NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 81-106

SPECTRAL ANALYSIS OF SOUND AND VIBRATION IN THIN VIBRATING PLANES CONTACTING FLUID

A.A. P o k r o v s k i

(Received March 1995)

Abstract. Sound and vibration of thin elastic objects contacting an acoustic medium is considered in the framework of operator theory. For a fluid-loaded homogeneous plate the self-adjoint operator describing the system’s oscilla­tions is derived from the energy. The method of the operator’s construction is generalized for all acoustic systems where fluid loading is essential. The operator thus obtained is a perturbation of the Laplacian in L2 -space, which simplifies its investigation. It is shown that the part of the spectral equation that describes vibrations is a linear stationary dissipative system and, there­fore, defines an operator-valued characteristic function with positive imaginary part.

The operators for infinite homogeneous elastic boundaries contacting half­space of acoustic medium in Rn are analyzed. It is proved that such an operator is the direct integral of the operators for the one-dimensional quantum particles interacting with zero-range potential with internal structure. Operators of this type are well studied, which makes it possible to perform complete spectral analysis of the system’s operator. The theorem of decomposition on the set of surface and scattered waves is proved, and the resolvent of the operator is obtained in the Fourier representation.

The results obtained are applied to the spectral analysis of oscillations of infinite membranes and plates (in both classical and Timoshenko-Mindlin models) contacting fluid.

1. Introduction

The present paper is a direct continuation of the previous author’s work [1, 2, 3]. We Consider acoustic problems with interaction of sound and vibration as a part of the theory of self-adjoint operators in mathematical physics [4, 5, 6]. In so far such systems were usually studied in the framework of the diffraction theory [7, 8 , 9], whereas we prove theorems about spectral properties of the correspond­ing operators. We formulate the general method of the operators construction and perform their spectral analysis for homogeneous systems. There also appear formal analogies with the theory of point interactions in quantum mechanics.

In Section 2 we derive the operator for a fluid-loaded infinite uniform plane from the expressions for the kinetic and the potential energy (we follow the ideas of Prof. V.M. Babich [10]). We consider the kinetic energy (on smooth functions) as the norm that naturally induces the Hilbert space for the system in question. Then we consider the potential energy as a quadratic form on this space. If the quadratic form is closable in this space (which requires separate discussion) then the general operator theory implies that there exists the unique operator corresponding to the

1991 AMS Mathematics Subject Classification: 35A05, 76Q05, 35Q.

82 A.A. POKROVSKI

quadratic form. The operator is self-adjoint in the Hilbert space and describes the oscillations of the system. Thus a formal mathematical procedure (derivation of the operator from a closed positive quadratic form) becomes physically important and gives rigorous formulation for the variational principle.

Of course, we can invert the order of the operators derivation; it is even more convenient in suitable coordinates. We can define the basic Hilbert space from potential energy. Then we consider the kinetic energy as the quadratic form of the operator for the system. It turns out that the operator obtained by this method is the simplest one - it acts in L2 space - while the first method leads to the Dirichlet space. We prove that the Z/2-operator coincides with the one obtained in [2] from the concepts of linearity and self-adjointness.

In Section 3 we generalize our method of operators construction for arbitrary systems with sound and vibration. We prove that the spectral problem for the operator in the subspace of functions describing vibrations is a linear stationary strictly dissipative system [1 1 ] and its characteristic function is uniquely defined by the operator of free vibrations. The results of the first two sections were partly presented in [2].

In Section 4 we analyze the operators for infinite homogeneous elastic planes contacting half-space of fluid in R n. Due to the translation invariance it is possible to decompose the operator into the orthogonal integral of one-dimensional operators depending on the tangential wave vector. Each of these operators formally describes a one-dimensional quantum particle interacting with a zero-range potential with internal structure [12, 13]. We use them in the algebraic form [14] rather than in terms of the extensions theory [12 ].

The parallel with the explicitly solvable quantum mechanical problems allows us to carry out detailed spectral analysis. We prove that the spectrum of scattered waves is absolutely continuous with constant multiplicity and fills the positive semi­axis. Analysis of the spectrum of surface waves is reduced to calculation of roots of the dispersion equation. We obtain in explicit form the surface and scattered waves and the reflection coefficient. One of the main results is the theorem of de­composition on the orthonormed set of surface and scattered waves. The theorem gives rigorous foundation for the generally used Fourier method of transition from the time-dependent setting of a problem to the stationary setting and vice versa. We also obtain the operator resolvent in the Fourier representation. The resolvents were studied extensively in terms of the stationary wave field caused by a point harmonic force [8 , 9], so we omit discussion of their properties.

In Section 5 we apply the results to the spectral analysis of a fluid-loaded plate (in both classical and Timoshenko-Mindlin models) and membrane. It turns out that the spectrum of surface waves (in both systems) is absolutely continuous with constant multiplicity and fills the semiaxis [0, +oo) (Theorem 9). The spectrum of surface waves for Timoshenko-Mindlin plate contacting fluid is more complicated and its multiplicity essentially depends on the parameters of the plate, such as density and cylindrical rigidity (Theorem 10).

Formulation of the scattering theory for homogeneous systems makes it possible to interpret head waves (or side waves, well known in seismology [15], radiophysics and acoustics [16]) in frames of Lax-Phillips scattering theory [5]; moreover, new

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 83

properties of resonances emerge for systems with head waves. This result will be presented in a separate paper [17].

2. Derivation of the Operator from the Energy Form

The procedure of the derivation of the operator is based on the following obser­vation concerning oscillatory mechanical systems. Let us consider a system with finite number of coordinates <&, i = 1,... , N. The term “oscillatory” indicates thatthe equations of the system’s dynamics are linear with respect to the coordinates»2qi and accelerations (ji = -gjffr. Therefore, the energy of the system may be writtenas

1 N 1 N dqE — 2 'y > QijQiQj + 2 y > ^ijQiQji Qi — > ( 1)

*>j= l i , j= l

where the first item is the potential energy and the second item is the kinetic energy of the system. Both these items should be non-negative for all qi, qi ^ 0. Therefore, the real matrices A = {d i j^ j -i and B are symmetric. We also suppose that the matrix B = { 6 } f j =1 is strictly positive. The new ( “kinetic” ) scalar product for x = is introduced as

N

2 / v -tjX{Xj, i,3= 1

so that it is positive and nondegenerate. Using this scalar product, the system’s energy can be written as

E = [B~1Aq,q]B -\-[q,q]B, (2)

so that the first item is the potential energy. The variational principle in this form gives us the evolution equations

= B ~lAq' (3) which is equivalent to the equation obtained directly from (1 ):

= Aq. (4)

The point is that the energy written in the form (2) (in terms of the “kinetic” scalar product) leads to the operator B ~ lA , which is symmetric with respect to the scalar product:

[B -lAx,y\B = {Ax,y) = (x,y)= (B ~ xB x , Ay) = (B x , B ~u Ay) = [x, B ~lAy]B,

where (x ,y ) = J2iLixiVi- Thus each scalar product induces the operator of evo­lution symmetric with respect to this product. This is true for an arbitrary scalar product, but the scalar product, associated with the kinetic energy, leads to the evolution equations where time derivatives do not mix with superposition of coor­dinates, as it was in (4). It is worth noticing that the same is true if we choose

84 A.A. POKROVSKI

the scalar product from the potential energy and swap the coordinates and the velocities, Qi = qi, Q{ — <&:

1 N • 1 NE = — ^ " dijQiQj + 2 ^ > bijQiQj

i,j—1 M=1

where [x, i /\a = (Ax, y ) = aijxiVj- The evolution equation obtained fromthe variational principle has the form

Q = A ~lBQ, (5)

where operator A ~lB is now symmetric with respect to the “potential” scalar product [ , ]A.

