THE DIFFRACTION OF SOUND WAVES BY AN ELASTIC HALF-PLANE ATTACHED TO A VERTICAL RIGID PLANE

Transcription

1 THE DIFFRACTION OF SOUND WAVES BY AN ELASTIC HALF-PLANE ATTACHED TO A VERTICAL RIGID PLANE By F. GHANDCHI (Department of Mathematical Sciences, The University, Dundee, DDl 4HN) [Received 14 June 1983] SUMMARY The sound field of a line source is investigated when a structure consisting of a semi-infinite thin elastic plate, attached at right angles to a rigid plate of infinite extent, is embedded in a stationary acoustic medium. A formal exact solution is presented and the distant sound field is explicitly derived, with emphasis on the free modes of the coupled fluid-structure system. The significance of the contributions associated with different parts of the structure is exposed by taking two distinct limits of the fluid loading parameter relevant in aerodynamic and underwater problems. 1. Introduction and formulation THE influence of boundary inhomdgeneities has been the subject of recent studies of sound interaction with flexible surfaces, and is of some practical importance in the field of noise control. Of particular interest to us are works (1 to 3) dealing with infinite planes with a variety of constraints, and (4,5) concerning acoustic edge scattering by a semi-infinite plane. The aim of this paper is to introduce an inhomogeneity due to a change in the structural configuration. The prototype model involves a semi-infinite thin elastic plate y = 0, x <0 (see Fig. 1), meeting an infinite rigid plate x = 0 at right angles. The whole is immersed in an inviscid, static, compressible fluid and is irradiated by a line source located at L(x 0, y 0 ) with x o <O, y o > 0. The harmonic time variation of angular frequency w is taken as exp (io)t), this factor being suppressed throughout. Within the fluid region the total velocity potential <p is subject to the reduced (Helmholtz) wave equation ^+k)<t>(x,y) 8(xx 0 )8(yy 0 ), x<0, (1.1) ay I where k = w/c is the fluid wave number, c is the speed of sound in the fluid, and 8 is the Dirac delta function. The potential <f> is related to the fluid pressure field P(x, y) through the identity ; M ) P ( ) (1.2) where p is the mass density of the fluid. [Q. Jl Mecb. appl. Math., Vol. 37, Pt. 4, 1984]

2 554 F. GHANDCHI y y=co = oo v= oo FIG. 1. The geometry of the problem The plate performs low-frequency small flexural vibration of amplitude TJ(X), governed by the differential equation (6) x<0, (1.3) where B = 2Eh 3 /3(l cr p ) is the bending rigidity, E is the Young modulus for the plate, h is the plate half-thickness, cr p is the Poisson ratio, p p is the mass density of the plate material, and the notation [P(x, 0)]t denotes the discontinuity of the pressure field across the plate. Combining equations (1.2) and (1.3) yields lax 4 la, x<0, (1.4) where fc p = (2p p hw 2 /B) J is the plate wave number and a = pa> 2 IB is the fluid loading parameter. Continuity of velocity across the elastic boundary implies that -iwtj(x) = ct>(x, y), y = 0, x<0, (1-5) dy and on the wall the normal component of the velocity vanishes; hence <M*, y) = 0, x=0. ox (1.6)

