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Scattering of sound by a finite non-linear elastic plate bounding a nearly resonant cavity
Journal
of Sound
and
Vibration (1989) M(3), 387-404
SCATTERING OF SOUND BY A FINITE NON-LINEAR ELASTIC PLATE BOUNDING A NEARLY RESONANT CAVITY I.
D.
Department of Applied Mathematics, 7%e
ABRAHAMS
University, Newcastle upon Tyne, NE 17 RU, UK
(Received 3 August 1988) A thin elastic plate of finite width is set in an infinite rigid bafIIe. A rectangular cavity with pressure release walls is appended to the underside of the plate, and a compressible inviscid fluid occupies the region above the batlIe and inside the cavity. The fluid is assumed to be light compared to the plate mass. Time-harmonic plane acoustic waves are incident onto the plate from above, and are of sufficient amplitude to necessitate the inclusion of a non-linear term (due to mid-plane stretching) in the plate equation. The plate deflection and scattered sound field are obtained for non-resonant frequencies, and are shown to increase in magnitude as a cavity resonance frequency is approached. The
method of multiple scales, involving two slow-time variables, is employed to obtain the leading-order asymptotic solution, and the orders of magnitude of the potentials and plate deflection are shown to agree with previous results for the fully linear problem. The plate non-linearity is found to introduce jump discontinuities in the scattered wave amplitude as it varies with frequency, incident-wave angle or incident-wave amplitude. Secondary and combination resonances are possible, but coupled primary and secondary resonances are shown to be impossible for the particular configuration chosen.
1. INTRODUCTION AND BACKGROUND There has been a good deal of interest shown in recent years on the interaction between sound waves in a fluid and flexible, or wave-bearing, surfaces. Early work was concentrated on the scattering of sound waves by infinite, semi-infinite and ribbed elastic structures (see, e.g., references [l-5]). These models were amenable to exact mathematical analysis and yielded interesting phenomena such as coupled fluid-structural waves and edge amplification. It is obviously more physically realistic to examine problems which contain finite elastic structures. Unfortunately, in these cases only approximate methods are available to solve the mathematical initial or boundary value problems. One procedure that could be employed is that of the method of modal expansions and truncation which leads to a finite algebraic system of equations. This is the so-called Galerkin approximation and has been extensively employed on finite elastic plate problems by workers in aerodynamic flutter [6,7]. However, in fluid-structural problems a physically useful quantity called the fluid loading parameter is conveniently large or small. This is the ratio of tluid mass in one wavelength to the mass of the elastic surface, and is often small in aeroacoustic situations and large in underwater applications. Hence, perturbation methods can be employed in either limit; for example, see the paper by Leppington [8] for the low loading case, and for heavy loading the method of matched asymptotic expansions was employed by the author [9]. Both these works showed that, for specific forcing frequencies, standing waves could form on the plate, and this led to an increase in magnitude of the plate deflection and also to an increase in the intensity of the scattered acoustic field. 387 0022460X/89/090387+
18 %03.00/O
@
1989 Academic Ress Limited
388
i.D. ABRAHAMS
All the studies involving finite elastic plates or membranes show that, at resonance, large plate deflections and large fluid fluctuation pressures occur. Researchers in aerodynamic flutter were the first to suggest that non-linearities in the plate and fluid equations may therefore become important. Some work has been performed with both non-linear fluid and plate equations [10, 11] but it can be argued that the dominant non-linearity occurs in the thin elastic plate equation [6], and is the result of in-plane tension caused by large deflections. Most of the flutter literature therefore incorporates this non-linearity only [12, 13]. The author [14, 15] has recently examined the problem of scattering of sound by a lightly loaded finite plate (Leppington [8] looked at the linear case) when the incident wave amplitude is large enough to justify inclusion of the non-linear tensional term. The main effect of the non-linearity is to distort the amplitude against frequency curve near resonance and to lead to jump phenomena and hystersis effects [16]. It was also shown that secondary and combination resonances could be excited so that the scattered field may contain terms which oscillate at frequencies different to that of the incident wave. This paper extends the above work [14, 15] by the inclusion of a soft-walled cavity behind the elastic plate (see Figure 1). The scattering problem has relevance to the sonar detection of underwater vehicles, and the radiation problem is an obvious simple model of a loudspeaker in which the speaker cone (modelled by a thin elastic plate) is forced at one or several specific frequencies. The linear case with a rigid-walled cavity has already been investigated in a rather ad hoc manner by Abrahams [17] and, as for plate resonances, the system gives rise to cavity resonances which produce large plate deflections and a large scattered field. In this study the method of multiple scales is employed (see the book by Nayfeh and Mook [16]) to solve the steady state boundary value problem in a rational and asymptotic fashion. The forcing is supplied by monochromatic incoming plane waves of small amplitude, yet it is assumed that the amplitude is of sufficient size to merit the inclusion of the non-linear tensional term in the plate equation. A mechanical vibration of the plate could equally well have been taken as the forcing, and this would not have altered the method of analysis.
T 1
t~
20
>
Figure 1. Physical configuration showing incident waves.
In section 2 the full initial-boundary value problem is specified. The system is nondimensionalized and the fluid loading parameter, e, is defined. For small e, and frequency of forcing well away from the cavity and plate eigenfrequencics, the solution is obtained to leading order in section 3. This solution breaks down within O(e) of a cavity resonance
SCATTERING
OF SOUND
BY AN ELASTIC
389
PLATE
frequency (called a second-order resonance) and so in section 4 new expansions are presented for the acoustic potentials and plate deflection. These potentials are also found to be unbounded near a particular frequency (within O(e2)) and so in section 5 another set of expansions is again introduced. Two slow-time scales are needed for the analysis and this leading-order resonance solution is obtained in terms of a complex function which satisfies a simple non-linear equation. The leading-order far-field acoustic potential is obtained. In section 6 it is shown that, for the particular geometry chosen in this paper, coupled primary and secondary resonances cannot occur. Concluding remarks are presented in section 7.
2. GOVERNING EQUATIONS A simply supported thin elastic plate of width 2a, and infinite length, is set in an infinite rigid baffle lying in the plane )7 = 0. The elastic plate lies on )7 = 0, [2[ < a, where (2,)7) are Cartesian co-ordinates which are perpendicular to the direction of infinite extent of the plate. An inviscid compressible fluid occupies the semi-infinite region above the plate, and a soft-walled rectangular cavity, of width 2a, height/~, and infinite length, is appended to the underside of the flexible surface (see Figure 1). The region inside the cavity is assumed to contain the same fluid as that above the plate, but in fact a liquid or gas with a different sound speed could easily be taken instead. Observing that all dimensional variables are denoted in this paper by an overbar, the plate deflection ~ is governed by the equation
BJ'--V-N024
(0~/02)2d2
o2---~+mo--~=-p(~,o)l+_, y=o,
12)
(2.1)
where B = Eb3/12(1 - v 2) is called the plate bending stiffness, N = Eb/4a, E and v are Young's modulus and the Poisson ratio, respectively, b is the plate thickness and m is the mass per unit area of the plate. Time is denoted by 5, and/~(~, 0)[_+ is the discontinuity in fluid pressure across the plate: namely, /~(2, 0)1+ _ --lim {/~(2, q)-/~(2, -q)}.
(2.2)
q-~0
A velocity potential ~(2, )7, t-) may be defined for the fluid in terms of 3,/~ as
t~(~, )7; t-) = - ( 1 / p ) p ( ~ , )7; t-),
~ ( ~ , 0; t-) = ~7~(~,0; t-),
(2.3, 2.4)
and for small disturbances of the compressible fluid this potential can be shown to satisfy the linearized wave equation ql3e~+ q ~ = (1/c2) q~rr.
(2.5)
The subscripts in equations (2.3)-(2.5), and henceforth, denote partial differentiation with respect to the given variables, p is the mean fluid density and 6 is the acoustic propagation speed. To specify fully the model problem it is necessary to enforce edge and radiation conditions, and to state the chosen forcing term. Firstly, the elastic plate is simply supported, and so the deflection and the bending moment vanish at each end: i.e., v = v~ = 0,
x = +a.
(2.6)
For mathematical convenience it is assumed that the plate is excited by a pair of symmetric, purely two-dimensional, plane acoustic waves, ~i say. These are incident from above
390
I.D. ABRAHAMS
and both travel in directions which subtend an angle 0 with the positive )7 axis. This may be written as q~, = l g c o s [/~07 cos 0 + £ sin 0 + ~t-)/~] +½gcos [~(fi cos 0 - ~ sin 0 + ~t-)/~]
(2.7)
where g is the amplitude and /2 is the angular frequency of the wave. The scattered potential above the plate is defined as 4g($; )7; t-) = • - qS, - q~,,
)7 > O,
(2.8)
where the reflected wave ~r is given by qbr(g, )7, t-) = ~,(~, --)7, t-),
(2.9)
and it is insisted that only outgoing or decaying waves are present in g~ as ~2+)72~ oo. Inside the cavity the scattered potential is written as
77(~;)7; t3= ~15(~,)7;t-),
[~[fi>-#~
(2.10)
It is required to obtain a steady state solution to the scattered acoustic field and plate deflection. It will. prove useful, before employing any asymptotic approximations, to nondimensionalize the initial-boundary value problem. This may be achieved by scaling the dependent and independent variables as follows: x=~/a,
y=)7/a,
t=x/{B/ma4}T,
(2.11,2.12)
v(x; t)= ax/{B/ma4}8-1~(x; t), ~b(x, y; t) = $-'4g(~, )7; t-),
(2.13)
qs(x, y; t) = g-'~(~, )7; t").
(2.14, 2.15)
On substitution into equations (2.1)-(2.10) these yield
qbxx+qbyy=(1/c2)~ptt, ~,xx+q,~y=(llc2)@,,, V~xx~- e~
-l v~ dx
y>O,
Ixl< 1,
O>y>-h,
(2.17)
v,~ + v,, = e{2D cos (Ox sin O/c) sin (Ot) -4~,(x,O;t)+O,(x,O;t)}, v, = 4~y= ~y ;
V=Vxx=O, x=+l; ~b=O,
(2.16)
Ix[
y=-h;
Ixl
Ix[ < 1, y = O,
y=O,
(2.18) (2.19)
d,,=o, Ixl>l, y=O,
(2.20,2.21)
~b=O,
(2.22,2.23)
x=+l,
O>y>-h.
The non-dimensionalized parameters appearing above are the acoustic wave speed, c = a.,/m/B~,
(2.24)
h = h/a,
(2.25)
D = a2~m/Bl~,
(2.26)
the cavity lengthscale,
the angular frequency,
SCA'VrERING OF S O U N D BY AN ELASTIC PLATE
391
the coefficient of the non-linear term,
Iz = a2 m2 Ng2 / ( B2p ),
(2.27)
e = pa/m.
(2.28)
and the fluid loading parameter, If the long-time behaviour of the system is sought (steady state solution) then equations (2.16)-(2.24), together with a radiation condition for ~(x, y; t) as x2+ y 2 , co above the plate, fully specify the boundary value problem. In this work the low loading limit will be employed, which (from Leppington [8]) is given as
p~ / mS] ~ O.
(2.29)
If the non-dimensionalized wavenumber of the forcing 12/c is of order unity,
f l / c = O(1),
(2.30)
then small fluid loading corresponds to e --, 0.
(2.31)
The further assumption is made that the remaining variables have order unity: i.e., c = O(1),
/2 = O(1),
/z = O(1).