Thus to obtain the simplest form of the dynamical equations one should hold fixed the “kinetic” or the “potential” scalar product (maybe, with non-degenerate change of variables of the type Qi = 'Yhij Uijqj, which leads only to the transfor­mations A I—> UAU~l , and B UBU ). This approach gives us simultaneously

(a) the operator that governs the evolution, [as in (5) and (3)] and(b) the Hilbert space where the operator is symmetric.The discussed ideas, being trivial for discrete systems, become meaningful for

continuous mechanical systems, in particular for problems of hydroelasticity. To avoid unnecessary complexity, we consider the simplest systems with interaction between sound and vibration: a fluid-loaded infinite elastic plate. The method may be applied to various systems with sound and vibration after evident modifications; its generalization is presented in the next section.

Let us choose the coordinate system (x ,y,z) in R 3 in such a way that the plate is situated in the plane z = 0 and the fluid fills the upper half-space ft = R \ = {(x ,y ,z ) (E R 3 \ z > 0}. Let uS ^(x, t), x — (x, y, z) denote the displacement of the small part of liquid situated at the point x in the moment t. Let w^°\x,y,t) denote the normal displacement of the plate at the point (x, y, 0) in the moment t. Here “displacement” evidently means “the small displacement from the state of equilibrium” . The equations of the evolution of the system, where sound and vibration are isolated, are

c2grad div D V 2w ^ = w^\

where index “t” denotes the partial derivative with respect to time, c2 is the speed of sound in liquid, M is the spatial density of the plate, D is the cylindrical rigidity of the plate, V = <92 + d 2 is the Laplacian in dQ, = R 2. The plate and the liquid are supposed to be homogeneous, so that their parameters are constant. The system’s energy E (without interaction) is given by

( E k — 1/2 f M \w f\2 dxdy + 1/2 f p\u{0 \2 dxdydz E = Ek + Ep, (6)

{E p = 1/2 f D\Vw^\2 dxdy + 1/2 f pc2\divu^\2 dxdydz,

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 85

where p is the density of the liquid. The energy form is defined for sufficiently smooth (C°°) functions. For convenience we introduce the new coordinates as

u = y/pu^Q\ w = \/Mw^0\

The energy of the interacting system is given by the same expression as for the non­interacting one (6), but it is defined only on the functions satisfying the condition of contact

if i. = u(0) = ^(0) =y / p Z =+0 Z Z=+0 s / M '

The Hilbert space, induced by the kinetic energy (6), is

L2(R l - c 3) ® L2{R 2). (7)

Its elements are the two-component vectors f Uw(xy) ) • space we have thequadratic form Q associated with the potential energy

Q ivw| dxdydz + w |2 dxdy, (8)

defined on smooth functions from the space (7) such that

uz(x, y, + 0) = J — w(x, y) for all (x, y). (9)

As far as the quadratic form (8, 9) is known, we should find the operator correspond­ing to this form. A semi-bounded quadratic form uniquely defines the corresponding self-adjoint operator if and only if the form is closed [18, 19]. Thus we should close our form from smooth functions and then obtain the operator from the closed form, which we denote by Q. The closure of a semi-bounded densely defined form always exists; however, we should check whether the condition of contact (9) holds on the domain of the closed form. In other terms, the domain of the closed form consists of the limits of all sequences of smooth ( ) satisfying (9) that converge to an element U = ( ^ ) from L2(R\ —> C 3) © L2( i l2) so that

Q

We should check the validity of (9) for all these U.

Lemma 1. The closure of the quadratic form Q (8,9), which we denote by Q, does not preserve the boundary condition (9).

Proof. We should find at least one vector from Dom(Q) for which the condition (9) is broken. Let us make use of the decomposition

unQ

uwn w

M R + C 3 ) — Wgrad © W rot j (10)

where 7igra.d is the closure of vectors u = grade/?, ip € Cq°(R \) in L2-norm, HTot is the closure of vectors u = rot^, ijj € Co°(JR .). Evidently HIOt € Dom(Q); on the other hand, for an arbitrary function from HTOt its trace on the plane 2 = 0 is not correctly defined, so that the condition (9) looses its sense for such vectors. □

86 A.A. POKROVSKI

We see that purely mathematical effects prohibit rigorous formulation of the contact conditions in the proposed form. The drawback has clear physical inter­pretation. The curl-like motions are prohibited in ideal fluid, since their potential energy is formally equal to zero according to (8); on the other hand, they are present in Dom(Q).

It turns out that elimination of such motions enables us to define the system’s operator on the functions satisfying boundary conditions. Let us consider the qua­dratic form (8) on infinitely smooth functions from "Hgrad ® L 2(R2)- We denote the new quadratic form by Q grad?

QgradW

I f 1 f D— 2 c2\dWu\2 dxdydz + - / — \Vw\2 dxdy,

Dom(Qgrad) = i ( “ ) : u <E C°° n ftgrad, w 6 C°° n L2(R 2),u2z = + 0

Lemma 2. The closure of the quadratic form Qgrad preserves the contact condition(9). The following estimate holds on Dom(Qgrad)-

J f L=+o dxdy - const( / ldivfll2 dxdydz + / \u\2 dxdydz'j .

Proof. The inequality may be found in standard textbooks [20, 21, 22]. It implies that the boundary values are Z/2-continuous on the domain of the form. □

Theorem 3. The operator R corresponding to the quadratic form Qgrad is given by

—c2grad divuR‘ u

ywJ \ c2 (div u)\ z = + u

The operator is defined on vectors ( “ ) € Hgra.d © L2(R2) such that

+ v ' D^z = + 0

* ( ! ) 6 ttgrad ® nz = J f a . (11)\w;

The operator is self-adjoint and positive in Hgra.d ® L2(R 2).

For the sake of brevity we omit the proof, which includes integration by parts and which is analogous to the calculations in [1]. Existence of the left part in the contact condition follows from Lemma 2. The property (divw)|^=+0 e L2(R 2) follows from the embedding theorems [20 , 22].

Operator R is essentially the same as in [23]. Here the additional smoothness u\z_ +Q e W2 {dCl) follows from the condition of contact in (11).