3 DIFFRACTION OF SOUND WAVES 555 The effects of the wall can be replaced by an image source symmetrically placed with respect to the wall. The total field is even in x and this symmetry property extends the thin plate equation (1.4) together with boundary condition (1.5) to all x, bearing in mind that the first and third derivatives of deflection are not necessarily continuous. Obviously, the field in x > 0 has no physical reality. This procedure leads to ^k)fi(x,y) {8(xx 0 ) + 8(x + x 0 )}8(yy 0 ), (1.7) dy / (74- ftpw) +-[*(*, 0)]- = C v 8(x) + C 2 8"(x), (1.8) VOX / O) ^ ^<t>(x,0), (1.9) ay where the prime denotes the operation d/dx, and C 2 and C t are the unknown jump discontinuities in TJ'(X) and TJ'"(X) at x = 0. With the radiation conditions at infinity to ensure outgoing scattered waves, the system (1.7) to (1.9) describes a well-posed boundary-value problem, and the Fourier integral representation of the field can be found in the form, V) = ~\ H [k{(x-x o ) 2 + (y + y o ) ]-^Hg»[fc{(x + x o ) A H [k{(x - x 0 ) 2 + (y - y o ) ] - \ H [k{(x + x 0 ) 2 + (y - 77 J +^ T77^ e ^> y >0 > C 1-10 ) fc y<o, (l.n) where HQ 2) and for real, is the Hankel function of the second kind, A( ) is defined by A( ) = 7(^4-kp)-2a, (1.12), (e-k = ue-k 2 )K \i(k 2 -Z 2 )K \z\>k, (1.13)

4 556 F. GHANDCHI A small negative imaginary part to the wave number k is assigned (fc = k 1 - ik 2, k 2 * s Q) in order that the radiation condition gives 4> to be exponentially small at infinity. Equivalently y may be regarded as a complex function of, with branch cuts from = ±fc to ±, with y = ik when = 0; on letting fc 2 > 0, the integration path of formulae (1.10), (1.11) becomes the real axis, indented above any poles on the positive half and below any on the negative half. The nature of the constants C t and C 2 depends upon the way the plate is fixed to the wall; when it is clamped horizontally so that at x = 0, TJ(X) = T)'(X) = 0, we have C 2 = 0 and C x = -2ia/ 1 /o)j 1 where f (1.14) The effects arising from other types of constraints will be discussed later. 2. The distant field To derive an explicit solution for the asymptotic behaviour of the total transmitted field, we specify a point of observation (x, y) by means of polar coordinates x = r, cos 0 y, y-y o = r, sin 0,-, / = 1, 2, with O s 0,^2"n'> r l = {x 2 + (y - y o ) 2 R r 2 = {x 2 + y 2 }\ and distort the contour of integration into the left-hand branch of the hyperbola = -kcos(0,- + it), -oo< T <oo. (2.1) We must, of course, take account of fields induced by the wave numbers of the free modes of the coupled fluid-plate system, namely possible residue contributions from any poles (A( ) = 0) captured in the deformation of the path of integration. On taking y as the variable in place of, A( ) = 0 reduces to (2.2) In a detailed analysis, Crighton (7) has shown that the roots of (2.2) can either be labelled as Yi>0, 72<0, 73<0, Y4 = 0i + i02, Y5 = Y3, with p t >0,0 2 >O, or as Tl>0, 72=^1 + 11*2, 73= ~Vi+iv 2, 74=Y2, Y5=7*> with jxx, M-2, v \i J / 2 > 0 the star indicating a complex conjugate. Let us consider the latter case and assume that the roots are of the form 7, = ik sin ( I + I'T,). TO each root 7,, satisfying I m 7 ^ 0, there corresponds two admissible poles in the -plane which can be determined from 2 = 7 2 +fc 2 ; these are t = ±k cosh T U fi = 0,T 1 <0, 7r, T 2 <0, ^7T, T 3>0.