(2.32)
Note that the non-linear and forcing terms can always be chosen to be of the same order (O(e), tz = O(1)) for any fluid/plate parameter values by increasing or decreasing the incident wave amplitude, ~ accordingly. 3. NON-RESONANT SOLUTION As the fluid loading parameter becomes small, e-* 0, the forcing term in the plate equation (2.18) becomes small also. Hence a trial solution may be posed as
¢b(x, y; t) = eckl(x, y; t) + E2¢~2(X,y; t) + ' ' ' ,
(3.1)
O(X, y; t) = eOI(X, y; t) + E202(X,
(3.2)
t) -I-" " • ,
y;
V(X; t)= eV~(X; t)+ e2V2(X; t ) + ' ' ' .
(3.3)
Substitution into the governing equations and collecting the leading order terms yields 1 v~.~x + v,1 = - i O cos (12x sin O~c) e-'a'+ c.c., D1
1
=v==0,
x=+l,
and
1_
1_
(3.4) I
4)y- 0 r - v,,
(3.5,3.6)
where c.c. denotes the complex conjugate. Similarly, the scattered potential ~bI satisfies equations (2.16) and (2.21), and 01 satisfies equations (2.17), (2.22) and (2.23). The plate deflection is now uncoupled from the fluid potential to leading order, and so the time-harmonic solution to equations (3.4) and (3.5) may be obtained immediately as
v~(x; t) = -ig(x; /2, /2 sin O/c) e-iat+c.c.,
(3.7)
g(x; a, ~) = d(a, ~) cos (~x)+ e(a, ~) cosh (al/=x)+f(a, ~) cos (~I/2x),
(3.8)
where
½ cos/3 d(a,/~) =/~4_ a2 ,
e(o~,/~) = (~2_1. o~) c o s h ot 1/2 ,
(3.9,3.10)
-½ cos/~
f(a, $) = (/32_ a ) cos al/2"
(3.11)
392
L D. ABRAHAMS
The particular frequency values /2 = (n +½)2~r2,
n = 0, 1, 2 , . . . ,
(3.12)
corresponding to plate resonances, will be excluded in this analysis (see reference [14]). Green's theorem may now be employed in the regions above and below the plate to obtain the ~b, ~ potentials. From equations (2.16), (2.21) and (2.19) the solution which has outgoing waves at infinity is
~ l ( x , y ; t ) = ½ i / 2 e -m'
_ g ( x l ; / 2 , / 2 s i n O / c ) H(ol)(/2R/x)dXl+C.C.,
y>O,
(3.13)
where H~1) is the Hankel function of the first kind and R = [ (x - x,)2 + y211/2.
(3.14)
Similarly, from equations (2.22), (2.23) and (2.19), the potential inside the cavity is given by
G(xl, 0; x, y)g(xl ;/2,/2 sin 0/c)
~l(x, y) = 12 e -iat
dx I +
c.c.,
1
[xl< 1,
0
(3.15)
in which the Green function, satisfying the boundary conditions (2.22), (2.23) and ~y = 0 o n Ixl < 1, y = 0, is given as
G ( x l , Yl ; x, y) = m., y~
2 COS (rmx) COS (rmXl) COS(say) COS (Sayl) h{( /2 / c) 2 - r,,2 _ s,}2
(3.16)
The double sum ranges over m = O, 1, 2 , . . . , n = O, 1, 2 , . . . , and r,, = (m +½)¢r,
s,=(n+½)cr/h.
(3.17,3.18)
Interchanging the order of summation and integration yields
4/22rm sin (rm) cos ((/2/c) sin O) cos (rmX) cos (say) ~b~(x,Y) =e-re' ~. , - 7 7 - 7 ~ , , _ 2 . . . . .---~-~,~-_--i-----~ t-c.c., ,,,n h ( r m - / 2 ) ( r m - ( ( / 2 / c ) s m 0) ){(/2/c) - r m - s n }
(3.19)
where it should be noted that all the odd modes in x are zero because of the symmetric forcing. The solution given by equations (3.7), (3.13) and (3.19) is satisfactory unless a plate or cavity resonance is approached. The plate resonances are precluded in this study, and for the cavity resonance the asymptotic solution (3.19) becomes unbounded when, for any n, m, ( / 2 / C ) 2 _ r m2 _ S n2 ._~ O "
(3.20)
This suggests that the potentials may sometimes be larger than the anticipated scalings presented in equations (3.1)-(3.3). In the following sections new expansions are introduced, together with slow-time variables, to investigate the solution in the neighbourhood of a cavity resonance.
SCATTERING
OF
SOUND
BY AN
ELASTIC
393
PLATE
4. SECOND-ORDER RESONANCE The frequency of the forcing is adjusted to lie within an order e of a cavity resonance frequency. Thus, a detuning parameter v is introduced such that
O=to+ev,
e -->0,
v = O(1),
(4.1)
where to = tOpq= cx/[((p + ½)~r)2+ ((q+½)Tr/h) 2]
(4.2)
for a particular pair of integers (p, q). It will be convenient to perform an asymptotic study by introducing a slow-time variable. Thus if,
To = t,
T~ = et,
e -> 0,
(4.3)
then sin (/2t) = sin [(to + ev)t] = sin (toTo+ ~,T1),
(4.4)
and the time derivative may be written as
O/Ot = O/OTo+ eO/OTl.
(4.5)
The sound field inside the cavity can be expected to be larger than that anticipated in section 3, and so a suggested expansion is
~b(x,y; To, TO=~b°(x,y; To, Tl)+e~bl(x,y; To, T I ) + ' " .
(4.6)
This potential is coupled to the plate deflection at O(e) and therefore the expansions in equations (3.1) and (3.3) for ~b and v are still assumed to be valid. Substituting the asymptotic expansions into the governing and boundary equations gives, to leading order, ~o+
o 2 0 ~byy-(1/c )~bToTo=0,
~by °, = 0
Ixl y > - h , = 0, Ixl < 1, y = - h ,
lxl < 1, y = 0;
(4.7) (4.8, 4.9)
x=+l,O>y>-h.
~/,° = o,
(4.10)
This unforced system has a general steady state solution, composed of the resonant eigensolutions, of the form 0 °= ~ Kmn(T1)
cos
e-i',.nTo-t-C.¢.,
(rmX) COS ( S a y )
(4.11)
m,n
where Kmn are as yet unknown functions of T1 and only the modes even in x have been included. The next order, O(e), system may be written as
V~,x+VroTol _ - - i t o [ c o s ( t o x s i n 0 / c ) e - i v r , - K ( T | ) c o s ( r p x ) ] e - i ~ r o +i
Y~ tom,Kin,cos (rmx) e-i'°~J°+ c.c.,
(4.12)
rtl, n
m,n ~p,q
where g ~ g~,
y=~by, 1
y-0,
to = topq.
tx[
Ixl> 1, y = 0 ;
ul
= v ,1= - 0, _
x=+l,
~bxx+ ~byy-(1/c)~broro=0,
(4.13, 4,14) y > 0,
(4.15,4.16)
1 1 ~bxx+ ~byr-(1/cE)~roro=(E/c2)~b°orl,
(4.17,4.18)
and ~bI satisfies the boundary conditions (4.9), (4.10). The solution of the plate equation, (4.12), satisfying the edge conditions (4.14), is v 1= - i [ g ( x ; to, to sin 0/c) e -i~rl - K ( Tl)to(r~- to2)-1 cos (rpx)] e -i*'T° +i ~. Kmn(T1)tom,(r 4 - t o m2 ) - i cos (rmx) e-iOJm Toq_c.c., (4.19) m,n
394
1. D. ABRAHAMS
and so the leading-order scattered potential above the plate may be found, as before, via Green's theorem. All that remains is to calculate the Kin,(T~) by examining the boundary value problem for $1. This potential must be bounded for all time and from Appendix A can only be achieved if
iK'n(TI)+j,,nKm,(T~)=O,
(m,n)#(p,q),
(4.20)
iK'(T~) +jK(Ti) - k e -t~r~ = 0,
(4.21)
where
k
=
(c2/h)
cos (rpx)g(x; to, to sin O/c) dx,
(4.22)
1
Jmn = c2tOmn/[h(r~ --tom,)], 2 •
J=j~,
K'(T~)=dK/dT~.
(4.23-4.25)
Integration reveals that
Kmn As,, exp (ijm, T~), (m, n) # (p, q),
(4.26)
K = A exp (ijT1) + k e-i~rl/(j + v),
(4.27)
=
where Amn are as yet undetermined constants. By examining the be shown that secular terms in /'1 can be avoided if and only if A,,n = 0,
O(e 2) problem
all (m, n).
it can (4.28)
Hence, from equation (4.19),
vl(x; To, 7"1)= - i [ g ( x ; to, to sin O/c) -k,w cos (rpx)/{(r4-to2)(j+ v)}] e-i(~ro+~T~)+ c.c.,
(4.29)
and so, to leading order, [g(xl ; to, to sin O/c)
~bl(x, y; t) = 2Litoe -iat ! -
ko cos (rpx~)/{(r~- to2)(j + v)}] H~ol)(~,g/c) dx~ + c.c.
(4.30)
The solution obtained in this section shows that near resonance (within O(e) of a resonant frequency) the scattered field above the plate is modified but does not change its order of magnitude. However, as j is real, the detuning parameter v may be adjusted so that the eigensolution coefficient K becomes large or even unbounded. This occurs when u=-j,
(4.31)
and suggests that there is an inner region of the local resonance range where the scalings must again be altered. Thus the region examined in this section is called the second-order resonant range, and it will be shown a posteriori that the leading-order resonant range lies within O ( e 2) o f /J = - - j . 5. LEADING-ORDER RESONANCE It will be assumed that the sub-region of the O(e) resonant range is of order e 2. To examine the solution in this region the forcing frequency is written as ~-2 ----to -- e j ' l " e 2 0 ",
(5.1)
SCATTERING
OF SOUND BY AN ELASTIC PLATE
395
where tr is the new detuning parameter and j is given in equation (4.23). Because of the e 2 term in the frequency it is necessary to introduce a second slow-time variable, namely
T2 = e2t,
e ~ O,
(5.2)
so that sin (Ot) = sin (toTo-jT1 + trT2).
(5.3)
The time derivative now becomes
O/at = a/aTo+ e O/OT1 + e 2 O/OT2
(5.4)
and the cavity potential is expanded as (cf. [17])
O(x,y; To, T1, T2)=e-ld/-~(x,y; To, TI, T2)+ 0 ° + e ~ l + "'" •
(5.5)
From the form of the coupling in equation (2.18) it is expected that the plate deflection and outer potential scale as
v(x; To, 7"1, T2) = v°+ evl+ "'" ,
and
4~(x, y; To, TI, T2) = 4~°+ ed~+ "'" • (5.6, 5.7)
These are again substituted into the full system (2.16)-(2.23), and the leading order potential (O(e-~)) is found to satisfy the governing equation and boundary conditions (4.7)-(4.10). Similarly, the O(1) potentials ~b°, 4,0 are defined by the system (4.9), (4.10), (4.14)-(4.18), whereas v ° is given by o
-q, --1 To,
l) xxx, x dl_ I) Oo To =
Ixl
(5.8)
It will also prove necessary to examine the O(e) terms, which are governed by the following equations: 6 ~ + ~b~r- (1/c2)6 ~-oro= (2/c2)d~°or,, 1
1
2
y > 0,
(5.9)
1
O,,x+~yr-(1/c )~roro =(2/c2)~b°roT +(1/c2)tbr~r + 2 VxVxdx
Vx=x Vroro=/Z
2 -1
Ixl
0
(5.10)
v ~ - 2 V r o r , - 2 t o c o s ( t o x s i n O / c ) sin(toTo-jTl+trT2)
1
o
o
+ 4, o- q' o-
Ixl
1 + vr,, o ~b~= ~b~= Vro
(5.11) Ixl< 1, y =0.