Acoustic problems are usually formulated in terms of scalar functions - the acoustic potential or the acoustic pressure - instead of the vectors of displacements. It turns out that it is possible to obtain the evolution operator in these coordinates. Let us introduce the new coordinates ip, £ as

grad^ = — u[°\ ipt = Cy/p divw^°\ (12 )

PC = ^ » t<0), C« = S D V w W , (13)

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 87

so that ip is the acoustic potential in fluid and ( is its analogue on the plate. In these coordinates the energy of the system (6) becomes

E — Ek + Ep,' E k = \ f jtf\'D(\2dxdy + \ f c2\gradip\2dxdydz

.EP = \ f |CtI2 dxdV + \ f M 2 dxdydz.

Now it is natural to choose the basic Hilbert space H induced by the potential energy,

H = L2(R 3) © L2(R 2) = L2(fi) © L2(dQ). (14)

We obtain the condition of contact in new coordinates taking the derivative with respect to time from the condition (9):

c dip y/p dz = d t { ui° ) L +o ) = d t w m = } w v cz=+0

The operator of the system should be obtained from the quadratic form of kinetic energy, which may be written, taking into account the condition of contact, as

2

Q'ip l r M e2 dipc. 2 Jdn P dz z=+0

dxdy + i / / | grad# dxdydz. (15)

For this quadratic form, initially defined on C°°” functions, the condition of contact is broken after the closure. However, it is possible to define the operator whose quadratic form coincides with (15) on the domain of the operator. In other terms, the contact condition is valid for functions from the operator’s domain, but not on the domain of its quadratic form. Thus the correspondence of the operator to the energy is weaker than for R ; we will also check the correspondence in the stationary setting.

The operator, which we denote by £, is given by

—c2

| )+ /3D2d ’(16)

where A denotes the Laplacian in fluid, A = d2 + d2 + d2, whereas the Laplace operator on the plate will be denoted hereafter by V, V = d2 + d2. The operator acts on the domain

Dom(£) == { u e n \ j c u e n ,a oz z=+0 z=+0

+ (17)

Theorem 4. The operator C is self-adjoint and positive in the Hilbert space (14). Rs quadratic form

Q = c2 / |grad |2 dxdydz + / lot (ip ) + y/{3T>(J n J d fi I z = + °

Dom (Q) = W } ( « )© W ! (0 n ) ,

dxdy, (18)

coincides with the kinetic energy (15) on Dom(£).

88 A.A. POKROVSKI

Proof of the theorem may be obtained using integration by parts, following sim­ilar calculations in [1 ], where the procedure is presented in details for the two- dimensional case. Comparison with (15) gives the values of a and /?:

/» = £ . N 2 = c ^ , (19)

so that the boundary condition in Dom(£) provides the transition from (18) to (15). □

Correspondence of the operator £ to the physical system in question can be verified without consideration of the energy form [1 ]. Indeed, the equations of the system’s oscillations in stationary setting have the form [7, 8 , 24]

c2 Aip(x) + cj 2ip(x) = 0, x e O , (20)

/ 2 M uj2\{ V — J

M cj2\ dip puj2 + - — ip

D= 0 . (21)

z = + 0z — + 0

Theorem 5. The spectral problem for the operator C with a and (3 given by (19),

is equivalent to the equations (20,21).

Proof of the theorem is evident. The Helmholtz equation corresponds to the first line in (22). The boundary-contact condition (21) follows from the second line in (22), taking into account the boundary condition from the operator’s domain. □

It is worth noticing that the functions from Dom(£) have the additional smooth­ness of the normal derivatives on the boundary, which does not follow from the embedding theorems [20, 21, 22]. Indeed, for G Dom(£) we have

dipdz

€ Wi(dQ),ail

while ip G W 2{0,) implies only dz^\dU G L2(dQ). This property is of special interest, because, as it follows from (2 1), dzi \ ) is proportional to the normal shift of the plate.

It turns out that the second component £ of a vector-function from H has no direct physical interpretation (such as normal displacement of the plate or its ve­locity). However, the same is true for the acoustic potential, which the variable ( formally resembles [25]. Moreover, the displacement can be easily found if ( is known.

3. The General Scheme of the Operator Construction

Let us consider vibrations of an arbitrary fluid-loaded (n—1)-dimensional smooth structure (shell, plate, etc.) that coincides with the boundary dQ, which divides R n into two separated parts. The fluid fills the interior of Q c R n. Both Q and

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 89

dft may be non-compact. When the fluid is absent, the free stationary vibrations of the elastic object with frequency u> are described by the equation

— u2 ( r j , A > 0, (23)

where A is a positive differential operator on dfl, w is the transversal shift of the elastic structure; (j) is the set of functions on <9ft that corresponds to all other degrees of freedom of the structure. Evidently $ is the tangential part of the vector- function ( ^) that describes the small displacement of the structure from the state of equilibrium. The division into the transversal and the tangential parts implies the decomposition

A = ( T . ? ' ) , (24)A n ^12 ^21 ^22

so that index “1 ” corresponds to the transversal part, index “2” corresponds to the tangential part. We suppose that the component A n acts in 1*2(00,)] therefore, the Hilbert space 7iv (index ‘V-’ denotes ‘vibration’) where the operator A acts is the orthogonal sum:

H v = L2(dft)©Ht, (25)

where the first item is the space of the transversal functions, the second item is the space of tangential functions on dft. The explicit form of operator A for various models of shells and plates may be found in [7, 8].

When the fluid is present, the acoustic pressure u(x) in the interior of ft satisfies the Helmholtz equation

A u(x) + uj2u(x) = 0. (26)

The boundary condition that connects the values of u(x) and dUQ ^ for x € <9ft is generated by the conditions that

(a) the value |9n coincides with the transversal shift w (here p is the density of the fluid), and

(b) — vu\d^ is the value of the normal force applied to the surface (u is the coupling constant, u = M _1, where M is the spatial density of the elastic boundary).

These conditions imply the following equation on u and on dft [7, 8 , 24, 25]:

(A - oj2) j ^ dn *> ) + ( VU « > ) = 0. (27)

After the elimination of the vector 0 from the last equation we obtain the explicit condition that connects u\QQ and §^\dn-

( A n —uj2)— 2 — ^ 12(^22 — ^2)- 1^ 2i — 2 £) + uu ~ 0- (28) puj2 o n dci pco* o n dn dn

90 A.A. POKROVSKI

The tangential component 4> is given by

0 = —(A22 — u 2) 1 A2\— o — (29)pu* dn\an

Let us find the operator that is equivalent to the boundary-contact problem in the form (26-28). The equivalence is formulated in terms of the reduced form of the boundary-contact condition (28) instead of the direct conditions (a), (b) and (23). Thus only the sound waves in fluid are the same for the constructed operator and for the initial problem given by (26), (23) and (a), (b). However, the solution of the initial problem can be easily obtained from the solution of the reduced one, using the relation (29). Therefore, investigation of the operator is equivalent to the investigation of the initial problem. Just the reduced form (26, 28) is commonly used in practice when one searches for the sound waves in fluid.

Theorem 6. For each boundary-contact value problem of the form (26), (28) with a positive differential operator A (24) there exists equivalent positive operator. The equivalence means that the spectral problem for the operator for all non-positive uj2

implies the equations (26), (28).