5 DIFFRACTION OF SOUND WAVES 557 r(oo) r(-oo) - _ t \-fc cos 0, FIG. 2. Deformation of the integration path The poles * are of the subsonic kind; the associated surface waves may generate a near-field confined to layers in the vicinity of the plate. Closing the contour of integration by arcs as depicted in Fig. 2 indicates that these poles are not captured in the deformation but their existence should be accounted for when 0,- ~ 0, in particular if gf ~ ±k. The poles captured in the deformation are for all 0,, 2 iffy >& and J if 0, < 3. It can easily be shown that the residue contributions corresponding to the poles 2 are exponentially small for all values of 0,, while J gives rise to waves which travel supersonically along the plate, and for a given x increase exponentially with y-y o and y, but they tend to decay ultimately as distances r, increase. For further details of these types of waves, often termed 'leaky waves', we refer to (7,8). It is seen that no physical significance can be attributed to the residue contributions in the acoustic far-field except near the plate where the contributions due to subsonic modes are to dominate. In fact, the main effect of the free modes of the coupled fluid-plate system is on C x. Thus, at sufficiently large distances from the plate, we have in x < 0, y > 0, 77 cos (~x o fc cos (0 t + h))ik sin -y(-k cos (0 t + iy))a(-fc cos (0j + JT)) ikr, cosh T dr + ik sin (0 2 + it)e- ikr * cosht dr. (2.3) A(-kcos(0 2 +it)) The integrals (2.3) are now amenable to asymptotic evaluation by the

6 558 F. GHANDCHI standard stationary-phase method when fcr,»l. We obtain -2a cos (x o k cos 0 t ) e- ikr,-i^ <f>(x,y)- ik sin 0x (k 4 cos 4 0 t - k 4 ) - 2a k sin _ 6 wc * e ~ a "'~ i * (2 4) 1 ik sin 0 2 (k 4 cos k 4 ) - 2a (2irkr 2 )*' " Note that if higher-order terms in the small quantity x o lr r are discarded, the distance between the point of observation and the source r becomes r = r 1 x 0 cos 0j. From the Appendix it follows that _ 2a f -ik sin 0 O t,-^*-** p ikr o^ «2 +" 2 +«o) x ~ <o \ik sin 0 o (k 4 cos 4 0 o -k$)-2a (2.5) for kr o»l. It is to be remarked that the solutions for <p and C x are not uniform in the polar angles 0,, for / = 0,1,2, since the possibility of 0, being near to the free wave angles i has been ignored. If, however, the possibility arises, the necessary modifications in terms of Fresnel integrals can quite simply be incorporated into the analysis (9). We may also have the common circumstances that the source or the point of observation are to lie near the surface of the plate (0, ~ 0), indicating that the stationary point is very close to the branch point k. More elaborate formulae in terms of parabolic cylinder functions can be derived (9). The field <(> given by (2.4) is in the form of inhomogeneous cylindrical waves emanating from (0, y 0 ) and (0,0). The first term, representing the superposition of the diffracted waves by the plate and reflection from the wall, is insensitive to any type of boundary conditions. The effects of the constraints emerge in the next component, corresponding to the field radiating from the 'junction' where the plate is fixed to the wall. Some interesting physical implications will be revealed in the following section, by making assumptions on the fluid loading parameter a. 3. Low and high fluid loading We may write a in the alternative form a = efcpfc, where the non-dimensional fluid loading parameter e = plhp p k gives a