(5.12)
The boundary conditions to this order are given in equations (4.9), (4.10), (4.14) and (4.17). To commence the solution, the work in section 4 reveals that
~b-1(x,y; To, TI, T2) = ~ KmI.(TI, T2) cos(r,,,x)cos(s.y)e-i%""r°+c.c.,
(5.13)
m, rl
with (from expression (5.8)) to,.. cos (r,x) T2) 7"--4"-'-T"7, e-'°'""r°+c.c.,
v°(x; To, T, T2)=i ~ K ~ ( T 1 '
m,n
"
(5.14)
(rm --torah)
and 2 i to.,,e ~b°(x, y; To, 7"1 T 2 ) = - ~ E
'
m,.
-i~
-- 1"ot- t- -m1 -t T l , T2) • 2
( r ' - to'.)
cos (r'x~) H~l~(to',R/c) dxl + c.c.,
x 1
y>0,
(5.15)
396
L D. ABRAHAMS
where R is given in equation (3.14) and Km~, are as yet unknown functions of T1 and T2. Hence equation (5.14), together with Appendix A, allows the second order potential inside the cavity ~b°, to be determined as
¢°(x,y;To, T1, T2) = ~ ~ t o m . ,.,n
sin (s.y) cos (rmX)jmnKmln e -i%~J°
SnC
+ ~ K ° , cos (s,y) cos (r.,x) e - i % J o + c.c.,
(5.16)
m,n
where Km. o is another matrix of as yet undetermined coefficients. The secularity condition (A14) gives K~,I, = A'~(T2) exp (ijm, T1),
all (m, n),
(5.17)
where A,,, are unknown functions of T2 and jm, are given in equation (4.23). Although only the leading order solution is required, it was necessary to examine ~/,o in order to determine K ~ as a function of 7"1. However, the leading potentials $-~, $o, v ° still contain the unknown functions A.,,. Thus it is now necessary to obtain the next order plate deflection term, v 1, and use this in the $1 boundary value problem. The imposition of the secularity condition (A14) should then specify the potentials to leading order. The plate deflection, v ~, is determined from equation (5.11) and is forced by several terms including the non-linear expression
/~
(i 1 oo)o
VxVxdx Vxx.
(5.18)
This may be written, from equation (5.14), as /~i Y~ r.,Kmnqm, -1 -1 Kot -1 e -i('~m"+~%'+o%')T° cos (rmX) ~ q~qvt{Kw ,., n
O,S, t
-2/(SIK~) e-itc°~"-°'°*+~%')T°+/~/~vt1 e-i("°~"-'%'-%")T°}+C.C.,
(5.19)
in which 4 2 qm, = rmtor,,/ ( rm -- tOm,)
(5.20)
and/(~,1 denotes the complex conjugate of K ~,~,. Although there are many terms in this expression, only those which oscillate (in the slow-time variable) at cavity resonance frequencies (4.2), tO,.,, will be present in the seculadty condition to be established. Thus, unless the duct height h is chosen specifically, the only terms from exp [ - i ( t o m , - tOys+ tOrt) To] which are important are those for which s = t or v = m, s = n. Similarly, tO.,, - tars tOrt contributes when v = m and s = n or t = n, but all terms like to,., +tO~s + to~, are irrelevant. Hence in the generic situation (for other cases see section 6), expression (5.19) may be written as
_ l t i ~ r m K - I,,,q,", cos (rmX) e-i=~"r°{ 3lKm, 2q2m,+6 ~ IKms[ --122qms+2 ~ IKv,--122} I qos .,,n
+.
s s#n
• .+
v,s v#ra
(5.21)
c.c.,
where " . . . " indicates terms which may be ignored for most h values and IK2I 2 = K ~) g L-).
(5.22)
The only other term on the right side of equation (5.11) which presents any difficulty is ~°0, because it contains a rather complex function of x. This may be simplified by noting
SCATTERING
OF SOUND
BY A N E L A S T I C P L A T E
397
that 4~°o is even in x, and as it is only required over the region Ixl < 1, y = 0, Fourier decomposition gives ~b°o(X,0; To, T~, T2) --1
2
Ixl
= - ~. Km"°J="qs"~.(bm"+idT")cos(rvx)e-i'°""r°+c.c., .,.. 2r,.
(5.23)
in which the real coefficients b~", d~" are given by bm"+id~" =
II
cos (rvx) cos (rmXl) H~ot)(tomJx- X , I / c ) dxl dx.
1
(5.24)
1
The plate deflection, v t, is now easily found from expression (5.11), with use of expressions (5.13)-(5.16), (5.21) and (5.23), and this may be substituted into equation (5.12) to yield the boundary condition on y = 0 for ~b~: namely, ~#~(x, 0; To, TI, 7"3)=
t o , to
--oJg(x;
sin O/c) e -i°~T°+ijT°-icrT: • mPI (b~t,l~l +ld~ ) c o s (fox) 4 2
Y~
-' ~ Kmnqmng°mn
~
ra,n
2 rm
v
+i
r v -
to,,, cos (rmx) f
+ ~
4
m,n
2
rm--(Dmn
e -ioj.,.T°
tO m n
o
~¢-°mnKmn+
~
2ir*mK~rl] 2
~
rm--tOmn
4
J
e-i%""T°
- I t ~. K mnq -, 2ran COS (r,.x) m, tl
x{3'K~,l.I 22q.,.+6 Y~ , K m , , q , .2s + 2 s
s~n
+ " " + c.c.,
. K ~Ts , oq ~2-} e 1- , o2, v,x
v#m
Ix I < 1.
(5.25)
As before, to is chosen to be the (p, q)th cavity eigenfrequency, where (p, q) is any particular pair of integers and 2 to = tom = cx/{rp2 + sq}.
(5.26)
The governing equation for et is given by expression (5.10), which may be written, upon using equations (5.13) and (5.16), as 1
1
~,x~+,y~-(1/c
2
1
)¢ToTo
-' . - , } COS (rmX) = (1/C ~) Y {KmnTtT , --21¢omnKmnT2
COS
( s . y ) e _ "°'"r°
m, tl
-(2i/c 4) ~. (y + h )( ~ozs./ s.)jm,K ..~r, cos (r.x) sin ( s.y ) e ~ ' J o m, n
-(2i/c2) E to,..K,..r, o cos (r,.x) cos (s.y) e-t°m"T°+ C.C.
(5.27)
m,n
Each (m, n) term of q/satisfies the boundary value problem given in equations (A1)-(A4) with A, B, C and f ( x ) chosen appropriately. Hence. the secularity condition (A14) may
398
L D. A B R A H A M S
be employed, which, after considerable simplification, gives
Amnt°mnC h2(r 4 - tom. )2
2
2 • m nm - d,~") - 2c 2 ( r m 4 " C O m n ) 4 . - " m htomn(lb 4 ( rm + CO2ra,) S~
-(2/h)Ara.l~c qra. 3qra.lAm.[
~
q~,lA,~l:+2 Z
$ s~n
qv.lAw
t~,$ t~m
+ 2iwmnAmnr2- 28mpSnq°~k e-i~T2 = --2i¢Omn~Tt ( K°" e-iJm"TQ'
(5.28)
where k is written in equation (4.22) and 8rap is the Kronecker delta. The left side (LHS) of this equation is purely a function of 7"2only, and so integration with respect to T~ yields o _
Kmn -
Bra. e +ij'..T, 4- i
{LHS of (5.28)} T~ e +ij'--T,,
(5.29)
2~mn
in which Bin. are undetermined functions of T2. However, it must be insisted that Kin.° remains bounded for all time and so cannot have a term proportional to 7"1. Therefore the left side of equation (5.28) must itself be identically zero. This is the amplitude equation for the leading-order terms. The solution is simply found by writing Am. in its modulus and phase form, namely
Ara. =/3 m. e i~".
(5.30)
Substituting into the homogeneous form of equation (5.28) and collecting the imaginary terms gives 3 2 ran 4 2 2 d/3ra./dT2+{tOm.C bra /2h(rraO,m.) }/3.. =0
(5.31)
for any (m, n) not equal to (p, q). Thus 3 2 ran 4 2 2 flra.oZexp{-tOm.C bra T2/2h(rm-O~m.) },
(m, n ) # ( p , q ) ,
(5.32)
where it is easy to show that b.~"> 0, and so, as a steady state solution is sought, these terms are all exponentially small and can be neglected. The only eigensolution which is non-zero is therefore the one which oscillates near the forcing frequency. The amplitude equation may be solved by writing Apq as A~ =/3 exp [iT - io'T2],
(5.33)
and collecting real and imaginary parts. This yields
dfl/dT2=-A~/3-ksin
T,
[3 dT/dT2=fl(A2+cr)-A3fl3-kcos
T,
(5.34,5.35)
where 2
3
AI=C ¢0
.,c 2
pq 4 bp /2h(rp-tO
fc2r
2 2
)
(5.36)
,
(g+,ob
}
A2=2h(r~_w2)2 [-~q - z c (r~ ¢02) -d~qho~ 2.,
(5.37)
4 3 4 A3-(3/h)l~c 2 rpoJ (rn-oJ 2 ) --4 .
(5.38)
Steady state solutions are obtained by setting d/3/dT2 and dT/dT2 to zero and both equations may be combined, on elimination of the phase factor % to give /32{A~+ [A2 + tr - A3/32]2} = k 2.
(5.39)
399
SCATTERING OF S O U N D BY AN ELASTIC PLATE
This is a simple equation, cubic in/~2, and has an identical form to that obtained for a primary plate resonance (see equation (3.35) o f reference [14]). As d is proportional to /~8, a typical plot of the amplitude/3g against the detuning parameter o, is shown in Figure 2 for a range of values of A3 (and hence a range of incident wave amplitudes g). Figure 3 displays the graph o f phase angle y against cr for a particular set of fluid/plate parameters and fixed/~ Note that the modulus of Apq tends to zero as or-* :~oo and has a maximum value of
/3 = Ikl/x,.
(5.40)
For a given physical problem, and incident wave angle, the constants a~, A2, A3 and k can be determined. Therefore/3 and y are obtained from equations (5.34), (5.35) or (5.59) and depend on the detuning parameter or. As the solution curve for/3 against cr i
I
O Detuned frequency, o"
Figure 2. Scattered wave amplitude against detuned frequency for a range of values of the incident wave
amplitude. The dashed line indicates the unstable part of the curve, and the unbroken line indicates the stable region. I
o.
0 Detuned frequency,
Figure 3. Phase of the scattered wave against detuned frequency for a fixed incident wave amplitude.
400
I.D.
ABRAHAMS
may be multi-valued, the actual value of/3 (i.e., the particular branch of the curve which is taken) can depend on the initial conditions for the problem (see concluding remarks). The leading-order solution near a cavity resonance has now been found, and from equations (5.13)-(5.15), may be written as ~b~(2fl/e)cos(Ot-y)cos(rex)cos(sqy),
Ixl~ 1,
v ~ flto(r 4 - to2)-~ cos (rex) sin (/2t - y), 4~-{-ito2~/2(r~-o)2)}ei'-m'
O>-y>~-h,
Ixt <~1,
cos(rexOH~l)(toR/c)dxl+e.c.,
(5.41) (5.42)
y>~O.