Proof. Let us construct the operator and demonstrate its equivalence to the initial problem. The global operator, which we will denote by £, acts as follows:

/ —c2Au(x) \

where A = aa* = a*a is the normal factorization of A, a is the coupling constant, c is the speed of sound in fluid, J[u] is the restriction of u on dQ. The operator C acts in Hilbert space

H = L2( ft)©Wv , (31)

where H v is given by (25). The operator C is defined on the vectors U = (jl'j E Hsuch that CJU E 'H and holds boundary condition

r'2 du -*= aJ[u] + a^w + aj 2<t>- (32)

aa\

ca dn dn

Here an, ai2 are the blocks in the decomposition of the operator a with respect to the transversal and the tangential shifts (25):

a = { «11 ^12^\a2i a22 J

The complex coupling constant q / 0 participates in the boundary condition in the way providing self-adjointness of the operator C. It is clear from the integration by parts (following [1 ]) that the quadratic form of the operator C (we denote it by Q) is given by

2Q\U] = c2 J |grad' 12d3x + aJ[u]

0 (33)

Dom(Q) = W ^fi) ©Dom(a*),

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 91

where || ||y denotes the norm induced by the scalar product in Hv- The quadratic form is closed because the operator J : W^O) — L2(dfl) is continuous [20, 22]. Thus the quadratic form (33) is closed and positive, implying self-adjointness and positiveness of the operator C.

Now let us prove the equivalence to the equations (26, 28). Consider the spectral problem for the operator C:

£U = XU, 3A ± 0. (34)

The Helmholtz equation (26) follows from the part of the equation (34) in L2 (O) if A = u,2. For the components in Hv, taking into account boundary condition (32), we have

aJ[u] 0 + a = a dn

_a21w a22 _Applying operator a* on the both sides of the last equation, we obtain

c2 du a dn = A

c2 du a dn = Xa*dn

\a21w "I" a22

The last formula may be rewritten as

= Ac du a dn — aJ[u]

dna2\W + a22$

[ A - X I )c du Xa dn dn + aJ[u]\ _

0 0,K(a2iW + a22(p)X 1>

and, introducing the function ip = a(Xpc2)~l (a21w 4- a22(p), we can write it as

( A - X I )1 du

pX dn

*P

dn 4-( * ? ) -

0,

which evidently coincides with (27) (and, therefore, with the condition (28)) if cj2 = A, |o:|2 = vpc2. □

It is worth noticing that for all U € Dom(£) we have 6 Dom(^41/2).This additional smoothness does not follow from the embedding theorems [20, 22], which provide only € L2(dCl).

The spectral equation for operator C in the space Hv = L2(fy © Ht may be interpreted as a stationary linear dynamic system [11]. Indeed, the equations have the form (x(A) = / 0°° e~xtx(t) dt, x(t) € H v , ^X > 0)

■AC(A) = AC(A) + aaJ[u(A)],(?_ du(X) I _a dn = cm(A) 4- [a*C(A)]±, aa* = a*a = A.

dn

Let us introduce the input (—) and the output (+) vectors <f>± as

au dn, du dn dn

= a(f>+ • (35)

92 A.A. POKROVSKI

Then ( is the vector of the internal state and equations (35) have the standard form

f- AC = - A C - aj[<t>-),

\ (P + — + 0 - •

Using for (36) the results from [11],we obtain:

Lemma 7. The linear system (36) is strictly dissipative with respect to the input and output vectors 4>±. The characteristic function is

0(A)0_ = [</>. + [a*(A + \)-1aJ[(l>-]}±Y

The standard matrix that controls the degree of dissipation [11] has the form

(0 0 \\0 A - 1/ 2aP±a*J ’

where P± is the orthoprojector in H y = L2 (dCl) ® 7it onto L2(dfl). The operator- matrix is strictly positive, thus providing the dissipative property of (36).

The choice of the operator a for a given A > 0 is non-unique; it is governed by the condition of the correct description of discontinuities (such as cracks) on dCl. The discontinuities also appear if the vibrating structure occupies only a part of dft. However, at least one operator a always exists: a = A1/2. For the classical model of the plate we have A = Tit = 0, and, evidently, a = —yjD /M V(with suitable boundary conditions, if the plate is only a part of dfl). In frames of this model of plates the operator a = a* is differential. For a fluid-loaded plate in Timoshenko-Mindlin theory [26, 27] the factorization in terms of differential operators is possible only at certain values of physical parameters of the plate. In the general case the factorization A = aa* = a* a for a Timoshenko-Mindlin plate leads to a pseudo-differential operator a.

The following theorem gives a simple proof of the condition of discreteness of the spectrum of finite resonator [23].

Theorem 8. Suppose that is compact and the embedding dCl is a C3-smooth surface. Then the spectrum of the operator C is discrete if and only if the spectrum of the operator A is discrete.

Proof. The operator’s spectrum is discrete if and only if the set of vectors, satis­fying

||W||2 + Q[U] < 1is compact. Consider an arbitrary sequence {Un} from this set. As it follows from the embedding theorems [20 , 2 1 ] we can choose a subsequence Uk = such that its spatial component and u\Qn converge in L2(0 ) and in L^dQ), respectilely. Using the expression for the quadratic form (33) we infer that the Hy-component of Uk is bounded in ^4-norm

II0H2 = ( A jJ ) .

Therefore, we can choose a subsequence from Uk such that its second component also converges in H y- □

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 93

4. The Operators for Plane Infinite Vibrating Boundaries

Let f2 = R ” = {(x,y) = (x\,x2, . . . ,x n- i ,y ) 6 R n : y > 0} denotes the half­space filled with fluid, and let dfl = R n_1 denotes its boundary. We consider the operator C in

H = L2{Cl)®Hv , (37)where Hy denotes the space of ra-component vector-functions on dfl that describes small displacement of the boundary. We suppose that operator A > 0 in Hv describes the free oscillations of the boundary and that Hv may be decomposed as

Hv = L2(dQ) © Ht, (38)where functions from L2 give the normal shift of the boundary. If the boundary is uniform, A commutes with the (n — l)-parameter unitary group of translations ((x) —> ((x+h ), ((x) G Hv- When the boundary is contacting fluid, their dynamics is governed by the operator

a i ^ c (n(x, y ) \ ( \

C(x) J \aaJ[u](x) + A£(x)J(39)

where u € L2(Q), c2 is the speed of sound in fluid, a is the coupling constant, J[u](x) is the restriction of functions u e W2 (tl) onto Hv given by

/M M = (40)

according to (38). In (39) a gives the normal factorization of A, aa* = a*a = A. The operator is defined on vectors U £ H such that CJU € H and

c2 dua dy

= audfl dCl

(41)

The coupling constant a is defined from the system’s parameters as

|a|2 = ( J i1 1 M

where p is the density of the fluid, M is the mass per unit for the upper layer of the boundary.

