Acoustic scattering by membranes and plates with line constraints

Journal of Sound and Vibration (1978) 58(3), 319-332

ACOUSTIC

SCATTERING WITH

BY M E M B R A N E S

AND

PLATES

LINE C O N S T R A I N T S F. G. LEPPINGTON

Department of Mathematics, Imperial College of Science and Technology, London SIV7 2BZ, England (Received 13 October 1977) A plane sound wave is incident upon an infinite membrane (or thin elastic plate) that is held fixed along two or more parallel lines. A formally exact solution is found and is investigated in the limiting cases of large and small fluid loading, with emphasis on resonance effects in the latter case. 1. INTRODUCTION AND GOVERNING EQUATIONS Problems of sound waves interacting with flexible surfaces occur frequently and have applications in aerodynamic noise theory and in underwater acoustics. Previous work has included exact analyses for infinite and semi-infinite membranes or plate s [I-5]. For the more realistic situation with finite flexible surfaces, approximate treatments have been given [6-8]; the possibility ofresonance is one ofthe interesting features in these cases. The model problem considered here is the two-dimensional one of a compressible fluid bounded on one side by a plane membrane (or elastic plate) that is constrained along one or more parallel lines. This provides a model for a surface supported by struts, and has the advantage that a formally exact solution can be found. For two-dimensional motions with simple-harmonic time dependence the total velocity potential can be written as Re {q~(x, y) exp (-io)t)}

( I. i)

and the time factor is henceforth suppressed. Cartesian co-ordinates (x,y) are chosen so that the flexible surface has the equilibrium position y = 0, with compressible fluid in the halfspace y > 0. In this region the potential is subject to the harmonic (Helmholtz) wave equation (a21ax 2 + a2/ay ~ + k s) ,i, = o, where k =

y > o,

(i .2)

o)/c is a wave number and c is the sound speed o f the fluid.

The presssure fluctuation p(x,y) and the surface deflection ll(x) are given in terms of by the relations

p = po io)O(x,y),

tKx) = (i/o)) O,(x, 0),

(1.3)

where the suffix y denotes partial differentiation and Po is the mean fluid density. The deflection of a membrane due to pressure loading p(x) is governed by the linearized equation

Ttl"(x) + too)2 'l = p(x, O)

y = O,

where T and m are the surface tension and mass per unit area; hence

(aZ/Ox2 + i t2) Or - coo = 0, y = 0, (1.4) 319 0022-460X/78/0608-0319 802.00/0 9 1978 Academic Press Inc. (London) Limited

320

F. G. LEPPINGTON

where (1.5)

p = t o ( m / T ) ll2

is the wave number for a membrane vibrating hz vacuo. ThefluM loading parameter ct is r = PO PZ] m ----112 k/;

(1.6)

where 2he is the ratio of fluid mass, within a fluid wavelength, to the membrane mass. Heavy or light fluid loading, relevant in underwater or aerodynamic problems, means that the dimensionless parameter e is large or small, respectively. Apart from radiation conditions at infinity, to ensure outgoing scattered waves, one needs to specify the nature of the constraints along the given lines (x,y) = (x,,0), n = l, 2 . . . . . Here the deflection is taken to be zero at these points so that Sy(x,0) = 0

when

x =x,.

(1.7)

For a thin elastic plate under flexural vibrations the boundary condition (1.4) is replaced by (a*lax" - I*~) % + ~ s = 0,

y = 0,

(1.8)

with la~ = moo~/B,

oq = PO Oj2] B = I"l]ke,

(1.9)

where B is the bending stiffness of the plate and m is its mass per unit area. Here the chosen constraints are those of zero deflection and zero deflection gradient; thus S,(x,0) = S,x(x,0) = 0 ,

x=x,.

(1.10)

In section 2 a detailed solution is given for the membrane with two line constraints, at x = +_L, so as to provide the simplest solution that includes resonance effects. The problem

is extended in section 3 to deal with an infinite periodic array of constraints and section 4 concerns the elastic plate with constraints at x = _+L where conditions (1.10) are imposed. In each case a solution is given in terms of a fundamental potential (~b or ~b~) that corresponds to the radiation by a line force. This potential is evaluated in integral form in Appendix A for both membrane and plate, and some approximate results are presented in limitingcases. 2. MEMBRANE WITH TWO LINE CONSTRAINTS A membrane in the planey = 0 is fixed along each ofthe lines ( x , y ) = (_+L,0) so that S,(x, 0) = 0,

x = _+L,

(2.1)

with the total potential S subject to equations (1.2) and (1.4). Suppose thesystem is excited by an incident wave of potential 4 , ( x , y ) e x p ( - i c o t ) ; for a plane wave at angle 0o to the plane, for example, ~ = exp {ik(x cos 0o - y sin 0o)}, (2.2) though the analysis carries through with any incident field. It is convenient to subtract out the wave potential S o that would be the solution in the absence of the line constraints; thus s = So(x,y) + ~,(x,y).