(5.43)
Finally, the far field form of the scattered potential 4~ is easily obtained by writing x = ! cos O, y = I sin O, 0 < O < or, where tol/c ~ oo and I is assumed to be I >>1. This gives q~ o,t/~
2to2rgfl COS [(to/c) cos O] sin rp e i(~°l/c-3~r/4+y-Dt) + c.c., (r~ - to4)ff(2"n'tol/c)(r2 - (to~c) 2 cos 2 O)
O # 0, ~r. (5•44)
It is a straightforward matter to employ the substitutions given in expressions (2.11 )-(2.15) to rewrite the potentials and plate deflection back in terms of the physical variables. For the sake of brevity these are not written here• 6. EXCLUSION OF COUPLED PRIMARY AND SECONDARY RESONANCES It was shown in a previous paper [15] that for specific values of the plate/fluid parameters, combined secondary and primary plate resonances could occur. This led to a scattered field consisting of two terms, one oscillating at the forcing frequency, and the other oscillating at three times or 1/3 of this frequency. However, for the particular plate/cavity configuration chosen in this paper, the coupling between two cavity eigenfrequencies is not possible• In order to illustrate this, a particular case is now examined. The duct height h is chosen to have the value x/2, so that (from expression (4.2)) tOoo= cx/(Tr2/4+ ~r2/8) = x/(3/S)c~r,
(6.1)
too2 = cff(zr2/4+25~r2/8) = x/(27/8)cTr,
(6.2)
and hence too: = 3tooo. The forcing frequency/2 may be chosen to be close to tooo or too2 ( w i t h i n o ( e 2 ) ) , and therefore the superharmonic or subharmonic secondary frequency (too2 or tOoo respectively) will be coupled through the nonlinear term in equation (5.1). Thus the two terms • ---1 2 --1 2 3/zlr0(Koo) Ko2 qooqo2 cos (foX) ei(2~°°-°~°2)r°+p.iro(K~ol)3q~ cos (foX) e -31°'°°r°, (6.3)
which were absorbed into " . . . " in equation (5.21), will now be included in the secularity equation as too2-2tooo = tooo and 3tooo= too2. The boundary condition for ~b1 is modified by the inclusion of ----1 2 --1 3 - - i t o "/~ --1 3 3 " 3bt(Koo) Ko2 qooqo2 cos (rox) e oo o+p,(Koo) qooqo2 cos (rox) e -1'°°2r°
(6.4)
in equation (5•25) and hence the right side of the secularity condition (5.28) becomes -(2/h)cE~q~oqo2{38moS,,oA02.~200 ei(Jo2-3Joo)r, + 8moS,,2A~o e-i(~o2-3Joo)r,} - 2ito,.. ~d--T=( K ° . e-iJ..r0.
(6.4)
The left side remains unaltered and, as •
•
2
3
4
2
4
2
Jo2 - 3joo = 6c tooo/h ( r o - 3tooo)(ro- tooo) # 0,
(6.5)
SCATTERING
OF
SOUND
BY AN
ELASTIC
PLATE
401
integration with respect to T1 gives K 0m~= Br,, e '""Jm"rL+
-t
i
2to,nn
{LHS of (5.28)} T1 e ij'"r~
C2tZq~oq°2 {38,,o8,oAo2.42 e i(j°2-ej°°)T'-8,.oS.2A 3e3%°T,}. hto,,, (jo2 - 3joo)
(6.6)
As before, Km, o cannot grow linearly with the slow time T~ and hence the left side of equation (5.28) must again be identically zero. Coupling between modes has not therefore occurred and equations (5.31), (5.34) and (5.35) are still valid. It is straightforward to show that for any value of h (and therefore for any pair of eigenfrequencies which have t0r,~ = 3aJst) the coupling does not occur between amplitude equations and so the solution written in equations (5.41)-(5.43) is always valid. 7. CONCLUSIONS The asymptotic values of the acoustic potentials inside and outside the cavity, and the plate deflection, have now been determined. Away from a cavity resonance, the elastic plate gives rise to a small scattered potential O(e), and the cavity field and plate deflection are also both O(e) in magnitude. As the incident wave is adjusted so that is approaches one of the cavity eigenfrequencies (section 4), the cavity potential becomes O(1) in size. Therefore, the scattered field above the plate (4.30) is altered, but does not change its order in the small parameter, O(e). However, within the region about an eigenfrequency (O(eZ)), there is a sub-region (section 5) in which the cavity potential further increases its magnitude to O(e-~), i.e., becomes very large. This results in the scattered potential increasing in size to O(1), and at this order both the plate non-linearity and radiation damping become important (these orders agree with previous linear results [ 17]). It should be noted that the combined potential ~b°+ eq~~ in section 5 (~b° given in expression (5.43) and 4~~ found from expression (5.25)) tends to the value of ~b~ given in expression (4.30) when or becomes very large. This is easily seen by letting tr = (~,+j)/e which, from equations (5.34), (5.39) gives/3 ei~ - ke/(~, +j). Similarly, both of these solutions tend to the non-resonant result (3.13) as v ~ O (1 / e). As has already been discussed, the plate non-linearity leads to a distortion of the modulus and phase against detuning parameter curves (Figures 2 and 3). This gives rise to the possibility of jumps in the potential as the frequency is varied. Related to this is the fact that the amplitude equation (5.39) contains multi-varied solutions for specific parameter regimes. The actual solution curve attained is determined by the initial conditions for the system (see reference [16]). Note that the jump phenomenon and multi-valued curves are also obtained when/38 is plotted against g, the incident wave amplitude, for fixed tr. Perhaps the most interesting effect of the non-linearity is the possibility of jump discontinuities in the modulus/3 (and hence in the far field scattered potential) as the incident wave angle, 0, varies. This occurs because k in expression (5.39) is a function of/9, and shows that reciprocity does not hold between 0 and 19. The jump discontinuities are illustrated, for specific values of the physical parameters in Figure 4. The variation in jump positions a s 0 is increased or decreased (hysteresis) should be noted. Also, the non-linearity,ean be seen to actually increase significantly the amplitude of the acoustic potential for some angles of O. This paper has illustrated that the non-linearity appearing in the thin elastic plate equation has a signitieant effect upon the acoustic field near a cavity resonance, A novel
402
t.D. ABRAHAMS r
1"
1
IrlA
2 7r18
o.
.E
i
g
¢)
w18
Ir12
Incident wave angle, e
Figure 4. Scattered wave amplitude against incident wave angle. The dotted line denotes the linear solution, the unbroken line denotes an increasing 0 path, and the dashed line denotes a decreasing 0 path.
feature o f the work is that whereas the secularity condition in multiple scales analyses is normally derived from a one-(space)-dimensional boundary value problem (e.g., [14, 16]), in this physical model the necessary condition is obtained from a two-dimensional problem. Finally, in section 6 it was proved that coupled primary and secondary resonances could not occur. However, if the amplitude of the incident wave was large, it can easily be shown that superharmonic or subharmonic secondary oscillations and combination resonances will exist [ 14]. Coupled resonances could occur if the cavity width was different from the plate width, or indeed if the cavity width became infinite (duct). This will b e an area of future study, as will the extension of these models to two-dimensional planar (circular and rectangular) or cylindrical elastic plates. ACKNOWLEDGMENTS This research was partly supported by the United States Air Force under Grant No. AFOSR-85-01-50, and by the United States Navy under Contract No. N000014-83-C-0518. REFERENCES 1. D. O. CRIGHTON 1971 Journal of Fluir,! Mechanics 47, 625-638. On acoustic beaming and reflection from wave-bearing surfaces. 2. P. A. CANNELL 1975 Proceedings of the Royal Society of London A347, 213-238. Edge scattering of aerodynamic sound by a lightly loaded elastic half-plane. 3. H. G. DAVIES 1974 Journal of the Acoustical Society of America 55, 213-219. Natural motion of a fluid-10aded semi-infinite membrane. 4. F. G. LEPPINGTON 1978 Journal of Sound and Vibration 58, 319-332. Acoustic scattering by membranes and plates with line constraints. 5. G. P. EATWELLand J. R. WILLIS 1982 IMA Journal of Applied Mathematics 29, 247-270. The excitation of a fluid-loaded plate stiffened by a semi.intnite array of beams. 6. E. H. DOWELL 1975 Aeroelasticity of Plates and Shells. Leyden: Noordhoff !nterx~ational Publishing. 7. Y.C. FUNG 1958 An Introduction to the Theory of Aeroelasticity. New York: John Wiley & Sons. 8. F. G. LEPPINGTON i976 ~arterly Journal of Mechanics and Applted Mathematies 29; 52%546. Scattering of sound waves by finite membranes and platas near r e s o n ~ .
SCATTERING
OF
SOUND
BY AN
ELASTIC
403
PLATE
9. I. D. ABRAHAMS 1981 Proceedings of the Royal Society of London A378, 89-117. Scattering of sound by a heavily loaded finite elastic plate. 10. J. H. GINSBERG 1978 Journal of Sound and Vibration 60, 449-458. A re-examination of the non-linear interaction between an acoustic fluid and a flat plate undergoing harmonic excitation. 11. A. H. NAYFEH and S. G. KELLY 1978 Journal of Sound and Vibration 60, 371-377: Non-linear interactions of acoustic fields with plates under harmonic excitations. 12. E. H. DOWELL 1966 Journal of the American Institute of Aeronautics and Astronautics 4, 1267-1275. Nonlinear oscillations of a fluttering plate. 13. C. H. ELLEN 1977 Journal of Sound and Vibration 54, 117-121. The nonlinear stability of panels in incompressible flow. 14. I. D. ABRAHAMS 1987 Proceedings of the Royal Society of London A414, 237-253. Acoustic scattering by a finite nonlinear elastic plate. I. Primary, secondary and combination resonances. 15. I. D. ABRAHAMS 1988 Proceedings of the Royal Society of London A418, 247-260. Acoustic scattering by a finite nonlinear elastic plate. II. Coupled primary and secondary resonances. 16. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: John Wiley & Sons. 17. I. D. ABRAHAMS 1982 Quarterly Journal of Mechanics and Applied Mathematics 35, 91-101. Scattering of sound by finite elastic surfaces bounding ducts or cavities near resonance. APPENDIX A In order to suppress any secular b e h a v i o u r o f ~bl(x, y; To, T~) in section 4, and ~b° and ~b~ in section 5, it is useful to examine the following problem: $~, + ~byy- (1/c2)$Zoro = {A cos [(q +½),ry/h] + B(y + h) sin [(q + ½)Try~h]} cos [(p +½),rx] e -i'°T°
[x]
x=+l,O
0
¢=O, Ix]<~l,y=-h,
(A1) (A2, A3)
Ixl ~< 1, y = 0,
(A4)
where
to = tom = cTrx/[(p + 1 ) 2 + ((q +½) ~r/h)2].
(A5)
It is convenient to m a k e the substitution ~ = 4~ + 2 r r ( q + ½ )
A + 2 ~ . ( q +½)
- ~ - (y + h) cos [(q +½)1ry/h]
sin[(q+½)~ry/h] cos [(p +l)Trx] e -i~T°,
(A6)
so that ~b satisfies the h o m o g e n e o u s wave equation, and the b o u n d a r y condition on y = 0 becomes 1 q~y = [ C f ( x ) - ( h / 2 ) { A - hB/2~r(q +½)} cos ((p +~)Trx)] e - i~ o To,
Ixl
1.