For the classical model of the plate Hv = L2(dQ,),r ! n 2n—1

E d2dx?A = (3

For the membrane Hv = L2(dQ),

A = cl

(3 = const > 0. (42)

n—1Y —h dx<

where cm is the speed of propagation of vibrations.For the Timoshenko-Mondlin model of plate [25, 26, 27] the equations are

presented in the separate section below. The equations for multilayered plates have a similar form [28].

94 A.A. POKROVSKI

4.1. Analysis of the Operator.

The main feature of the operator £ is its unitary equivalence to the orthogonal integral of operators for one-dimensional quantum particle interacting with zero- range potential with internal structure. These operators were first introduced and studied by Pavlov [12, 13] and applied to the investigation of resonance phenomena in complex systems [13, 29, 30]. In this Section we omit proofs of the theorems, which follow from the known results in [12, 13, 19] applied to the operators in question.

Since the boundary is homogeneous, £ commutes with translations u(x,y) —> u(x + h, y), ((x) —► C(x + h), h e it” -1 . We introduce the Fourier transform Fy in H with respect to the variable x:

fu(x ,y)\ _ fu(k,y)\ , c111, CM ) ~ V C(*) ) ' k’x£R - (43)

where

cm =The operator A in the Fourier representation acts as matrix-function A(k) = F^AF^1. We suppose that the scalar product in H v is also translation- invariant and, therefore,

<CW(*)> = / <C»,C>))(fc) dh,where ( , )(*.) denotes the scalar product in C m induced by the scalar product in H v for the functions elkxQ with ( e C m. Denoting C m with the scalar product ( ? )(fc) by H y \ we see that A(k) is a hermitian matrix with respect to this product. Both A(k) and are defined for almost all k. According to (38) the first component of H y (corresponding to the normal shift) is C 1, so that for the vector e± = (1,0,... ,0) we have (e*j_,ej_)(fc) = 1 for almost all k. We supposethat purely tangential vibrations are absent, that is, \JAl(k)e± = H y for almost

iall k. We require that a and a* commute with translations and FuaF^ 1 = a(k), Fija^Fjp1 = a*(k) (the conjugation of the matrix a(k) is taken with respect to thescalar product in H y ).

We use the operator for a particle interacting with zero-range potential with internal structure in the ‘algebraic’ form [14]:

Ck = ( - c V '(3') + c2W2u(*') \ (44)\ w \au(+0)a(fc)ei + A(k)£ J

Dom(Ck) = K ^ ) : « S W22(0 ,+ o o ),^ U'(+0) = qU(+0) + (C,?i>W } . (45)

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 95

Ck is self-adjoint in L2(0, + 00) 0 Hy^ [12, 13]. In the following theorem C is decomposed into the orthogonal integral according to the decomposition of Hilbert space H

n = L2(0 ) © H v = J ® { L 2(R\) © n {y ]}dk. ... (46)

Theorem 9. Fourier transformation (43) of the operator C may be represented as the direct integral of operators Ck, associated with (46) :

£ = F|, ( J ® £ kd k j F ^ . (47)

Proof follows from commutation of C with the (n — l)-parametric unitary group of shifts and from the theorems on commutative operators [19].

Thus we need to analyze each operator Ck separately. Its explicit analysis was performed in [12, 13]. Below we reproduce the results of the analysis in our nota­tion, in order to make the paper self-contained.

Lemma 10. The spectrum of the operator Ck consists from the absolutely contin­uous branch aac = [c2|fc|2, +oo) of multiplicity 1 and I (0 < I < m) eigenvalues in the interval (0, c2|fc|2).

Proof. Let Ck,o denote the operator Ck given by (44, 45) with a = 0. It was proved in [13] that the difference

(Ck - z y 1 - (Ck,o - z y 1,

is an m-dimensional operator. Therefore, their absolutely continuous spectra coin­cide and Ck has no more than m eigenvalues. They are non-negative, since Ck > 0. Noticing that purely tangential vibrations are absent, we conclude that the eigen­values are less than c2|fc|2 and coincide with the solutions of the equation

c2V W ~ V c 2 = \a\2R(\,k) = \a\2X((A(k) - A)- 1ej., e±)(k), (48)

which evidently has no more than m roots within (0, c2\k\2). We have

^ * ) - Ag 1<?A ^ f - ? )|a' (49)

where ips are the eigenvectors of A(k):

A(k)$s(k) = A ?(k)rj>a(k), <&(*).&(*0>(fc) = L

All the items in the right part of (49) are non-zero, since e± is supposed to be a generating element for all A(k). The left side of (48) is strictly decreasing for A € (0, -f oo); its right side is strictly growing on this interval for A ^ As(/c), where it has no more than m simple poles. The condition A(k) > 0 for almost all k proves that (48) has at least one solution. □

Everywhere below we will denote the eigenvalues of Ck by {Aj(fc)}^f^ so that Aj(k) < \j+i(k) and N(k) is the total multiplicity of the discrete spectrum.

96 A.A. POKROVSKI

Lemma 11 ([12, 13]). The scattered waves Up^(y) for the operator Ck satisfying CkUp,k = c2(p2 + \k\2)UPtk have the form

f up k(y)\Up,k(y) = \ ? , Up.fc(y) = A(p, k) sinpy + B(p, k) cospy, p > 0, (50)

where

Cp,k

a m = , r - - , (si)\J\a\2R2(c2{p2 + \k\2),k) + {c2p)2/ !

B{p,k) = 7 PC' ■ , (52)y/|a:|2.R2(c2(p2 + |A:|2),A:) + (c2p)2

CP,fc = ~aB(p,k) [A(fc) - c2(p2 + |fc|2)] 1 a(k)e±, (53)and R(p,k) is given by (48,49). The eigenvectors Ujyk of the operator Ck corre­sponding to the eigenvalues {X j{k )}^ ^ given by (48) have the form

_1/2 f uj,k{y)\Uj,k =

where

1 + M 2|[A(fc)-Aj ] - 1 a(fc)e1 |w[2Vj(k) Cj,k

(54)

uj,k{y) = e VjWv, Cj,k = - a [j4(fc) - Xj{k)] 1 a(k)e±, (55)and

Xj (k) = c2m 2 - V j (kf) , Vj{k) > 0. (56)

Let { , ) denote the scalar product in £ 2(0, + 00) © Hy^ and = 6* an or- thonormed basis in C l. Then the mapping

L2(0, + 00) © H P L2(0, + 00) © C NW

given by~ = / (U,Up,k) \

\T,?±l? (v ,u j,k)m t j ) 'is unitary. Operator Ck in this representation coincides with

fc 2(p2 + \k\2) 0 \\ 0 diag{Ai(fc),... , \N(k)(k)})'

Apart from the trivial shift, an operator Ck at fixed k coincides with the operator for a 1 -dimensional quantum particle interacting with ‘breathing bag’ with finite number of states. This operator was introduced and investigated in [1 2 , 13]. The results presented in these papers include also the results of this Lemma, although in different notation.