(2.3)

The reflection field So satisfies equations (1.2) and (1.4). For the plane incident wave (2.2), So is readily found to be So = qS, + R exp {ik(x cos 0o + y sin 0o)},

(2.4)

PLATE SCATTERING WITH LINE CONSTRAINTS

321

R = [ik sin Oo(k2 cos 2 0o - p2) _ cQ/[ik sin 0o (k 2 cos 2 0o - 1~2)+ a],

(2.5)

with and r can be regarded as being known in principle for any given incident field. Evidently the diffracted potential $, in formula (2.3), accounts for the effect of the constraints. Now the deflection t/(x) = (i/w) ey(x,0) is given to be continuous (in fact zero) at the constraint points, while the derivative r/'(x) may be discontinuous there. This feature can be effectively incorporated into the boundary condition (1.4) by adding Dirac delta functions on the right-hand side; thus (02/ax2 + p2) $ y _ ct $ = ,41 ~(x + L) + ,4, 6(x - L),

(2.6)

where At and ,42 are (unknown) constants. Integrating across the singular point x - - L shows that `42 corresponds to the jump discontinuity in Sy~(x,0) at x = L, and similarly for At. Now introduce the wave potential ~ that satisfies a radiation condition at infinity, and with (O2/Ox2 + p2) $~ _ c~$ = 6(x), y = 0. (2.7) The function q~ is the potential due to a time-harmonic line force applied to the membrane at the origin. It is evaluated, in Fourier integral form, in Appendix A and can be regarded as known exactly. The diffracted field $ is now readily expressed in terms of~b, for by inspection of expressions (2.6) and (2.7)

r = At O(x + L,y) + `42 c~(x - L,y)

(2.8)

satisfies equation (2.6) and it remains only to evaluate A x and A2. In the derivation of equation (2.8) so far only the fact that Ov is continuous at +L has been used and one can now impose the condition that its value is zero there; hence $,(x,0) = -Ooy(X,0)

at

x = _+L

are the conditions to determine At and A2. It is found that 2At.2 =

{*o,(L, 0) + r 0)} {~'o,(L, O) - *o,(-L, 0)) {tky(0,0) + q~y(2L,0)} + {4,(0,0) - q~y(2L,0)} '

(2.9)

where the + and - signs refer respectively to At and A2. Evidently there is the possibility of even (or odd) resonant modes if the denominator 4,y(0,0) + Sv(2L,0) (or Sy(0,0)- t~v(2L,0)) is small, and this is investigated later in this section. The solution (2.8) is the analogue of that due to Konovalyuk [2] for the elastic plate, though the derivation is somewhat different. It is interesting to compare expressions (2.8) and (2.9) with the corresponding solution r

= *o(x,y) -- [r

0)/~b,(0, 0)1 ~b(x + L, y)

obtained by the same method for a single constraint at x = - L ; evidently expression (2.8) does not follow in a simple way from its single-constraint counterpart. 2.1. FAR FIELD

The formal solution (2.8) for the diffracted potential $ is exact, with At, ,42 and ff'o given by equations (2.9) and (2.4) and with t~ having the integral representation (A7) of Appendix A. In the distant sound field, r - (x 2 + y2)V2 ~ co, the potential $(x,y) has the form of a radially spreading wave t#R together with a surface wave t#s confined near the plane y = 0.

322

F. G . L E P P I N G T O N

Upon collecting together results for the large fluid loading limit (~, --~ oo), formulae (AI2), (A20) and (A29) combine to show that q~ ~ q~o + ~R + ~Zs as

r --+ 0%

(2.10)

+cosOo)] + i(i + l / V " 3 ) c o s [ k L ( c o s O - c o s O o ) ] } ,

(2.11)

where

0.~

12k 2 sin 0 sin Oo exp (ikr - kin) o?/3(2nkr) I/2 x {exp (2i:d/3L)c~176

exp(4i:V~L) + (i + I/V'3) 2 s ~ 4k~ -us sin 0o exp (4-i'y*/s x - ~/Sy) 3 -x/z cos (~va L -T- k L cos 0o) + sin (2=mL) exp (i~*/SL 4- i k L cos 0o) x

exp (4iq'/aL) -t- (i +

1/a/~):

