ + (1 + 2 2 + 4 ) 3 = 0 2 d d

12[2 ( - 1)u3 + (-4 + 2 - 2 2 )u3 + 2 ( - 1)u2 + ^1 ^3 ^2 ( - 2)(1 + 2 ) du2 ^2 ] + 8 ( - 1)w + (-20 + 19+ ^ d (A 26)

dw ^ 6 (1 - ) - 32 2 + 31 2 ) + (-24 (1 + 2 ) + 2 d 21 (1 + 2 )) d2 w ^ d3 w ^ + (3 - 4)(1 + 2 2 + 4 ) 3 = 0. 2 d d

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Slender elastic wedge in liquid

19

The values u3 , u3 , u2 and du2 /d can be excluded from the equations (A 24) ^1 ^3 ^2 2 (A 26). The result is the equation for w: ^ (1 + 2 2 + 4 ) d3 w ^ d2 w ^ + 6 (1 + 2 ) 2 + 3 d d
2 dw ^ kT - 1)] = 0. 2 2 d

(A 27)

6[(1 + 2 ) + (1 - )(

^^ ^ Thus, we have obtained the equation L[w] = 0 with the operator L in the form (2.8).

References
Hladky-Hennion, A.-C., Langlet, P. & de Billy, M. 1997 Finite element analysis of the propagation of acoustic waves along waveguides immersed in water. J. Sound and Vibration. 200, 519530. Krylov, V.V. 1989 On the possibility of application of geometrical acoustics to the problem of wave propagation in thin elastic wedge. Sov. Phys. Acoust., 35, 294301. Krylov, V.V. 1994 Propagation of wedge acoustic waves along wedges imbedded in water. IEEE Ulrasonic Symp., 793796. Krylov, V.V. 1998 On the velocities of lokalized vibration modes im immersed solid wedges. J. Acoust. Soc. Am. 103, 767770. Lagasse, P.E. 1973 Higher-order finite element analysis of topographic guides supporting elastic surface waves. J. Acoust. Soc. Am., 53, 11161122. Maliuzhinetz, G.D., 1955 Radiation of sound by oscillating faces of an arbitrary wedge. Acoustical Journal, 1, 144164, 226234 (In Russian). McKenna, J., Boyd, G.D. & Thurston, R.N. 1974 Plate theory solution for guided flexural acoustic waves along the tip of a wedge. IEEE Trans. on Sonics and Ultrasonics, SU-21, 178186. Moss, S.L., Maradudin, A.A. & Cunningham, S.L. 1973 Vibrational edge modes for wedges of arbitrary interior angles. Phys. Rev. B8, 29993008. Mozhaev, V.G. 1989 Ray theory of wedge acoustic waves. Proc. Moscow State University (Phys.,Astron.), 30, 4045 (1989) (In Russian). Nayfeh, A. 1981 Introduction to perturbation tecnique. New York: J.Whiley. Noble, B. 1958 Methods based on the Wiener-Hopf technique. Oxford: Pergamon Press. Osipov, A.V. & Norris, A.N. 1997 Acoustic diffraction by a fluid-loaded membrane corner. Proc. R. Soc. Lond. A., 453, 4364. Parker, D.F. 1992 Elastic wedge waves. J. Mech. Phys. Solids. 40, 15831593. Shanin, A.V. 1996 On wave excitation in a wedge-shaped region. Acoust. Phys. 42, 696 701. Shanin, A.V. 1997 Excitation and scattering of a wedge wave in an obtuse elastic wedge close to 180o . Acoust. Phys. 43, 344349. Shanin, A.V. 1997 Excitation of wave field in a triangle area. Proc. of Int. Sem. "Days on Diffraction'97". S.Pb., June 35, 205214. Timoshenko, S. & Woinovsky-Krieger, S. 1959 Theory of plates and shel ls. New York: McGraw-Hill.

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