Below we will denote the spectrum of surface waves by

S . = U w - j,k

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 97

Let us find the resolvent of the operator Ck. We write the solution of the non- homogeneous spectral equation

(C k - z ) ( “ ^ = ( t t y ) € L 2 (0, + 0 0) © «{?> (57)

with given / G in the formr OO

w(y) = / G«(y,y,)/(i/)dy/ + <^,(5*(y))(fc), (58)Jo

r OO

C = / Gz(y/)/(y ')dy/ + ^ » (59)Jo

where gz is a m x m matrix (an operator inLemma 12 ([12]). Solution of the resolvent problem (57) /or 2 £ cr(£fc) given by (58,59) with

Gz{y,y') = G(0)(y,y') + G(? ){y,0)s(z,k)Gi°){0,y'),G*z(y) = --Of[A(fc) - z ] _1a(A:)ej.G(o)(0,y){l + s(2,/:)},Gz(y') = a[A(fe) - z ^ 1 a(k)e±G ^\y',0){l + s(z,k)},

gz = [A(fc) - z]_1

- (a [i4(fe) - z]*1 a(k)e±)(k)a[A(k) - z]_1 a(fc)£LG«0)(0,0){1 + s(z,k)},

where G^\y,y') is the resolvent of the unperturbed problem,,n. e i\y -y '\ y /z/ c2 - \ k \2 _____________

Gz ( y , y') — --------. — — , 5$y/z/c2 — \k\2 > 0,2 Ky y ) elic2\Jz/c2 — |fc|2 V ' 1 1 “

and s(z, k) is the off-shell scattering matrix,

= ic2y/z/c2 - \k\2 - \a\2R(z, k)ic2y/z/c2 — |A;|2 + |o:|2-R( , A:)

Using these results about zero-range potentials with internal structure and the representation (47) one can easily obtain the orthonormed basis of scattered and surface waves.Theorem 13. The orthonormed basis of generalized eigenfunctions for operator C consists of the scattered waves

11 __ ______ \______ A k x r j

P'k ~ (27r )^ "1)/2 P,fc’and the surface waves

U U )* = (27r)(n-1)/2 elkXUj'k'

where Up , Ujtk are given by Lemma (11). The eigenfunctions satisfy

(Up,k,Uf,k‘) = S(p - p ’) «(* - *0 . (Uu),h’u m,k) = 6i w ~ k’)<

(Up,k,U(j),k) = 0, CHp,k — c2(p2 + \k\2)l/Pik, = ^]{k)M(j),k,

98 A.A. POKROVSKI

where Aj(k) are given by (48).

The explicit form of the operator’s eigenfunctions is an essential part of the spectral analysis of the operator. In particular, completeness of the orthonormed basis of surface and scattered waves gives operator theory background for widely used in acoustics Fourier method of transition from the time-dependent equations to the stationary ones.

Remark 14. It turns out that the H-function (49) is connected with the input conductivity (the inverse input impedance, see [25, 31]) Z ~ l in a simple way:

C ^ = — ia2R(uj2,u) sin 0),Z(v,0)

where u> > 0 is the frequency of oscillations, 6 is the angle of the incoming plane wave.

Theorem 15. The branch of spectrum corresponding to the scattering waves is absolutely continuous with constant multiplicity and fills the half-axis [0, +oo).

Sketch of the Proof. This branch of spectrum is generated by subspace HScat, whose elements may be represented in the form

U — j UPtk<t>(p,k) dpcF^k. (60)

As it follows from Lemma 5 and Theorem 7, the partial isometry L2(Rn) —> H, (f) i—► IA given by the relation (60) has the spectral property

c2(|A:|2 + p 2) i-+ C.

Since H scat is by definition the image of L2(Rn) after mapping (60), our operator on this space is unitary equivalent to multiplication by a free variable in an L2-space with absolutely continuous measure.

As it follows from Theorem 15, the singular spectrum can arise only from the eigenfunctions of Ck■ If for each A from the spectrum of the surface waves the set of k such that A(k) = const > 0 has zero Lebesgue measure in R n_1 then the operator C has no point spectrum. The condition providing constant multiplicity and absence of singularly continuous spectrum for the surface waves can be easily obtained for the operators invariant with respect to rotations around Y-axis.

Lemma 16. Suppose that A(k) = -A(|fc|), a(k) = a()fc|) are smooth matrix-functions of parameter Ifcl. If

dXj(\k\) d\k\

for almost all |fc| > 0 from the interval I C R +, then each branch of the spectrum(j j) = IJ Aj(|fc|) has constant multiplicity. To ensure this it is sufficient that

kei^•i?(A, |fc|) < 0 for A < c2|fc|2. In particular, this holds if ^yA(|fc|) is positive.

Proof. Starting with equation (48), we write it in the form

F ( A , W ) = A - 1i J ( A , | f c | ) - ^ p = 0.

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 99

Evidently for A = Aj(|fc|)

dX dF f dFdF f dF

2|a|2Av/|A:|2 — A/c21 > 0

for almost all \k\ >0 . On the other hand,

AM V|A:|2 - A/c2 < 0

if < 0. We can re-write it as

which proves the lemma. □It is worth noticing that various branches <7(j)(|fc|), being of constant multiplicity

by themselves, may overlap one another, thus changing the total multiplicity of the spectrum of the surface waves.

5. Surface Waves for Membranes and Plates (Classical Models)

5.1. Classical Models of Plates and Membranes.

A fluid-loaded membrane is the simplest physically reasonable system with in­teraction between sound and vibration. It was widely investigated in literature [7 , 8 , 9], but, to the best of our knowledge, the corresponding theorems concern­ing the operators’ spectrum were never formulated. We unite investigation of the classical model of the plate with the investigation of the membrane, since methods and results of their spectral analysis are very similar. We omit repetition of results concerning the scattering waves, since their spectral properties were discussed in Theorem 15.

Operator of a membrane’s free vibrations is

and it acts in the space Hv = L^dQ). The operator describing sound and vibration of the membrane coupled with the half-space y > 0 filled with fluid is given by

where A denotes the Laplacian in Q, aa* = a*a = A. It is self-adjoint in the space of pairs U = («C) from H = L2(f2) © Hv, taking into account its domain

—c2Au aau(x, +0) — A(,{x)

(61)

Dom(£) = <U £ H : CU € H a n d -------- = au

100 A.A. POKROVSKI

where a is a complex coupling constant and the boundary condition in the domain ensures the correct coupling of sound and vibration, see [1, 2, 3].

Consider a fluid-loaded plate in frames of the classical theory. Now the operator of the free vibrations is

Ak = 0n—1

L»=l0 > 0

in L2(dQ). The operator C describing the plate coupled with the half-space Q filled with liquid (with speed of sound c) acts in 7i = L2(fl) © L2(dft). It is given by the expression

'u(:r,y)\ _ / —c2Au \C(x) J I Y% = 1 d^.u(x, +0) - A k C(x)) '

(62)

It is self-adjoint in the domain of pairs U = ) from H such that

C (u() £H , = ^ = au\ + dx A x )-v ' a ay y=+o ly=+o •1=1Using the results of the previous sections, we can analyze the operators’ spectral

properties.