7 DIFFRACTION OF SOUND WAVES 559 measure of the relative mass of fluid to plate; thus 2TT is the ratio of the mass of fluid within one fluid wavelength, against the mass of the plate beneath the fluid. In the aeronautical applications, effects of the fluid-elastic boundary are small while such effects are considerably significant in many underwater problems. We may also define a Mach angle 0 M = arc cos (kjk), and a coincidence frequency <o c such that k = k p. In the low fluid loading limit, i.e. a» 0, the distant field reduces to 2ia cos (x o k cos flj e-"" -*"* k sin O^cos 4 0 X - k 4 ) where from the Appendix (part (a)) it follows that V2ae 4ifc 3 ik i k9 ClL= Z l -Uk2-kpK 0 sin ik p r 0 cos 6 0 -> +me M -e o) ^-^ }. 0.2) The above approximations remain valid near the Mach angle if the factor {k 4 cos 0,-k 4 }- 1 is replaced by ik sin ^{ik sin 0,(k 4 cos 4 0,-k 4 )-2a}~\ for j = 0,1, 2. A vivid description of the field demands a careful examination of the three factors appearing in C 1L. The first one exhibits an algebraic decay with (kr o )~i whereas the second term is exponentially attenuated except when 0 O is near \-n, that is, when the source is close to the wall but far from the junction. The final term is of a plane-wave form with no attenuation, but it persists only if 9 0 is less than the Mach angle. We see that although the components of the field < L in (3.1) are both of order a, they are comparable, subject to the conditions 0 O ~2"' or 0 O<^M; otherwise the decay factor (kr o )~* ensures that signals emitted from junction make an insignificant contribution to the acoustic intensity. When a = 0, we have the expected result that the plate behaves like a rigid boundary. However, this fails near the Mach angle where 2a{ik sin 0,(k 4 cos 4 0,-k 4 )-2a}~ 1 = -1, no matter how small the fluid loading parameter. An interesting feature of this property arises when 0 o ~~d M ; the propagation centred at the junction is then independent of the fluid loading parameter and has a large peak when the observation angle 0 2 is also in the direction 0 M. The roles of the angles 0 O and 0 2 may be interchanged. It should be noted that the above mentioned Mach envelope can be formed when the excitation frequency is greater than coincidence frequency &> c. For a typical air and steel problem, «c = l-2xl0 4 h~ l and

8 560 F. GHANDCHI 6 M = arc cos [1-1 x 10 2 (o>h)"5]. With h = 0-2 cm, and w = 6-5 x 10 4 Hz, we find that 0 M = In the high fluid loading limit, that is as a», the far field becomes 1 ) 2. [,, iksine 1 (fc 4 cos 4 e 1 -k 4 )l e-^-i^ 4>H ~ 1 + T ~ cos (x o k cos 00, + L 2a I (2TrJcr (3.3) 2a {2 where C 1H takes the form (see the Appendix, part (b)) ^33 [-i exp{-i(2a)ir o e- w o} + (^ o) x sin { TT + (2a)*r 0 sin (^TT - 0 O )}J- (3.4) As expected, the leading-order term of 4>H is the field experienced in the absence of the plate. Thus the presence of the plate is expressed by the remaining two parts, one consisting of the transmitted waves through the plate and the other the effects arising from the way it is fixed to the wall. Because C 1H ~ O(a*), one expects the scattering from the junction to produce a sound field more intense than that transmitted through the plate though the final outcome also depends on the exponentially attenuated nature of C 1H. This matter is, however, more clearly established when the source is close to the plate and far from the junction; C 1H now behaves as (2a)5exp{-(2a)=(ix o +y o )}, indicating a small exponential decay. Note that the field induced by this contraint is of order a~k 4. Concluding remarks The generalization of the problem to three-dimensional geometries is quite straightforward and will not be pursued here. This has practical advantages since point sources are regarded as better substitutes for real sources than line sources. In examining extreme cases of the fluid loading parameter, we realized that the surface inhomogeneity is not, in general, an efficient mechanism of sound production. It is interesting to see whether this position changes markedly if the constraint at the rigid wall is changed. Take the situation in which the edge of the plate is simply supported at the wall; then TJ(X) = TJ"(X) = 0 at x = 0, and we arrive at l 2ia (IJt-hM w \J X J,-J\r 2 2ia <o X, where I u / t are given by (1.14) as before and, for n = 1, 2, -