(2.12)

as ~ ~ co, with upper or lower signs according as x > 0 o r x < 0. The variables (r,0) are the usual polar co-ordinates. Corresponding results for small fluid loading (~ --+ 0) can also be written down, by using formulae (2.8), (2.9), (AI2), (AI7), (AI9) and (A28). It is found that the radiating part o f the far field has the form 8k 2/~ sin 0 sin Oo(2nkr) -~/2 exp (ikr - kin) [ik sin Oo(k 2 cos 2 0o -/~2) + a][ik sin O(k 2 cos 2 0 - I?) + a]

•

s n(kLc_osOo)s (kLcosO) I (

1 + e "l"L - M+

+

1 - e "j"L - a L

J'

(2.13)

where ilt I+_ = --rr

f

1 4- e -21sL (s 2 _ k2),/2 (s 2 _ / ? ) 2

ds,

(2.14)

--oo

along a contour that passes just above the negative real axis and below the positive axis, and the square root function (s 2 - k2) 1/2 is defined by equation (A5). In most cases the terms involving~ in the denominators ofexpression (2.13) can be neglected in the limit ~ -~- 0. Exceptions occur when 0 (or 0o) is near a grazing angle (0 or n) or near the coincidence angles cos-~(+_p/k) for It < k. Further, the terms M+ and cz/_ are significant only near the respective resonance conditions/tL = (n + 89 n and/tL = nm The imaginary part o f / + ( o r / _ ) simply adjusts the resonant value o f p by a term of order ~, while the real part of I+ (or I_) gives a measure of the radiation damping. Thus i f p has its resonant value near (n + 89 and if0 = n - 0o is at the backscatter angle, then ~kR~

- 8 k 2 tl sin 2 0o cos 2 ( k L cos 0o) exp (ikr - kin) (2nkr)l/2 {ik sin Oo(k 2 cos 2 0o -/~2) + a} 2 Re I+)

(2.15)

provided cos2(kLcosOo) is not as small as O(a). If 0o is also near the coincidence angle, for example, then the factor cos2(kLcosOo) in the numerator is small and one must include the other term in the curly bracket of expression (2.13). Evidently the radiated potential ~'R can be relatively large near resonance, and varies critically near coincidence angles, where the full expression (2.13) should be used.

323

PLATE SCA'I-I-ERING WITH LINE CONSTRAINTS 3. P E R I O D I C A L L Y

SUPPORTED

MEMBRANE

A similar analysis can be used for any number, m, of parallel line constraints. In such a case there are m constants A1, A2 . . . . . A,, in place of the pair (A1,A2) that appear in formula (2.6), and the m constants have to be determined from the m • m system of linear algebraic equations obtained by applying the boundary condition (2.1) at each constraint point. Although this determines the A~ in principle, an explicit evaluation is not generally feasible when m is large. One special case that can be dealt with exactly is that of a plane wave incident upon a membrane with constraints at x = nL where n takes all integer values from -co to +oo. With such an infinite array there is clearly some question as to the existence of a solution since even a single line force produces surface waves that travel outwards towards infinity without attenuation. Accordingly some dissipation is initially introduced into the system here, by assigning small positive imaginary parts to the wave numbers # and k, which then have the forms IL = lq + i~u2, k = kt + ik2, with 1~2 = k2, and then finally letting k2 -+ +0. The effect is to make the outward travelling waves (A 12), (AI 7) and (A 19) decay exponentially with distance. Because of the nature of the geometry and of the incident wave (2.2) one is led to seek a solution of the periodic form

~(x + nL, y) = cb(x,y) exp (inkL cos 0o)

(3. I)

for any x and any integer n. As before, the potential is written as a sum (2.3) ofreflected wave plus a diffracted potential ff that is subject to the wave equation (1.2) and periodicity requirement (3.1). With constraints at the points x = nL the membrane equation can be expressed as

+1~2 0 , - c t O = A

~ ( x - n L ) e ~L'~176

y=0,

(3.2)

where A has to be found. On comparing equation (3.2) with formula (2.7) for the known function r the solution is identified as

~ ( x , y ) = A ~. qb(x--nL,y)e'"kLc~176 /Ig--

(3.3)

cO

T o determine A, apply the constraint condition r

0) = - q,o~(x, 0)

at

x =0

(3.4)

to get 2i=k sin 0o

I

A = {ik sin Oo(k2 cos 2 0o -/12) + c~} ~ ~b,(nL, 0) e I~Lc~176

(3.5)