Theorem 17. The operators given by (61) and (62) are self-adjoint and posi­tive. Their spectrum consists of the two branches: the waveguide and the scat­tering ones. Each of these branches is absolutely continuous and fills the semi-axis [0, + 00). Corresponding generalized eigenfunctions form the orthonormed basis in H.Multiplicity of the waveguide branch of the spectrum equals 2.

Proof of these results follows from the Theorems 13, 15 and Lemma 16. We see that for the membrane

= c2JA;|2 - A ’and for the plate

To prove the theorem it is sufficient to substitute them in (49) and recall Lemma 5 and Lemma 6. □

5.2. Surface Waves for Timoshenko-Mindlin Plate.

5.2.1. The Operator of Free Vibrations.

We start with the two-dimensional plate contacting three-dimensional fluid. Vi­brations of a free plate in frames of Timoshenko-Mindlin theory [25, 26, 27] are

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 101

given by the equations (the plate is two-dimensional with coordinates (x, y))

ph3 d2* x D~12 dt2 2

ph3 d2Vy DU dt2 2

dw\~ d i)'

(\ -v )V ^ !y + (\ + v) ^ ^

(63)

d2w dSfcph^ = p'h{Vw + * )+ q , $ =dt2 +dx dy

(65)

where w = w(x, y\t) is the normal shift of the plate at the moment t, z^x and z^y are equal to the shift of the plate’s point (x ,y ,z ),h > z ~ 0 in the X — and In­directions, respectively, h is the thickness of the plate, p is the density of the plate, v is the Poisson’s coefficient, pi is the corrected shift modulus, D is the cylindrical rigidity, q is the normal external pressure. We denote by V the Laplace operator on the plate, V — d2 + d2 (above we used A for three-dimensional Laplace operator).

The right part of (63-65) gives the self-adjoint operator, which is analyzed be­low. The operator evidently commutes with translations. It is defined only when dimfi = 3, but it may be re-written in terms of the operations of vector analysis and thus generalized for dimf2 > 3. The equations for multilayered plates have a similar form [28].

Introducing new functions ^ x,y = -A=\|/x y we obtain the equations of motion in the form

' dlw ' ' W "

= c * x

*y.

so that in theby

uLx>p

d_p h dx

3'aj1 d_p h dy

C 3) given

12ph.-

vL JLp h dx

a

adxdy

12ph3

Introducing the vector-function* = ( ! : )

one can write the operator in the form

A q = —aV y/ab(V, •)

-VabV £ 0P - 6 + CV(V,-)

where

a = , 1 2 / / h2 p ’

C = (1 + y) 12 D 2 ph3 ’ Bn =

( I - v ) 12D 2 ph3

(66)

(67)

102 A.A. POKROVSKI

Lemma 18. The operator Ao, given by (66) on Cq°-functions is essentially self- adjoint in L2{R2) © ^ ( R 2 C 2). Its quadratic form is given by

A I WI = J IVa V w 4 - Vb$\2 dxdy

+ B0 J (|V$ X|2 + |V$y|2) dxdy + C J |(V,tf)|2 dxdy.

Therefore, Aq > 0.Proof. Self-adjointness follows from the positivity and closability of the quadratic form on the dense subspace of smooth functions [19]. For such functions the qua­dratic form can be calculated from integration by parts. □

The operator A$ has an invariant subspace of purely tangential displacements, so we should consider it on the orthogonal complement to the subspace to satisfy the conditions of non-degeneracy. The subspace is the set of all vectors of the form

( ~ ° AVV x ’ $ G C 0°° .

On this subspace the operator acts as (—BqD + b). The operator in the orthogonal complement may be written as

aV \fabT>—y/ab BV — 6Atm

where Vx = B = B0 + C. ATM acts in Hv = L2(R2) © HD(R2), where HD is the space induced by the Dirichlet norm

\x \d = J |Vxl2 dxdy.

Lemma 19. The operator A t m on Cq°-functions is positive and symmetric.

P roof of the lemma immediately follows from the positiveness of the operator’s quadratic form in Hv-

A™ ( x ) ’ (x)) = / + 2dxdy + B J |A*|2 dxdy.

□Everywhere below we will use the same notation Atm for the Friedrichs extension

of the symmetric operator [18, 19]. The domain of the closed operator is

Dom(ArM) = j f “ ) e Hv : Vw € L2(R2),T>x e ^ ( f l 2) J .Now it is clear that the operator Atm may be defined in the spaces of higher dimensions n > 2 using the operations of vector analysis.

To apply the spectral theory from the previous sections, we take the Fourier transform of the operator At m • It acts as the orthogonal integral (with respect to k) of matrices

a|fc|2 \/ab\k\2 \ y/ab B\k\2 + b)

ATM(k) =

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 103

in two-dimensional spaces Hy with the scalar product

(fc)’fc))w = H2 + l W - The dispersion equation for the plate in vacuum connects the wavenumber \k\ of a generalized eigenfunction with the corresponding eigenvalue A. For our system the equation

aB\k\4 - X\k\2(a + B) - bX + A2 = 0 (68)is of second order with respect to both A and \k\2 and may be analyzed explic­itly. Although the following results are well-known, we present them to make our considerations independent.Lemma 20. For each |fc| > 0 exist two solutions of the dispersion equation (68)

*±(1*1) = i[(a + B)|*|> + * i { l ± ^ - [(a + g S 4+612 } ■

When | A:| ~ 0, we have

A+ (|fc|) = (> + (a + B)!fc|2 + 0(|fc|4), A_(|/!|) = ^|*:|4 + O P |6). (69)

When \k\ —> oo,A+(|fc|) = |fc|2 • max{a, B } -(- 0(1), (70)A_(|fc|) = |A:|2 • min{a,B } + 0(1),

so that in the high-frequency limit the phase velocities are close to those with 6 = 0. The qualitative behavior of A±(|fc|) is shown on Figure 1.

F igure 1. The dispersion curves for the two types of shear waves in free Timoshenko-Mindlin plate.

Lemma 21. The spectrum of the operator for free vibrations fills the semi-axis [0, + 00). In the interval [0,6) its multiplicity is 1, in the interval [b, -foe) the multiplicity is 2 .Proof. It is easy to demonstrate that A±(|fc|) are monotonous functions; therefore each generates a branch of the spectrum with multiplicity 1. Together with the estimates (69, 70) this property proves the lemma. □

104 A.A. POKROVSKI

5.2.2. The Spectrum of Surface Waves under Fluid Loading.

Now we can analyse the spectrum of surface waves. According to Theorem 13, the spectrum of surface waves (we denote it by Ew) coincides with the set

U U V(W)>j |fc|>o

where (AJ(|fc|)} are the solutions of the equation

c V W 2 - A /c 2 B\k\2 + b - \H2A ~ aB\k\*-\{\k\2{a + B) + b} + \2'

We derive it substituting ArM(\k\) in the dispersion equation (49). It may be re-written as

c V W 2 - A / c 2 B\k\2 + b - \|a|»A [A-A+ (|fc|)][A-A_(|fc|)]’ V >

where A±(|fc|) are the squared frequencies of the plate’s vibration in vacuum with the wave number k. Thus the spectrum of surface waves in the system with fluid loading is closely connected with the spectrum of the plate’s vibrations in vacuum.