(A7)
This m a y be expressed as the Fourier series thy = C ~
a~ cos [(m +½)~rx] e -i°'ro ,
(A8)
where am =
I
+l
cos [(m +½)Trx]f(x) dx,
m # p,
(A9)
--1
a p = ( - h / 2 C ) { a - h B / 2 z c ( q + ½ ) } + I ~ cos [(p+l)*rx]f(x) dx. 1
(A10)
404
i.D. ABRAHAMS
The solution to the steady state boundary value problem may now easily be found as
4' = C ~ am cos [(m +½)¢rx] sin [bm(y + h)] e-i°'ro-~ h(y + h) .1=o bm cos (bmh) 2~r(q+ 1) x[\~:A+ 27r(q +½)]hB~ sin
[(q+l)~'y/h]-B(y+h)
cos [(q+½)zry/h]}
x cos [(p + ½)¢rx] e-i~'r°,
(A11)
where bm= x/[(to/c) 2 - ((m +½) ¢r)2].
(A12)
However, from equation (A5), b~ = (q + 1)7r/h and so the denominator of the pth term vanishes. Therefore 4' can be bounded only if a, =0,
(A13)
which corresponds to A
hB 2¢r(q+½)
-
2C Ih
cos [(p +½)~rx]f(x) dx.
(A14)
This is the secularity condition which can be used to evaluate the amplitudes of the resonant cavity eigensolutions. Finally, it should be noted that equation (All) is the particular solution of 4' for the steady state boundary value problem. Complementary functions, or eigensolutions, as written in equation (4.11), may be added to this solution.
of Sound
and
Vibration (1989) M(3), 387-404
SCATTERING OF SOUND BY A FINITE NON-LINEAR ELASTIC PLATE BOUNDING A NEARLY RESONANT CAVITY I.
D.
Department of Applied Mathematics, 7%e
ABRAHAMS
University, Newcastle upon Tyne, NE 17 RU, UK
(Received 3 August 1988) A thin elastic plate of finite width is set in an infinite rigid bafIIe. A rectangular cavity with pressure release walls is appended to the underside of the plate, and a compressible inviscid fluid occupies the region above the batlIe and inside the cavity. The fluid is assumed to be light compared to the plate mass. Time-harmonic plane acoustic waves are incident onto the plate from above, and are of sufficient amplitude to necessitate the inclusion of a non-linear term (due to mid-plane stretching) in the plate equation. The plate deflection and scattered sound field are obtained for non-resonant frequencies, and are shown to increase in magnitude as a cavity resonance frequency is approached. The
method of multiple scales, involving two slow-time variables, is employed to obtain the leading-order asymptotic solution, and the orders of magnitude of the potentials and plate deflection are shown to agree with previous results for the fully linear problem. The plate non-linearity is found to introduce jump discontinuities in the scattered wave amplitude as it varies with frequency, incident-wave angle or incident-wave amplitude. Secondary and combination resonances are possible, but coupled primary and secondary resonances are shown to be impossible for the particular configuration chosen.
1. INTRODUCTION AND BACKGROUND There has been a good deal of interest shown in recent years on the interaction between sound waves in a fluid and flexible, or wave-bearing, surfaces. Early work was concentrated on the scattering of sound waves by infinite, semi-infinite and ribbed elastic structures (see, e.g., references [l-5]). These models were amenable to exact mathematical analysis and yielded interesting phenomena such as coupled fluid-structural waves and edge amplification. It is obviously more physically realistic to examine problems which contain finite elastic structures. Unfortunately, in these cases only approximate methods are available to solve the mathematical initial or boundary value problems. One procedure that could be employed is that of the method of modal expansions and truncation which leads to a finite algebraic system of equations. This is the so-called Galerkin approximation and has been extensively employed on finite elastic plate problems by workers in aerodynamic flutter [6,7]. However, in fluid-structural problems a physically useful quantity called the fluid loading parameter is conveniently large or small. This is the ratio of tluid mass in one wavelength to the mass of the elastic surface, and is often small in aeroacoustic situations and large in underwater applications. Hence, perturbation methods can be employed in either limit; for example, see the paper by Leppington [8] for the low loading case, and for heavy loading the method of matched asymptotic expansions was employed by the author [9]. Both these works showed that, for specific forcing frequencies, standing waves could form on the plate, and this led to an increase in magnitude of the plate deflection and also to an increase in the intensity of the scattered acoustic field. 387 0022460X/89/090387+
18 %03.00/O
@
1989 Academic Ress Limited
388
i.D. ABRAHAMS
All the studies involving finite elastic plates or membranes show that, at resonance, large plate deflections and large fluid fluctuation pressures occur. Researchers in aerodynamic flutter were the first to suggest that non-linearities in the plate and fluid equations may therefore become important. Some work has been performed with both non-linear fluid and plate equations [10, 11] but it can be argued that the dominant non-linearity occurs in the thin elastic plate equation [6], and is the result of in-plane tension caused by large deflections. Most of the flutter literature therefore incorporates this non-linearity only [12, 13]. The author [14, 15] has recently examined the problem of scattering of sound by a lightly loaded finite plate (Leppington [8] looked at the linear case) when the incident wave amplitude is large enough to justify inclusion of the non-linear tensional term. The main effect of the non-linearity is to distort the amplitude against frequency curve near resonance and to lead to jump phenomena and hystersis effects [16]. It was also shown that secondary and combination resonances could be excited so that the scattered field may contain terms which oscillate at frequencies different to that of the incident wave. This paper extends the above work [14, 15] by the inclusion of a soft-walled cavity behind the elastic plate (see Figure 1). The scattering problem has relevance to the sonar detection of underwater vehicles, and the radiation problem is an obvious simple model of a loudspeaker in which the speaker cone (modelled by a thin elastic plate) is forced at one or several specific frequencies. The linear case with a rigid-walled cavity has already been investigated in a rather ad hoc manner by Abrahams [17] and, as for plate resonances, the system gives rise to cavity resonances which produce large plate deflections and a large scattered field. In this study the method of multiple scales is employed (see the book by Nayfeh and Mook [16]) to solve the steady state boundary value problem in a rational and asymptotic fashion. The forcing is supplied by monochromatic incoming plane waves of small amplitude, yet it is assumed that the amplitude is of sufficient size to merit the inclusion of the non-linear tensional term in the plate equation. A mechanical vibration of the plate could equally well have been taken as the forcing, and this would not have altered the method of analysis.
T 1
t~
20
>
Figure 1. Physical configuration showing incident waves.
In section 2 the full initial-boundary value problem is specified. The system is nondimensionalized and the fluid loading parameter, e, is defined. For small e, and frequency of forcing well away from the cavity and plate eigenfrequencics, the solution is obtained to leading order in section 3. This solution breaks down within O(e) of a cavity resonance
SCATTERING
OF SOUND
BY AN ELASTIC
389
PLATE
frequency (called a second-order resonance) and so in section 4 new expansions are presented for the acoustic potentials and plate deflection. These potentials are also found to be unbounded near a particular frequency (within O(e2)) and so in section 5 another set of expansions is again introduced. Two slow-time scales are needed for the analysis and this leading-order resonance solution is obtained in terms of a complex function which satisfies a simple non-linear equation. The leading-order far-field acoustic potential is obtained. In section 6 it is shown that, for the particular geometry chosen in this paper, coupled primary and secondary resonances cannot occur. Concluding remarks are presented in section 7.
2. GOVERNING EQUATIONS A simply supported thin elastic plate of width 2a, and infinite length, is set in an infinite rigid baffle lying in the plane )7 = 0. The elastic plate lies on )7 = 0, [2[ < a, where (2,)7) are Cartesian co-ordinates which are perpendicular to the direction of infinite extent of the plate. An inviscid compressible fluid occupies the semi-infinite region above the plate, and a soft-walled rectangular cavity, of width 2a, height/~, and infinite length, is appended to the underside of the flexible surface (see Figure 1). The region inside the cavity is assumed to contain the same fluid as that above the plate, but in fact a liquid or gas with a different sound speed could easily be taken instead. Observing that all dimensional variables are denoted in this paper by an overbar, the plate deflection ~ is governed by the equation
BJ'--V-N024
(0~/02)2d2
o2---~+mo--~=-p(~,o)l+_, y=o,
12)
(2.1)
where B = Eb3/12(1 - v 2) is called the plate bending stiffness, N = Eb/4a, E and v are Young's modulus and the Poisson ratio, respectively, b is the plate thickness and m is the mass per unit area of the plate. Time is denoted by 5, and/~(~, 0)[_+ is the discontinuity in fluid pressure across the plate: namely, /~(2, 0)1+ _ --lim {/~(2, q)-/~(2, -q)}.
(2.2)
q-~0
A velocity potential ~(2, )7, t-) may be defined for the fluid in terms of 3,/~ as
t~(~, )7; t-) = - ( 1 / p ) p ( ~ , )7; t-),
~ ( ~ , 0; t-) = ~7~(~,0; t-),
(2.3, 2.4)
and for small disturbances of the compressible fluid this potential can be shown to satisfy the linearized wave equation ql3e~+ q ~ = (1/c2) q~rr.
(2.5)
The subscripts in equations (2.3)-(2.5), and henceforth, denote partial differentiation with respect to the given variables, p is the mean fluid density and 6 is the acoustic propagation speed. To specify fully the model problem it is necessary to enforce edge and radiation conditions, and to state the chosen forcing term. Firstly, the elastic plate is simply supported, and so the deflection and the bending moment vanish at each end: i.e., v = v~ = 0,
x = +a.
(2.6)
For mathematical convenience it is assumed that the plate is excited by a pair of symmetric, purely two-dimensional, plane acoustic waves, ~i say. These are incident from above
390
I.D. ABRAHAMS
and both travel in directions which subtend an angle 0 with the positive )7 axis. This may be written as q~, = l g c o s [/~07 cos 0 + £ sin 0 + ~t-)/~] +½gcos [~(fi cos 0 - ~ sin 0 + ~t-)/~]
(2.7)
where g is the amplitude and /2 is the angular frequency of the wave. The scattered potential above the plate is defined as 4g($; )7; t-) = • - qS, - q~,,
)7 > O,
(2.8)
where the reflected wave ~r is given by qbr(g, )7, t-) = ~,(~, --)7, t-),
(2.9)
and it is insisted that only outgoing or decaying waves are present in g~ as ~2+)72~ oo. Inside the cavity the scattered potential is written as
77(~;)7; t3= ~15(~,)7;t-),
[~[
(2.10)
It is required to obtain a steady state solution to the scattered acoustic field and plate deflection. It will. prove useful, before employing any asymptotic approximations, to nondimensionalize the initial-boundary value problem. This may be achieved by scaling the dependent and independent variables as follows: x=~/a,
y=)7/a,
t=x/{B/ma4}T,
(2.11,2.12)
v(x; t)= ax/{B/ma4}8-1~(x; t), ~b(x, y; t) = $-'4g(~, )7; t-),
(2.13)
qs(x, y; t) = g-'~(~, )7; t").
(2.14, 2.15)
On substitution into equations (2.1)-(2.10) these yield
qbxx+qbyy=(1/c2)~ptt, ~,xx+q,~y=(llc2)@,,, V~xx~- e~
-l v~ dx
y>O,
Ixl< 1,
O>y>-h,
(2.17)
v,~ + v,, = e{2D cos (Ox sin O/c) sin (Ot) -4~,(x,O;t)+O,(x,O;t)}, v, = 4~y= ~y ;
V=Vxx=O, x=+l; ~b=O,
(2.16)
Ix[
y=-h;
Ixl
Ix[ < 1, y = O,
y=O,
(2.18) (2.19)
d,,=o, Ixl>l, y=O,
(2.20,2.21)
~b=O,
(2.22,2.23)
x=+l,
O>y>-h.