Note that the periodicity of ~, ensures that condition (3.4) also holds at each of the points

x =nL. 3.1. FAR FIELD

To find the distant form of the diffracted potential (3.3), use the Poisson sum formula to get the alternative representation =

~q~ --a0

9

,y

exp - i x

(3.6)

324

F.G. LEPPINGTON

where ~(s,y) is the Fourier transform of q~ and is given by formula (A4) of Appendix A. Now the factor exp{-(s 2 - k2)l/2y} is exponentially small a s y --~ o% when k = kx + ik2 has a small positive imaginary part k2; but for vanishingly small k2 the exponential attenuation is minimal for s < k~ and is significant when s > kl. Thus for k = k~ + i0 the only propagating modes o f the sum (3.6) are those for which -N~ ~
2nNl < kL(l - cos 0o),

2nN2 < kL(l + cos 0o).

(3.7)

On using the expression (A4) for the transform q~ one obtains ~P"

A ~

~

exp{ik(xcosO, + ysinO,)} 2

"

~

.

~

~ ~

(3.8)

--N I

as y --~ +oo, with A given by equation (3.5) and with 0, defined by cos 0, = cos 0o - 2nn/kL

(3.9)

in the range 0 ~ 0, ~ n. In particular ifkL < ~zthen, for any incidence angle 0o, the only propagating mode is the fundamental one n = 0; hence

A

exp{ik(xcosOo + ysin0o)}

~ L ( - i k sin Oo(k~cos 2 0o - Its) _ ~}'

(3.10)

which represents a simply reflected wave. The general expression-(3.8) includes a wave travelling at the principal angle o f reflection 0o together with secondary and higher order reflections, just as in the classical electromagnetic diffraction grating problem. Note that near-resonance occurs if kL(l ~ cOS0o) is an integer N: then sin0u = 0 and ~ = O(1[~) is large if the fluid loading is small. 3.2. LOW FLUID LOADING

Formula (3.3) is formally exact, though A is expressed in the form o f an infinite sum (3.5). This can be simplified by using the Poisson formula again to get an alternative sum similar to that in expression (3.6), but with x = 0 and with an extra factor inside the sum" in particular this leads to an estimate for A when ct ~ 0. In this low fluid loading limit, however, it is simpler to proceed directly from expression (3.5), using expressions (A23) and (A25) to get

dp,(x, O) ~ - (i/2p) exp (iit Ix I) + o (~)

(3.11)

for any fixed x. This formula is not uniform with respect to x, as is shown clearly from the far field limit (A19) for Ix I ~ co with ct small and fixed. But the approximation (3.11) is nevertheless suitable for estimating the sum (3.5), since q~,(x,0) is exponentially attenuated at large Ixl when It and k have small positive imaginary parts. Thus

dp,(nL, O)eJ,~Lr ~... -

t

exp {iln]itL + inkLcosOo} = 2it{cositL - cos(kLcosOo)}'

oo

A~

4i:ckit sin 0o

cos (kL cos 0o) - cositL

ik sin 0o(k2 cos 2 0o - It2) + ~

sin p L

(3.12)

The resonant conditions, pL = mrr, are excluded here; near these values the terms O(c0 in expression (3.11) have to be included.

325

PLATE SCAI IERING WITII LINE CONSTRAINTS

4. ELASTIC PLATE WITH TWO LINE CONSTRAINTS A brief account is presented here for the case of a thin elastic plate whose governing equation is equation (1.8) in place of equation (1.4), and with constraints along each of the lines (x,y) = (+L,0). With an incident wave (2.2), the total potential is subject to the Helmholtz equation (1.2), a radiation condition at infinity and the boundary condition (1.8) foi x # +L. The chosen constraints are those of zero deflection and deflection gradient, so that qs,(• O) = ~,~(• O) = O. (4.I)

Since the deflection has (unknown) discontinuities in its second and third derivatives at x = • the plate equation can be written as (a'lOx" - ~ ) % + ~, 9 = .,t,1,5(x + L) + A2~ 6(x - L) + A~2 6'(x + L) + A2z 6'(x - L),

(4.2)

where the Aij are unknown for the moment, and with lq and cq defined by equations (1.9). The suffix 1 is used in this section to distinguish the present problem from the membrane problems in sections 2 and 3. The exact solution can now be expressed in terms of a reflection field ~x, which is the solution in the absence of constraints, together with a diffracted potential qb~ that corresponds to a line force radiated field. Dealing firstly with ~1 one has ~ = q~l + Rx exp{ik(xcos 0o + ysin 0o)},