Let us analyze the equation (71). Qualitative behavior of its parts (with constant \k\) is presented on Figure 2.

F igure 2. Graphic solution of the dispersion equation for fixed k.

We see that for all \k\ exists the solution that is less than A_(|fc|); we call it the slow surface wave and denote by As (|/c|). Evident estimates show that

Asz(|fc|) —► 0 if |*|->0,

andA.,(|k|)-A_(|k|) = 0 ( - L ) , |fc| _ oo.

If B > c2, then AS/(|A:|) is the single solution of (71). If B < c2, then for sufficiently large \k\ (such that B\k\2 + B < c2\k\2) exist another surface wave. Therefore, the following theorem holds.

SPECTRAL ANALYSIS OF SOUND AND VIBRATION 105

Theorem 22. The spectrum of slow surface waves fills the semi-axis [0,+00). If B < c2, then exists another branch of surf ace waves with multiplicity 1, which fills the semi-axis [6/(1 — B/c2) ,+ 00).

On Figure 2 we present qualitative behavior of the right and the left parts of (71).

Thus the perturbation of the spectrum of surface waves under fluid loading is essential in Timoshenko-Mindlin theory. In real physical systems always B » c2, so that “quick” surface waves are absent.

Acknowledgements. The author is very thankful to Professor B.S. Pavlov, Pro­fessor Yu. A. Kuperin and Professor S.N. Naboko for stimulating discussions. I am grateful to Dr A.V. Badanin, whose questions and suggestions were very help­ful. I am thankful to the referee for careful correction of numerous mistakes and misprintings.

The author thanks the Commission of the European Communities for financial support in the framework of the EC-Russia collaboration Contract ESPRIT P9282 ACTCS. This work was done under partial financial support from International Science Foundation, under Grant NX-1300 and from the Russian Foundation for Fundamental Research, under Grant #93-01-00266.

References

1. A.A. Pokrovski and A.V. Badanin, On the linear operator for a boundary- contact value acoustic problem, New Zealand Journal of Mathematics, 24 (1995), 65-79.

2. A.A. Pokrovski and A.V. Badanin, Lax-Phillips theory for scattering by thin elastic bodies, to appear in Journal of Mathematical Physics.

3. A.V. Badanin and A.A. Pokrovski, On the model of a plate, supported by stiff- ener, Vestnik of St. Petersburg University, Ser. 4, No. 3(18) (1994), 94-97.

4. M. Reed and B. Simon, Methods of Modem Mathematical Physics, Vol. 3, Scattering Theory, Academic Press, New York, London, 1979.

5. P.D. Lax and R.S. Phillips, Scattering Theory, Academic Press, New York, 1967.

6. E.C. Titchmarsh, Eigenfunctions Expansions Associated with Second-Order Differential Equations, V. 1, 2. Clarendon Press, Oxford, 1946 and 1958.

7. S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959.

8. M.C. Junger and D. Feit, Sound, Structures and their Interaction, The MIT Press, Cambridge, Mass., and London, England, 1972.

9. D.G. Crighton, The 1988 Rayleigh medal Lecture. Fluid loading - the interac­tion between Sound and Vibration, Journal of Sound and Vibration 133 (1) (1989), 1-27.

10. V.M. Babich, Stationary oscillations of mechanic systems and spectral Theory of self-adjoint operators, Zap. Nauchn. Sem. LOMI 218 (1994), 12-17.

11. D.Z. Arov, Passive linear stationary dynamic systems, (in Russian), Siberian Math. Journ. 20 No. 2 (1979).

12. B.S. Pavlov, Extensions theory and solvable models, Russian Math. Surveys 42 No. 6 (258), 99-131.

106 A.A. POKROVSKI

13. B.S. Pavlov, Model of zero-range potential with internal structure, Theor. Math. Phys. 59 No. 3 (1984) 345- 353.

14. Yu. A. Kuperin, K.A. Makarov, S.P. Merkuriev and A.K. Motovilov, The alge­braic version of the extensions theory in few-body quantum problem with internal structure, Yad. Fiz (Sov J. Nucl. Phys.) 48 No. 2 (1988) 358-370.

15. E.F. Savarensky and D.P. Kirnos, Elements of Seismology and Seismometry, 2nd ed., (in Russian), Moscow, 1957.

16. L.M. Brekhovskih, Waves in Layered Media, (in Russian), AN SSSR Press, Moscow, 1957.

17. A. A. Pokrovski, Lax-Phillips theory for sound waves scattering by thin infinite elastic planes, to appear in International Journal of Mathematics and Comput­ers.

18. M. Sh. Birman and M.Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, (in Russian), LGU Press, Leningrad, 1980.

19. N. Dunford and J. Schwartz, Spectral Theory. Self-adjoint Operators in Hilbert Space, Academic Press, New York, 1966.

20. O.A. Ladyzenskaya, Boundary Problems in Mathematical Physics, 2nd ed., Nauka, Moscow, 1973.

21. V.I. Smirnov, A Course of Higher Mathematics, Vol. 5 Pergamon Press, 1964.22. S.L. Sobolev, Some Applications of Functional Analysis in Mathematical

Physics, (in Russian), LGU Press, 1950.23. A.G. Aslanyan, D.G. Vassiliev and V.B. Lidskii, Frequencies of Free Vibrations

of a Thin Shell Interacting with Fluid, Functional Analysis and its Applications, (in Russian), 15 (1981), 157-164.

24. H. Henl, A. Maus and K. Westpfal, Diffraction Theory, (in Russian), Mir, Moscow, 1964.

25. E. Skudrzyk, The Foundations of Acoustics, Vol. 1, Springer-Verlag, Wien, New York, 1971.

26. S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Phil. Mag. Ser. G 41 (1921), 744-746.

27. R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plate, J. Appl.Mech. 18 No. 1 (1951), 31-35.

28. B.Q. Vu, Field equation for the transverse motion of a two-layered plate, Jour­nal of Sound and Vibration 99 (2) (1985), 267-273.

29. Yu. A. Kuperin, K.A. Makarov and A.K. Motovilov, An extensions Theory setting for scattering by breathing bag, Journ. of Math. Phys. 31 No. 7 (1990).

30. B.S. Pavlov and A.A. Pokrovski, A solvable model of Mdssbauer scattering, (in Russian), Theor. Math. Phys. 95 No. 3 (1993), 439-451.

31. M.A. Isakovich, General Acoustics, (in Russian), Nauka Press, Moscow, 1973.

A.A. Pokrovski ,Laboratory of the Complex Systems TheoryInstitute for PhysicsSt.Petersburg State UniversityUlyanovskaya 1St. Petersburg 198904RUSSIApokrovs@snoopy.niif.spb.su

andInternational Solvay Institute of Physics Chemistry1050 Boulevard de TriompheBrusselsBELGIUMalexis@solvayins.ulb.aic.be