9 DIFFRACTION OF SOUND WAVES 561 The above integrals can be evaluated by employing the methods presented in the Appendix; we do not wish to display them explicitly. It is easily deduced that C ll, C 2L ~ O(a), C 1H ~ O(at) and C 2H ~ O(al). Therefore, in the low fluid loading limit, the fields generated at the junction are of the same order as that in the previous case with the obvious change in the directivity patterns. However in the high fluid loading limit the order of the dominant term remains unaltered but the additional field behaves as O(a~*). It appears that the fields induced at the junction are more sensitive to the boundary conditions in the high fluid loading limit than in low limit. Further evidence is provided by assuming the edge of the plate to be free at the wall, that is TJ"(X) = TJ'"(X) = 0, at x = 0, indicating that C\ = 0, C 2L ~ O(a), and C 2H ~ O(a$), which show that there is a drastic reduction in the intensity of the sound field coming from the junction, in the high fluid loading limit. Evidently, the influence of the junction is minimal when the plate is allowed to scrape past the wall with no constraint (C\ = C 2 = 0). It is hoped that this paper may provide some insights into investigations of more complex models of right-angled structural elements. Acknowledgements I should like to thank Professor D. S. Jones for his helpful advice on this work, and the Ministry of Defence for the financial support of a Research Fellowship. REFERENCES 1. F. G. LEPPINGTON, Ministry of Defence, A.U.W.E., Tech. Note No. 548/77, J. Sound Vib. 58 (1978) M. TEYMUR, Ministry of Defence Final Report, University of Dundee, P. A. CANNELL, Proc. R. Soc. Lond. A 347 (1975) , ibid. 350 (1976) L. CREMER, M. HECKL and E. E. UNGAR, Structure-borne Sound (Springer, Berlin, 1973). 7. D. G. CRIGHTON, J. Sound Vib. 63, (1979) , J. Fluid Mech. 47 (1971) D. S. JONES, SIAM Review 14 (1972) D. G. CRIGHTON, J. Sound Vib. 20 (1972) APPENDIX Evaluation of the unknown Q The constant C\ is expressed in terms of two integrals I, and J u defined in (1.4). The integral I t can be evaluated asymptotically by following the method (hyperbolic path) of section 2. It is found that, for fcr o»l,, ~'fc sin e 0 /^V^-ikv-h* +? v m 1 ksi0(fe 4 4 0O2\k/

10 562 F. GHANDCHI where I\ is the residue field: T' _ - 1 -,,** <*» «2 +it 2 + «0> _ H ( e 0~& -ikr0o 1 A'( + ) A'(S) rncos(6n_ t<+ here x o = -r o cos0o, yo=r o sin0o, O<0 o <37r,»b = (xo + y<>)*> and H is the Heaviside unit step function. The integral J, may be interpreted as the line-force admittance of the plate at the junction. If we write then the asymptotic expansion of (3) in the limit as x * 0 can be investigated. The technique is given in full in (10) and need not be repeated here. The result is! are the roots of M( ) = 0 with Im f > 0 implied, where and 3^ = M' (a) Low fluid loading limit When a» 0, A( ) may be approximated by Hence I t and J t reduce to (4) A( )~Y( 4 -O- (5) The first integral can be evaluated asymptotically for kr 0»1; we obtain IiL~( I 4 4 r+2mi\ L (8) \fcr 0 / (k cos 0 0 fc p ) where I\ L, the residue contribution, is given as On the other hand, a simple residue calculation shows that

11 DIFFRACTION OF SOUND WAVES 563 (b) High fluid loading limit Asa^M, with fixed, we have A( )- 2a, this being valid unless is large. But if is large then wave numbers fc andfc p can be neglected, that is, 7~ and 4 kp~ 4, and we arrive at a uniformly valid approximation Thus A( )~\t\?-2a, asa^oo. (11) The value of (13) is readily obtained as while (12) converts to exp{i(2a)'r 0 e ie } dfc (12) 2TT ^ ) - i }, (14) after a simple change of variable. The integrals in (15) can now be evaluated asymptotically, as a=r 0», by deforming onto the upper and lower imaginary axes, and using Watson's lemma. It is found that IIH = -77TTJ exp {i(2a)>r o e" ie»}- -^-i HCn>ir- 0 o )exp {- (2a)' cos (tu- 6 0 )y 0 }x 5(2a) s 5(2a)! Xsin[ 7r-(2a)'r 0 sin(e 0 --rett)]. (16)