The non-dimensionalized parameters appearing above are the acoustic wave speed, c = a.,/m/B~,
(2.24)
h = h/a,
(2.25)
D = a2~m/Bl~,
(2.26)
the cavity lengthscale,
the angular frequency,
SCA'VrERING OF S O U N D BY AN ELASTIC PLATE
391
the coefficient of the non-linear term,
Iz = a2 m2 Ng2 / ( B2p ),
(2.27)
e = pa/m.
(2.28)
and the fluid loading parameter, If the long-time behaviour of the system is sought (steady state solution) then equations (2.16)-(2.24), together with a radiation condition for ~(x, y; t) as x2+ y 2 , co above the plate, fully specify the boundary value problem. In this work the low loading limit will be employed, which (from Leppington [8]) is given as
p~ / mS] ~ O.
(2.29)
If the non-dimensionalized wavenumber of the forcing 12/c is of order unity,
f l / c = O(1),
(2.30)
then small fluid loading corresponds to e --, 0.
(2.31)
The further assumption is made that the remaining variables have order unity: i.e., c = O(1),
/2 = O(1),
/z = O(1).
(2.32)
Note that the non-linear and forcing terms can always be chosen to be of the same order (O(e), tz = O(1)) for any fluid/plate parameter values by increasing or decreasing the incident wave amplitude, ~ accordingly. 3. NON-RESONANT SOLUTION As the fluid loading parameter becomes small, e-* 0, the forcing term in the plate equation (2.18) becomes small also. Hence a trial solution may be posed as
¢b(x, y; t) = eckl(x, y; t) + E2¢~2(X,y; t) + ' ' ' ,
(3.1)
O(X, y; t) = eOI(X, y; t) + E202(X,
(3.2)
t) -I-" " • ,
y;
V(X; t)= eV~(X; t)+ e2V2(X; t ) + ' ' ' .
(3.3)
Substitution into the governing equations and collecting the leading order terms yields 1 v~.~x + v,1 = - i O cos (12x sin O~c) e-'a'+ c.c., D1
1
=v==0,
x=+l,
and
1_
1_
(3.4) I
4)y- 0 r - v,,
(3.5,3.6)
where c.c. denotes the complex conjugate. Similarly, the scattered potential ~bI satisfies equations (2.16) and (2.21), and 01 satisfies equations (2.17), (2.22) and (2.23). The plate deflection is now uncoupled from the fluid potential to leading order, and so the time-harmonic solution to equations (3.4) and (3.5) may be obtained immediately as
v~(x; t) = -ig(x; /2, /2 sin O/c) e-iat+c.c.,
(3.7)
g(x; a, ~) = d(a, ~) cos (~x)+ e(a, ~) cosh (al/=x)+f(a, ~) cos (~I/2x),
(3.8)
where
½ cos/3 d(a,/~) =/~4_ a2 ,
e(o~,/~) = (~2_1. o~) c o s h ot 1/2 ,
(3.9,3.10)
-½ cos/~
f(a, $) = (/32_ a ) cos al/2"
(3.11)
392
L D. ABRAHAMS
The particular frequency values /2 = (n +½)2~r2,
n = 0, 1, 2 , . . . ,
(3.12)
corresponding to plate resonances, will be excluded in this analysis (see reference [14]). Green's theorem may now be employed in the regions above and below the plate to obtain the ~b, ~ potentials. From equations (2.16), (2.21) and (2.19) the solution which has outgoing waves at infinity is
~ l ( x , y ; t ) = ½ i / 2 e -m'
_ g ( x l ; / 2 , / 2 s i n O / c ) H(ol)(/2R/x)dXl+C.C.,
y>O,
(3.13)
where H~1) is the Hankel function of the first kind and R = [ (x - x,)2 + y211/2.
(3.14)
Similarly, from equations (2.22), (2.23) and (2.19), the potential inside the cavity is given by
G(xl, 0; x, y)g(xl ;/2,/2 sin 0/c)
~l(x, y) = 12 e -iat
dx I +
c.c.,
1
[xl< 1,
0
(3.15)
in which the Green function, satisfying the boundary conditions (2.22), (2.23) and ~y = 0 o n Ixl < 1, y = 0, is given as
G ( x l , Yl ; x, y) = m., y~
2 COS (rmx) COS (rmXl) COS(say) COS (Sayl) h{( /2 / c) 2 - r,,2 _ s,}2
(3.16)
The double sum ranges over m = O, 1, 2 , . . . , n = O, 1, 2 , . . . , and r,, = (m +½)¢r,
s,=(n+½)cr/h.
(3.17,3.18)
Interchanging the order of summation and integration yields
4/22rm sin (rm) cos ((/2/c) sin O) cos (rmX) cos (say) ~b~(x,Y) =e-re' ~. , - 7 7 - 7 ~ , , _ 2 . . . . .---~-~,~-_--i-----~ t-c.c., ,,,n h ( r m - / 2 ) ( r m - ( ( / 2 / c ) s m 0) ){(/2/c) - r m - s n }
(3.19)
where it should be noted that all the odd modes in x are zero because of the symmetric forcing. The solution given by equations (3.7), (3.13) and (3.19) is satisfactory unless a plate or cavity resonance is approached. The plate resonances are precluded in this study, and for the cavity resonance the asymptotic solution (3.19) becomes unbounded when, for any n, m, ( / 2 / C ) 2 _ r m2 _ S n2 ._~ O "
(3.20)
This suggests that the potentials may sometimes be larger than the anticipated scalings presented in equations (3.1)-(3.3). In the following sections new expansions are introduced, together with slow-time variables, to investigate the solution in the neighbourhood of a cavity resonance.
SCATTERING
OF
SOUND
BY AN
ELASTIC
393
PLATE
4. SECOND-ORDER RESONANCE The frequency of the forcing is adjusted to lie within an order e of a cavity resonance frequency. Thus, a detuning parameter v is introduced such that
O=to+ev,
e -->0,
v = O(1),
(4.1)
where to = tOpq= cx/[((p + ½)~r)2+ ((q+½)Tr/h) 2]
(4.2)
for a particular pair of integers (p, q). It will be convenient to perform an asymptotic study by introducing a slow-time variable. Thus if,
To = t,
T~ = et,
e -> 0,
(4.3)
then sin (/2t) = sin [(to + ev)t] = sin (toTo+ ~,T1),
(4.4)
and the time derivative may be written as
O/Ot = O/OTo+ eO/OTl.
(4.5)
The sound field inside the cavity can be expected to be larger than that anticipated in section 3, and so a suggested expansion is
~b(x,y; To, TO=~b°(x,y; To, Tl)+e~bl(x,y; To, T I ) + ' " .
(4.6)
This potential is coupled to the plate deflection at O(e) and therefore the expansions in equations (3.1) and (3.3) for ~b and v are still assumed to be valid. Substituting the asymptotic expansions into the governing and boundary equations gives, to leading order, ~o+
o 2 0 ~byy-(1/c )~bToTo=0,
~by °, = 0
Ixl
lxl < 1, y = 0;
(4.7) (4.8, 4.9)
x=+l,O>y>-h.
~/,° = o,
(4.10)
This unforced system has a general steady state solution, composed of the resonant eigensolutions, of the form 0 °= ~ Kmn(T1)
cos
e-i',.nTo-t-C.¢.,
(rmX) COS ( S a y )
(4.11)
m,n
where Kmn are as yet unknown functions of T1 and only the modes even in x have been included. The next order, O(e), system may be written as
V~,x+VroTol _ - - i t o [ c o s ( t o x s i n 0 / c ) e - i v r , - K ( T | ) c o s ( r p x ) ] e - i ~ r o +i
Y~ tom,Kin,cos (rmx) e-i'°~J°+ c.c.,
(4.12)
rtl, n
m,n ~p,q
where g ~ g~,
y=~by, 1
y-0,
to = topq.
tx[
Ixl> 1, y = 0 ;
ul
= v ,1= - 0, _
x=+l,
~bxx+ ~byy-(1/c)~broro=0,
(4.13, 4,14) y > 0,
(4.15,4.16)
1 1 ~bxx+ ~byr-(1/cE)~roro=(E/c2)~b°orl,
(4.17,4.18)
and ~bI satisfies the boundary conditions (4.9), (4.10). The solution of the plate equation, (4.12), satisfying the edge conditions (4.14), is v 1= - i [ g ( x ; to, to sin 0/c) e -i~rl - K ( Tl)to(r~- to2)-1 cos (rpx)] e -i*'T° +i ~. Kmn(T1)tom,(r 4 - t o m2 ) - i cos (rmx) e-iOJm Toq_c.c., (4.19) m,n
394
1. D. ABRAHAMS
and so the leading-order scattered potential above the plate may be found, as before, via Green's theorem. All that remains is to calculate the Kin,(T~) by examining the boundary value problem for $1. This potential must be bounded for all time and from Appendix A can only be achieved if
iK'n(TI)+j,,nKm,(T~)=O,
(m,n)#(p,q),
(4.20)
iK'(T~) +jK(Ti) - k e -t~r~ = 0,
(4.21)
where
k
=
(c2/h)
cos (rpx)g(x; to, to sin O/c) dx,
(4.22)
1
Jmn = c2tOmn/[h(r~ --tom,)], 2 •
J=j~,
K'(T~)=dK/dT~.
(4.23-4.25)
Integration reveals that
Kmn As,, exp (ijm, T~), (m, n) # (p, q),
(4.26)
K = A exp (ijT1) + k e-i~rl/(j + v),
(4.27)
=
where Amn are as yet undetermined constants. By examining the be shown that secular terms in /'1 can be avoided if and only if A,,n = 0,
O(e 2) problem
all (m, n).
it can (4.28)
Hence, from equation (4.19),
vl(x; To, 7"1)= - i [ g ( x ; to, to sin O/c) -k,w cos (rpx)/{(r4-to2)(j+ v)}] e-i(~ro+~T~)+ c.c.,
(4.29)
and so, to leading order, [g(xl ; to, to sin O/c)
~bl(x, y; t) = 2Litoe -iat ! -
ko cos (rpx~)/{(r~- to2)(j + v)}] H~ol)(~,g/c) dx~ + c.c.
(4.30)
The solution obtained in this section shows that near resonance (within O(e) of a resonant frequency) the scattered field above the plate is modified but does not change its order of magnitude. However, as j is real, the detuning parameter v may be adjusted so that the eigensolution coefficient K becomes large or even unbounded. This occurs when u=-j,
(4.31)
and suggests that there is an inner region of the local resonance range where the scalings must again be altered. Thus the region examined in this section is called the second-order resonant range, and it will be shown a posteriori that the leading-order resonant range lies within O ( e 2) o f /J = - - j . 5. LEADING-ORDER RESONANCE It will be assumed that the sub-region of the O(e) resonant range is of order e 2. To examine the solution in this region the forcing frequency is written as ~-2 ----to -- e j ' l " e 2 0 ",
(5.1)
SCATTERING
OF SOUND BY AN ELASTIC PLATE
395
where tr is the new detuning parameter and j is given in equation (4.23). Because of the e 2 term in the frequency it is necessary to introduce a second slow-time variable, namely
T2 = e2t,
e ~ O,
(5.2)
so that sin (Ot) = sin (toTo-jT1 + trT2).