(4.3)

R~ = [ik sin Oo(k+cos +0o - p~) - ct]/[ik sin 0o (k +cos + 0o - p~) + ct],

(4.4)

with on substituting equation (4.3) into the homogeneous equation (1.8). As for the potential if1, it has no incident forcing, but is specified by the boundary condition (a'lax + - l,I)

+ =,

=

y

=

0,

(4.5)

together with the wave equation (1.2) and radiation condition at infinity. Its exact solution, in Fourier integral form, is given in Appendix B. A comparison of formulae (4.2) and (4.5) now shows that 9 has the solution

= ~ +A~tqb~(x+L,y)+A~2dp~=(x+L,y)+ A21:P~(x-L,y)+A22~pI~(x-L,y),

(4.6)

and the A tj are to be determined from the final conditions (4. I) which lead to the linear system

(!

b

0 -c\[A~\

a

c

0c d e

[~,(-L,O)\

0 / [A2t| =_/*,,(+L,0)/, e Ila,~! ~,,,(-t,O)/ d ]\A,2] \qh,~(+L,O)/

(4.7)

where a = q~l,(0,0),

b = ~b,,(2L,0),

c = ffa,~(2L,0),

d = ~b~,~,~,(0,0),

e = ~1,,~(2L,0) (4.8)

are all known constants. The solution is

A,,\ [-dP+eQ A2,|= 1 [ eP-dQ A12] - "A \ -cQ A22/ cP

eP-dQ -dP+eQ -cP cQ

cQ

-ce ] [ ,,,(-L,o) \

cP

-cQ

- a e - bQ bP + aQ

bP+aQ

l ~,,(+L,O) |

(4.9)

- a P - b Q / \ :PI,~(+L, O)]

where

P = c z + a d - be,

Q = ae - bd;

A = Q2 _ p2.

(4.10,4.1 I)

326

F.G. LEPPINGTON

4.1. LOW FLUID LOADING In the limit ce --+ 0 the results of Appendix B for the matrix elements in equation (4.7) lead to the estimates

811~ Q = i(e 2ju'L - - e -2"U') + O(0~),

81t~P = (1 - e-2Ua- e zj#'L) + O(ce).

(4.12)

In particular, it is found that (even and odd) resonances occur at the conditions t a n h p L = -T-tanltL.

(4.13)

Near these values of BL proper account must be taken of the small radiation damping effect by including the terms of order ce in formulae (4.12), as in the membrane problem of section 2. Corresponding results can be obtained similarly for different geometries and different boundary conditions (see references [2] and [9], for example).

ACKNOWLEDGEMENT I am grateful to Dr A. Freedman, A.U.W.E. Portland, for several interesting conversations on this and related subjects.

REFERENCES 1. M. C. JUr~ER and D. FElT 1972 Sotald, Structures and Their Interaction. M.I.T. Press. 2. I. P. KOXOVALYt/K1969 Soviet Physics, Acoustics 14, 465--469. Diffraction of a plane sound wave by a plate reinforced with stiffness membranes. 3. D. G. C/~GnTO,',/ 1971 Journal ofFhdd A~echanics 47, 625-638. On acoustic beaming and reflexion from wave bearing surfaces. 4. P. A. CANNELL 1975 Proceedings of the Royal Society London A347, 213-238. Edge scattering of aerodynamic sound by a lightly loaded elastic half-plane. 5. P. A. CANNFLL 1976 Proceedings of the Royal Society London A350, 71-89. Acoustic edge scattering by a heavily loaded elastic half-plane. 6. P. A. CANNELL1977 (unpublished notes). 7. F. G. LEPI'INGTON1976 Quarterly Journal ofAlechanics and Applied Alrathematics 29, 527-546. Scattering of sound waves by finite membranes and plates near resonance. 8. D. C. HANDSCOMB1977 Joltrnal of the Institute ofAfathematics andIts Applications 20, 183-189. Vibrations of a submerged window. 9. F. G. LEPPZ/qGTON1977 A.U. IV.E. Technical Note 548]77. Scattering by an infinite eleastic plate constrained along a line. 10. P. C. CLEMMOW1966 The.Plane Wave Spectrunl Representation of Electromagnetic Fields. New York: Pergamon Press.