(5.3)
The time derivative now becomes
O/at = a/aTo+ e O/OT1 + e 2 O/OT2
(5.4)
and the cavity potential is expanded as (cf. [17])
O(x,y; To, T1, T2)=e-ld/-~(x,y; To, TI, T2)+ 0 ° + e ~ l + "'" •
(5.5)
From the form of the coupling in equation (2.18) it is expected that the plate deflection and outer potential scale as
v(x; To, 7"1, T2) = v°+ evl+ "'" ,
and
4~(x, y; To, TI, T2) = 4~°+ ed~+ "'" • (5.6, 5.7)
These are again substituted into the full system (2.16)-(2.23), and the leading order potential (O(e-~)) is found to satisfy the governing equation and boundary conditions (4.7)-(4.10). Similarly, the O(1) potentials ~b°, 4,0 are defined by the system (4.9), (4.10), (4.14)-(4.18), whereas v ° is given by o
-q, --1 To,
l) xxx, x dl_ I) Oo To =
Ixl
(5.8)
It will also prove necessary to examine the O(e) terms, which are governed by the following equations: 6 ~ + ~b~r- (1/c2)6 ~-oro= (2/c2)d~°or,, 1
1
2
y > 0,
(5.9)
1
O,,x+~yr-(1/c )~roro =(2/c2)~b°roT +(1/c2)tbr~r + 2 VxVxdx
Vx=x Vroro=/Z
2 -1
Ixl
0
(5.10)
v ~ - 2 V r o r , - 2 t o c o s ( t o x s i n O / c ) sin(toTo-jTl+trT2)
1
o
o
+ 4, o- q' o-
Ixl
1 + vr,, o ~b~= ~b~= Vro
(5.11) Ixl< 1, y =0.
(5.12)
The boundary conditions to this order are given in equations (4.9), (4.10), (4.14) and (4.17). To commence the solution, the work in section 4 reveals that
~b-1(x,y; To, TI, T2) = ~ KmI.(TI, T2) cos(r,,,x)cos(s.y)e-i%""r°+c.c.,
(5.13)
m, rl
with (from expression (5.8)) to,.. cos (r,x) T2) 7"--4"-'-T"7, e-'°'""r°+c.c.,
v°(x; To, T, T2)=i ~ K ~ ( T 1 '
m,n
"
(5.14)
(rm --torah)
and 2 i to.,,e ~b°(x, y; To, 7"1 T 2 ) = - ~ E
'
m,.
-i~
-- 1"ot- t- -m1 -t T l , T2) • 2
( r ' - to'.)
cos (r'x~) H~l~(to',R/c) dxl + c.c.,
x 1
y>0,
(5.15)
396
L D. ABRAHAMS
where R is given in equation (3.14) and Km~, are as yet unknown functions of T1 and T2. Hence equation (5.14), together with Appendix A, allows the second order potential inside the cavity ~b°, to be determined as
¢°(x,y;To, T1, T2) = ~ ~ t o m . ,.,n
sin (s.y) cos (rmX)jmnKmln e -i%~J°
SnC
+ ~ K ° , cos (s,y) cos (r.,x) e - i % J o + c.c.,
(5.16)
m,n
where Km. o is another matrix of as yet undetermined coefficients. The secularity condition (A14) gives K~,I, = A'~(T2) exp (ijm, T1),
all (m, n),
(5.17)
where A,,, are unknown functions of T2 and jm, are given in equation (4.23). Although only the leading order solution is required, it was necessary to examine ~/,o in order to determine K ~ as a function of 7"1. However, the leading potentials $-~, $o, v ° still contain the unknown functions A.,,. Thus it is now necessary to obtain the next order plate deflection term, v 1, and use this in the $1 boundary value problem. The imposition of the secularity condition (A14) should then specify the potentials to leading order. The plate deflection, v ~, is determined from equation (5.11) and is forced by several terms including the non-linear expression
/~
(i 1 oo)o
VxVxdx Vxx.
(5.18)
This may be written, from equation (5.14), as /~i Y~ r.,Kmnqm, -1 -1 Kot -1 e -i('~m"+~%'+o%')T° cos (rmX) ~ q~qvt{Kw ,., n
O,S, t
-2/(SIK~) e-itc°~"-°'°*+~%')T°+/~/~vt1 e-i("°~"-'%'-%")T°}+C.C.,
(5.19)
in which 4 2 qm, = rmtor,,/ ( rm -- tOm,)
(5.20)
and/(~,1 denotes the complex conjugate of K ~,~,. Although there are many terms in this expression, only those which oscillate (in the slow-time variable) at cavity resonance frequencies (4.2), tO,.,, will be present in the seculadty condition to be established. Thus, unless the duct height h is chosen specifically, the only terms from exp [ - i ( t o m , - tOys+ tOrt) To] which are important are those for which s = t or v = m, s = n. Similarly, tO.,, - tars tOrt contributes when v = m and s = n or t = n, but all terms like to,., +tO~s + to~, are irrelevant. Hence in the generic situation (for other cases see section 6), expression (5.19) may be written as
_ l t i ~ r m K - I,,,q,", cos (rmX) e-i=~"r°{ 3lKm, 2q2m,+6 ~ IKms[ --122qms+2 ~ IKv,--122} I qos .,,n
+.
s s#n
• .+
v,s v#ra
(5.21)
c.c.,
where " . . . " indicates terms which may be ignored for most h values and IK2I 2 = K ~) g L-).
(5.22)
The only other term on the right side of equation (5.11) which presents any difficulty is ~°0, because it contains a rather complex function of x. This may be simplified by noting
SCATTERING
OF SOUND
BY A N E L A S T I C P L A T E
397
that 4~°o is even in x, and as it is only required over the region Ixl < 1, y = 0, Fourier decomposition gives ~b°o(X,0; To, T~, T2) --1
2
Ixl
= - ~. Km"°J="qs"~.(bm"+idT")cos(rvx)e-i'°""r°+c.c., .,.. 2r,.
(5.23)
in which the real coefficients b~", d~" are given by bm"+id~" =
II
cos (rvx) cos (rmXl) H~ot)(tomJx- X , I / c ) dxl dx.
1
(5.24)
1
The plate deflection, v t, is now easily found from expression (5.11), with use of expressions (5.13)-(5.16), (5.21) and (5.23), and this may be substituted into equation (5.12) to yield the boundary condition on y = 0 for ~b~: namely, ~#~(x, 0; To, TI, 7"3)=
t o , to
--oJg(x;
sin O/c) e -i°~T°+ijT°-icrT: • mPI (b~t,l~l +ld~ ) c o s (fox) 4 2
Y~
-' ~ Kmnqmng°mn
~
ra,n
2 rm
v
+i
r v -
to,,, cos (rmx) f
+ ~
4
m,n
2
rm--(Dmn
e -ioj.,.T°
tO m n
o
~¢-°mnKmn+
~
2ir*mK~rl] 2
~
rm--tOmn
4
J
e-i%""T°
- I t ~. K mnq -, 2ran COS (r,.x) m, tl
x{3'K~,l.I 22q.,.+6 Y~ , K m , , q , .2s + 2 s
s~n
+ " " + c.c.,
. K ~Ts , oq ~2-} e 1- , o2, v,x
v#m
Ix I < 1.
(5.25)
As before, to is chosen to be the (p, q)th cavity eigenfrequency, where (p, q) is any particular pair of integers and 2 to = tom = cx/{rp2 + sq}.
(5.26)
The governing equation for et is given by expression (5.10), which may be written, upon using equations (5.13) and (5.16), as 1
1
~,x~+,y~-(1/c
2
1
)¢ToTo
-' . - , } COS (rmX) = (1/C ~) Y {KmnTtT , --21¢omnKmnT2
COS
( s . y ) e _ "°'"r°
m, tl
-(2i/c 4) ~. (y + h )( ~ozs./ s.)jm,K ..~r, cos (r.x) sin ( s.y ) e ~ ' J o m, n
-(2i/c2) E to,..K,..r, o cos (r,.x) cos (s.y) e-t°m"T°+ C.C.
(5.27)
m,n
Each (m, n) term of q/satisfies the boundary value problem given in equations (A1)-(A4) with A, B, C and f ( x ) chosen appropriately. Hence. the secularity condition (A14) may
398
L D. A B R A H A M S
be employed, which, after considerable simplification, gives
Amnt°mnC h2(r 4 - tom. )2
2
2 • m nm - d,~") - 2c 2 ( r m 4 " C O m n ) 4 . - " m htomn(lb 4 ( rm + CO2ra,) S~
-(2/h)Ara.l~c qra. 3qra.lAm.[
~
q~,lA,~l:+2 Z
$ s~n
qv.lAw
t~,$ t~m
+ 2iwmnAmnr2- 28mpSnq°~k e-i~T2 = --2i¢Omn~Tt ( K°" e-iJm"TQ'
(5.28)
where k is written in equation (4.22) and 8rap is the Kronecker delta. The left side (LHS) of this equation is purely a function of 7"2only, and so integration with respect to T~ yields o _
Kmn -
Bra. e +ij'..T, 4- i
{LHS of (5.28)} T~ e +ij'--T,,
(5.29)
2~mn
in which Bin. are undetermined functions of T2. However, it must be insisted that Kin.° remains bounded for all time and so cannot have a term proportional to 7"1. Therefore the left side of equation (5.28) must itself be identically zero. This is the amplitude equation for the leading-order terms. The solution is simply found by writing Am. in its modulus and phase form, namely
Ara. =/3 m. e i~".
(5.30)
Substituting into the homogeneous form of equation (5.28) and collecting the imaginary terms gives 3 2 ran 4 2 2 d/3ra./dT2+{tOm.C bra /2h(rraO,m.) }/3.. =0
(5.31)
for any (m, n) not equal to (p, q). Thus 3 2 ran 4 2 2 flra.oZexp{-tOm.C bra T2/2h(rm-O~m.) },
(m, n ) # ( p , q ) ,
(5.32)
where it is easy to show that b.~"> 0, and so, as a steady state solution is sought, these terms are all exponentially small and can be neglected. The only eigensolution which is non-zero is therefore the one which oscillates near the forcing frequency. The amplitude equation may be solved by writing Apq as A~ =/3 exp [iT - io'T2],
(5.33)
and collecting real and imaginary parts. This yields
dfl/dT2=-A~/3-ksin
T,
[3 dT/dT2=fl(A2+cr)-A3fl3-kcos
T,
(5.34,5.35)
where 2
3
AI=C ¢0
.,c 2
pq 4 bp /2h(rp-tO
fc2r
2 2
)
(5.36)
,
(g+,ob
}
A2=2h(r~_w2)2 [-~q - z c (r~ ¢02) -d~qho~ 2.,
(5.37)
4 3 4 A3-(3/h)l~c 2 rpoJ (rn-oJ 2 ) --4 .
(5.38)
Steady state solutions are obtained by setting d/3/dT2 and dT/dT2 to zero and both equations may be combined, on elimination of the phase factor % to give /32{A~+ [A2 + tr - A3/32]2} = k 2.
(5.39)
399
SCATTERING OF S O U N D BY AN ELASTIC PLATE
This is a simple equation, cubic in/~2, and has an identical form to that obtained for a primary plate resonance (see equation (3.35) o f reference [14]). As d is proportional to /~8, a typical plot of the amplitude/3g against the detuning parameter o, is shown in Figure 2 for a range of values of A3 (and hence a range of incident wave amplitudes g). Figure 3 displays the graph o f phase angle y against cr for a particular set of fluid/plate parameters and fixed/~ Note that the modulus of Apq tends to zero as or-* :~oo and has a maximum value of
/3 = Ikl/x,.