APPENDIX A: LINE FORCE POTENTIAL FOR THE MEMBRANE Exact solutions for the problems described in this work have all been expressed in terms of the potential q~ (or ~b~) that corresponds to the radiation by an infinite membrane (or plate) with a line force at ( x , y ) = (0,0). The Fourier integral analysis is similar for both membrane and plate and is now described in detail for the former case. A brief treatment for the plate follows in Appendix B. The potential q~(x,y) satisfies the wave equation (1.2), an outgoing wave condition at infinity and the boundary condition (a~/ax ~ + i, ~) 4,~ - ~,b = ~(x),

y = o.

(AI)

PLATE SCATTERINGWITH LINE CONSTRAINTS

327

Defining the Fourier transform $(s,y) one has o0

$(s,),) = f dp(x,y)e"Xdx,

~(x,y) = (2n)-'

f(b(s,y)

e -''~ds

(A2, A3)

--oo

--ca

expression (A3) being the inversion integral. It is mathematically convenient to assign, initially, a small imaginary part to the wave number k (k = kl + ik2) in order that the radiation condition gives ~b to be exponentially small at infinity; one then finally lets k2 ~ +0. Under Fourier transformation the wave equation has a solution of the form

~9 = { l[P (s)} exp (-)'y),

(A4)

where

)' = (s z - k2) 112

(AS)

has branch cuts from +k to +co along straight lines with constant arguments and with )' = - i k when s = 0. This choice o f square root function ensures that Re), >10 for all real s, so that expression (A4) is the solution with the required behaviour at infinity. The function P(s) is determined from the boundary condition (AI), whose transform leads at once to the result P(s) = (s 2 - p2)7 _ ct (A6) and ~b is given exactly as

- isx) ds. dp(x,y) = (2n) -t f exp (-)'y ~

(A7)

On letting k2 ~ +0, the integration path in equation (A7) lies just below the positive real axis and just above the negative axis. An alternative form due to Clemmow [I0] is obtained from the substitution s = - k cos O (A8) f r o m the s-plane into the complex O-plane. Then

I f k s i n O e x p { i k r c o s ( O - 0)} dO ik sin O(k 2 cos z O - IL2) + ct

(A9)

~b(r, 0) = - ~

r along the path F of Figure 1, where (r,0) are polar co-ordinates such that x = rcos0,

y = rsin 0.

rr 8+z"

s(o)

F

Figure 1. The integration contours Fand S(O) in the complex O-plane, with k < p.

(AI0)

328

r.G. LEPPINGTON

Since ~bis obviously an even function of x, and hence of(89 - 0), one may confine attention to the region 0 ~ 0 ~<89 At great distances from the origin one can anticipate that the main contributions to the integral (A9) wiI1 arise from the saddle point 6) = 0 and also from those values of O where the denominator of the integrand vanishes. These poles correspond to coupled waves of the fluid/membrane system and might be triggered by the line force excitation. Following Clemmow [10], one now finds a far field estimate by deforming the integration path on to the steepest descents contour S(O) through the saddle point O -- 0. If O is written as a sum of real and imaginary parts, (9 = igr § iOl, then the steepest descents contour S(O) is given by cos ((9, - 0) cosh (91 = I (A11) (see Figure 1) and an increase in 0 simply shifts S(O) bodily to the right. The shaded regions in Figure I show the "valleys" where the integrand is exponentially small as k r --~ oo. It is readily found that the contribution to the integral (A9) from the vicinity of the saddle point is a radiated wave potential ~R =

- k sin 0

exp (ikr - kin)

{ik sin O(k 2 cos 20 - pz) + c~}

(2nkr) 1/2

,

(AI2)

which has the expected form for a cylindrically spreading wave. In addition one must take account of possible residue contributions from any poles that are crossed in the deformation from F to S(O). A detailed analysis has been given by Clemmow [10] and by Crighton [3] for a similar membrane problem. The poles Correspond to the zeros of the function P(s) defined by expression (A6); in terms of 7 = " i k sin (9, these are solutions of the cubic equation 73 + (k 2 _/~2)), _ ~ = 0, (A13) with Im? ~<0 to accord with the choice of branch cuts in the s-plane. Although equation (A13) could be solved exactly, here only approximate solutions for small or large a, with k and/z fixed, are considered. SMALL FLUID LOADING

When ~ ---)-0 the approximate locations of the roots are shown in Figure 2 and the relevant poles are given asymptotically for fixed p > k by (91 "" i cosh -1 (1a/k) + i~/2/l(p 2 - k2),

(A14)

(92 ~ cos-'(Mk) - i~/2p(k 2 - / ? ) ,

(A15)

(93 ~ i ~ / k ( k 2 - ~2),

(A16)

and for fixed tt < k by as

- - + 0.

There are also corresponding poles at points (n - (91) and at the points shown as 4 and 5 in Figure 2. The estimates (A14)-(A16) are not uniformly valid when k and/l are nearly equal. k >/~

9

2

P-o

k
7r

b3

4,

5

Figure 2. The poles in the complex 8-plane, Poles with the same labelcorrespond to the same value of 7.

PLATE SCATTERING WITH LINE CONSTRAINTS

329

SURFACE WAVE FOR/t > k T h e o n l y relevant pole, for/~ > k, is the one labelled (gt a n d this is c a p t u r e d b y the deform a t i o n when 0 is less than the critical value 0c such that S(Oc) passes through (91. Its residue

contribution is the surface wave q~ts~-,- {i/2/,(p 2 - k2) '/2} exp {_y~2 _ k2),,2 +

it, Ix[}.

(A17)

Since this is exponentially small wheny increases it is negligible for 0 > 0c and can be included without harm for all 0, whence q~ " q~R + ~b(s'),

as

(A18)

kr --+ oo,

when/t > k, c~ small, with q~R and q~(sl) given by expressions (A12) and (A17). SURFACE WAVE FOR p < k

In this case the far field is a little more complicated. The contribution from the pole 02 has already been described in detail in a similar problem by Crighton [3]. At sufficie.ntly large distances from the origin its residue contribution is found to be exponentially attenuated whenever 02 is captured: it is seen from Figure 1 that 02 is always within the shaded (exponentially damped) region if it is captured. But the decay is arbitrarily small when the radiation damping constant #, ~ 0, so that its influence can persist to large distances, just below the coincidence angle c o s = ~ / k ) , but decays ultimately as k r ~ ~o, ct fixed. The surface wave corresponding to pole 03 is captured when 0 is less than, approximately, ~{k(k 2 - p2)}-x and persists to arbitrarily large distances near the surface, so that q5 ~ dpR + {i~]k(k 2 - p2)2} exp{_[~tyl(k 2 _ / F ) ] +

ikl.~l}

(m19)

as r ~ ~, with q~a given by expression (A12). LARGE FLUID LOADING

When ~ is large (compared with [k 2 - U213z2) the cubic equation (A13) has the approximate solutions Yt ,,,#,t/3 and ~'2 ~ ~|/3exp(--'~ni) in the lower half-plane, with the corresponding values (91 ~ i log (2~I/3/k), O2 = ~ n - i log (2cd/~/k) and additional poles n - O~. Only the pole Ot is crossed in the deformation from F to S(O) with the result that ~b ,-, 4)R + (i/3,~2/3) exp {-~V3y + i,,'Z~lxl}, r --+ co, (A20) for large cq and the surface wave is confined to a very thin layer of thickness y = O(~-u~). THE CONSTANTS ~ t AND .4 2

The constants Ax and .42, that appear in the solution (2.8), are given by equations (2.9), where the numerators are -4ictk sin 0o cos ( k L cos 0o) 9 oy(L, O) + ~oy(-L, 0) = ik sin 0o(k 2 cos 2 0o -/,t 2) + ct' (A21) q)o,(L,0) - q)oy(-L, 0)

=

4ctk sin 0o sin ( k L cos 0o) , iksinOo(k2cos20o-#2) +a

(A22)

upon using equations (2.4) and (2.5). The denominators in formula (2.9) are given according to expression (A7) as O._ - ~b,(O,0) + qb,(2L, 0) = -(2re) -t

f

T(l + e -21a') r(s 2 p2) _ ~ ds.

(A23)

330

F. G . L E P P I N G T O N

LOW FLUID LOADING

When ~ ~ 0 one can use the identity ? r(s~-t~2)-~

1 ~x - + s~-~ ~ (s'-~){r(s~-~2)-~}

(A24)

'

where the second term can be ignored; for it is negligible except near s =/~ and near the zeros of P, and the contour can be deformed away from these points. Thus one has

D+ =-(il2lO (I + e 2'"') + O(o0

(A25)

after an elementary integration. There is clearly a resonance phenomenon, at small fluid loading, if /~L = (n + 89 n

or

pL = nrr,

(A26, A27)

where n is an integer. The first set of values (A26) correspond to resonant modes even in x and the set (A27) correspond to odd modes. Near these resonant values the (small) radiation damping has to be included and one must retain the second term of expression (A24), to get -i

• e21UL)_ ct f

1 _+ e -21~t"

(A28)

with an error of order c~2. The integration path is just above the negative axis and below the positive axis. This result shows that the amplitude constants A1 and A2 remain bounded near resonance, because of the radiation damping of course. The form of expression (A28) is similar to the approximateresults obtained by Leppington [7] for the problem of a finite baffled membrane, for which an exact solution is not yet available. HIGtl FLUID LOADING

When ~ --~ ~ the integrand of expression (A23) is O(l/c0 for any fixed s, and it follows that the contribution from any finite integration range is O(1/,). The term y(s 2 - it 2) in the denominator can not be neglected for large s, but here one can replace ~ by - s s g n ( I m s ) so that, for ,, ~ 0%

D+ ,.. - n~?/3

~ o

tdt

1 -T- 2nctl/------S

~

t

{exp (2itcd/3 L) + exp(-2it~ ~/3L)} dt

o

after a simple change of variable, where the path passes below the pole at t = 1. The first integral can now be evaluated exactly, while the others can be estimated asymptotically, as c~1/3L --~ 0% by deforming on to the upper and lower imaginary axes and using Watson's Lemma. It is found that 3:d/3 D . ~ - i - I / V"3 -T- iexp(2i~l/SL). It follows finally from expressions (2.9), (A21) and (A22) that { - i cos (kL cos 0o) sin(kLcosOo)} A1"2 ~ 6k~/3sin0~ i + I/V"3+ ie 21~u~L + - i - 1/V~ + ie 2~'~'L a s o~il3L ~

oo.

(A29)

331

PLATE SCATTERING WITH LINE CONSTRAINTS

APPENDIX B: LINE FORCE POTENTIAL FOR THE ELASTIC PLATE The fundamental potential qbl has to satisfy the wave equation (1.2) with a radiation condition at infinity and the forced plate equation

(a'lax" - ~;)4,,, + ~, 4,1 = ,~(x),

y = 0,

(BI)

with lq and cq defined by expression (1.9). An exact solution can again be found by Fourier transformation with respect to x. It is found that - D ' - isx) c~l(x,y) = (2n)-' f exp (Pl(s) ds,

(B2)

--oo

where y is given by expression (A5) and

P,(s) = - y ( s " - ll~*)+ ct;

(B3)

the integration path of equation (B2) is just above the negative real axis and just below the positive axis. LOW FLUID LOADING The matrix elements a, b, c, d and e of formulae (4.7)-(4.9) can now be estimated when ct --+ 0. A typical element is b = (2n) -1 f

exp (-2iSL) d s

),exp(-2isL) ds = (2n) -1 ~(s' - ld) - ~

--oo

s ' - ~d

+ O(~)

--co

as - --+ 0 with all other parameters fixed; hence

41~]b = (i e 2'"'L - e -2"'L) + O(cQ.

(B4)

4p~ a = (i - 1) + O(c0.

(BS)

41~]c = - ( e 21",L - e -2~,L) + O(cQ,

(B6)

4pl d = - ( i + 1) + O(00,

(B7)

41q e = - ( i e 21"'L + e -2~'L) + O(c0.

(B8)

Setting L = 0 gives Similarly it is found that

FAR FIELD

The distant field form for ~b1, as r ~ 0% is obtained by a steepest descents analysis like that of Appendix A. An integral representation in the complex O-plane is obtained by the substitution (A8) and the path F is deformed on to the steepest descents contour S(O) of Figure 1, with account taken of any residue contributions from the poles of Pl that are captured in the path deformation. When the fluid loading parameter is small the relevant poles of P~ (formula (B3)) are near s = +k{l + a2/2k2(k 4 -.//~)2} s = +lq{l + ~[4y~(y]

-

.if

k2) 1/2}

k >

y~

and

if k < / q .

(B9) (BI0)

It is found that q~, ~ ~biR + ~b,s

as

r ---> 0%

(Bl

l)

332

F.G. LEPPINGTON

where the radiated field ~blg is from the saddle point at O = 0 and has the f o r m ~bln =

k sin 0

exp (ikr - 88

{ik sin O(k 4 cos 4 0 - p~) + ~}

(BI2)

( 2 n k r ) 112

The surface wave potential ~b. arises from a residue contribution from a pole (B9) or (B10) and is given by dp,s ... k(k-ZZ--p4) 2 exp

+

iklxl

/

,

k > pt,

(BI3)

or

-i qS~s ... 4p](p] - k2) ~/2 exp {-(g~ - kZ)'/Zy + ipa[x[},

as ct ~ O, with Pl and k fixed and unequal.

k < p,

(BI4)