(5.40)
For a given physical problem, and incident wave angle, the constants a~, A2, A3 and k can be determined. Therefore/3 and y are obtained from equations (5.34), (5.35) or (5.59) and depend on the detuning parameter or. As the solution curve for/3 against cr i
I
O Detuned frequency, o"
Figure 2. Scattered wave amplitude against detuned frequency for a range of values of the incident wave
amplitude. The dashed line indicates the unstable part of the curve, and the unbroken line indicates the stable region. I
o.
0 Detuned frequency,
Figure 3. Phase of the scattered wave against detuned frequency for a fixed incident wave amplitude.
400
I.D.
ABRAHAMS
may be multi-valued, the actual value of/3 (i.e., the particular branch of the curve which is taken) can depend on the initial conditions for the problem (see concluding remarks). The leading-order solution near a cavity resonance has now been found, and from equations (5.13)-(5.15), may be written as ~b~(2fl/e)cos(Ot-y)cos(rex)cos(sqy),
Ixl~ 1,
v ~ flto(r 4 - to2)-~ cos (rex) sin (/2t - y), 4~-{-ito2~/2(r~-o)2)}ei'-m'
O>-y>~-h,
Ixt <~1,
cos(rexOH~l)(toR/c)dxl+e.c.,
(5.41) (5.42)
y>~O.
(5.43)
Finally, the far field form of the scattered potential 4~ is easily obtained by writing x = ! cos O, y = I sin O, 0 < O < or, where tol/c ~ oo and I is assumed to be I >>1. This gives q~ o,t/~
2to2rgfl COS [(to/c) cos O] sin rp e i(~°l/c-3~r/4+y-Dt) + c.c., (r~ - to4)ff(2"n'tol/c)(r2 - (to~c) 2 cos 2 O)
O # 0, ~r. (5•44)
It is a straightforward matter to employ the substitutions given in expressions (2.11 )-(2.15) to rewrite the potentials and plate deflection back in terms of the physical variables. For the sake of brevity these are not written here• 6. EXCLUSION OF COUPLED PRIMARY AND SECONDARY RESONANCES It was shown in a previous paper [15] that for specific values of the plate/fluid parameters, combined secondary and primary plate resonances could occur. This led to a scattered field consisting of two terms, one oscillating at the forcing frequency, and the other oscillating at three times or 1/3 of this frequency. However, for the particular plate/cavity configuration chosen in this paper, the coupling between two cavity eigenfrequencies is not possible• In order to illustrate this, a particular case is now examined. The duct height h is chosen to have the value x/2, so that (from expression (4.2)) tOoo= cx/(Tr2/4+ ~r2/8) = x/(3/S)c~r,
(6.1)
too2 = cff(zr2/4+25~r2/8) = x/(27/8)cTr,
(6.2)
and hence too: = 3tooo. The forcing frequency/2 may be chosen to be close to tooo or too2 ( w i t h i n o ( e 2 ) ) , and therefore the superharmonic or subharmonic secondary frequency (too2 or tOoo respectively) will be coupled through the nonlinear term in equation (5.1). Thus the two terms • ---1 2 --1 2 3/zlr0(Koo) Ko2 qooqo2 cos (foX) ei(2~°°-°~°2)r°+p.iro(K~ol)3q~ cos (foX) e -31°'°°r°, (6.3)
which were absorbed into " . . . " in equation (5.21), will now be included in the secularity equation as too2-2tooo = tooo and 3tooo= too2. The boundary condition for ~b1 is modified by the inclusion of ----1 2 --1 3 - - i t o "/~ --1 3 3 " 3bt(Koo) Ko2 qooqo2 cos (rox) e oo o+p,(Koo) qooqo2 cos (rox) e -1'°°2r°
(6.4)
in equation (5•25) and hence the right side of the secularity condition (5.28) becomes -(2/h)cE~q~oqo2{38moS,,oA02.~200 ei(Jo2-3Joo)r, + 8moS,,2A~o e-i(~o2-3Joo)r,} - 2ito,.. ~d--T=( K ° . e-iJ..r0.
(6.4)
The left side remains unaltered and, as •
•
2
3
4
2
4
2
Jo2 - 3joo = 6c tooo/h ( r o - 3tooo)(ro- tooo) # 0,
(6.5)
SCATTERING
OF
SOUND
BY AN
ELASTIC
PLATE
401
integration with respect to T1 gives K 0m~= Br,, e '""Jm"rL+
-t
i
2to,nn
{LHS of (5.28)} T1 e ij'"r~
C2tZq~oq°2 {38,,o8,oAo2.42 e i(j°2-ej°°)T'-8,.oS.2A 3e3%°T,}. hto,,, (jo2 - 3joo)
(6.6)
As before, Km, o cannot grow linearly with the slow time T~ and hence the left side of equation (5.28) must again be identically zero. Coupling between modes has not therefore occurred and equations (5.31), (5.34) and (5.35) are still valid. It is straightforward to show that for any value of h (and therefore for any pair of eigenfrequencies which have t0r,~ = 3aJst) the coupling does not occur between amplitude equations and so the solution written in equations (5.41)-(5.43) is always valid. 7. CONCLUSIONS The asymptotic values of the acoustic potentials inside and outside the cavity, and the plate deflection, have now been determined. Away from a cavity resonance, the elastic plate gives rise to a small scattered potential O(e), and the cavity field and plate deflection are also both O(e) in magnitude. As the incident wave is adjusted so that is approaches one of the cavity eigenfrequencies (section 4), the cavity potential becomes O(1) in size. Therefore, the scattered field above the plate (4.30) is altered, but does not change its order in the small parameter, O(e). However, within the region about an eigenfrequency (O(eZ)), there is a sub-region (section 5) in which the cavity potential further increases its magnitude to O(e-~), i.e., becomes very large. This results in the scattered potential increasing in size to O(1), and at this order both the plate non-linearity and radiation damping become important (these orders agree with previous linear results [ 17]). It should be noted that the combined potential ~b°+ eq~~ in section 5 (~b° given in expression (5.43) and 4~~ found from expression (5.25)) tends to the value of ~b~ given in expression (4.30) when or becomes very large. This is easily seen by letting tr = (~,+j)/e which, from equations (5.34), (5.39) gives/3 ei~ - ke/(~, +j). Similarly, both of these solutions tend to the non-resonant result (3.13) as v ~ O (1 / e). As has already been discussed, the plate non-linearity leads to a distortion of the modulus and phase against detuning parameter curves (Figures 2 and 3). This gives rise to the possibility of jumps in the potential as the frequency is varied. Related to this is the fact that the amplitude equation (5.39) contains multi-varied solutions for specific parameter regimes. The actual solution curve attained is determined by the initial conditions for the system (see reference [16]). Note that the jump phenomenon and multi-valued curves are also obtained when/38 is plotted against g, the incident wave amplitude, for fixed tr. Perhaps the most interesting effect of the non-linearity is the possibility of jump discontinuities in the modulus/3 (and hence in the far field scattered potential) as the incident wave angle, 0, varies. This occurs because k in expression (5.39) is a function of/9, and shows that reciprocity does not hold between 0 and 19. The jump discontinuities are illustrated, for specific values of the physical parameters in Figure 4. The variation in jump positions a s 0 is increased or decreased (hysteresis) should be noted. Also, the non-linearity,ean be seen to actually increase significantly the amplitude of the acoustic potential for some angles of O. This paper has illustrated that the non-linearity appearing in the thin elastic plate equation has a signitieant effect upon the acoustic field near a cavity resonance, A novel
402
t.D. ABRAHAMS r
1"
1
IrlA
2 7r18
o.
.E
i
g
¢)
w18
Ir12
Incident wave angle, e
Figure 4. Scattered wave amplitude against incident wave angle. The dotted line denotes the linear solution, the unbroken line denotes an increasing 0 path, and the dashed line denotes a decreasing 0 path.
feature o f the work is that whereas the secularity condition in multiple scales analyses is normally derived from a one-(space)-dimensional boundary value problem (e.g., [14, 16]), in this physical model the necessary condition is obtained from a two-dimensional problem. Finally, in section 6 it was proved that coupled primary and secondary resonances could not occur. However, if the amplitude of the incident wave was large, it can easily be shown that superharmonic or subharmonic secondary oscillations and combination resonances will exist [ 14]. Coupled resonances could occur if the cavity width was different from the plate width, or indeed if the cavity width became infinite (duct). This will b e an area of future study, as will the extension of these models to two-dimensional planar (circular and rectangular) or cylindrical elastic plates. ACKNOWLEDGMENTS This research was partly supported by the United States Air Force under Grant No. AFOSR-85-01-50, and by the United States Navy under Contract No. N000014-83-C-0518. REFERENCES 1. D. O. CRIGHTON 1971 Journal of Fluir,! Mechanics 47, 625-638. On acoustic beaming and reflection from wave-bearing surfaces. 2. P. A. CANNELL 1975 Proceedings of the Royal Society of London A347, 213-238. Edge scattering of aerodynamic sound by a lightly loaded elastic half-plane. 3. H. G. DAVIES 1974 Journal of the Acoustical Society of America 55, 213-219. Natural motion of a fluid-10aded semi-infinite membrane. 4. F. G. LEPPINGTON 1978 Journal of Sound and Vibration 58, 319-332. Acoustic scattering by membranes and plates with line constraints. 5. G. P. EATWELLand J. R. WILLIS 1982 IMA Journal of Applied Mathematics 29, 247-270. The excitation of a fluid-loaded plate stiffened by a semi.intnite array of beams. 6. E. H. DOWELL 1975 Aeroelasticity of Plates and Shells. Leyden: Noordhoff !nterx~ational Publishing. 7. Y.C. FUNG 1958 An Introduction to the Theory of Aeroelasticity. New York: John Wiley & Sons. 8. F. G. LEPPINGTON i976 ~arterly Journal of Mechanics and Applted Mathematies 29; 52%546. Scattering of sound waves by finite membranes and platas near r e s o n ~ .
SCATTERING
OF
SOUND
BY AN
ELASTIC
403
PLATE
9. I. D. ABRAHAMS 1981 Proceedings of the Royal Society of London A378, 89-117. Scattering of sound by a heavily loaded finite elastic plate. 10. J. H. GINSBERG 1978 Journal of Sound and Vibration 60, 449-458. A re-examination of the non-linear interaction between an acoustic fluid and a flat plate undergoing harmonic excitation. 11. A. H. NAYFEH and S. G. KELLY 1978 Journal of Sound and Vibration 60, 371-377: Non-linear interactions of acoustic fields with plates under harmonic excitations. 12. E. H. DOWELL 1966 Journal of the American Institute of Aeronautics and Astronautics 4, 1267-1275. Nonlinear oscillations of a fluttering plate. 13. C. H. ELLEN 1977 Journal of Sound and Vibration 54, 117-121. The nonlinear stability of panels in incompressible flow. 14. I. D. ABRAHAMS 1987 Proceedings of the Royal Society of London A414, 237-253. Acoustic scattering by a finite nonlinear elastic plate. I. Primary, secondary and combination resonances. 15. I. D. ABRAHAMS 1988 Proceedings of the Royal Society of London A418, 247-260. Acoustic scattering by a finite nonlinear elastic plate. II. Coupled primary and secondary resonances. 16. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: John Wiley & Sons. 17. I. D. ABRAHAMS 1982 Quarterly Journal of Mechanics and Applied Mathematics 35, 91-101. Scattering of sound by finite elastic surfaces bounding ducts or cavities near resonance. APPENDIX A In order to suppress any secular b e h a v i o u r o f ~bl(x, y; To, T~) in section 4, and ~b° and ~b~ in section 5, it is useful to examine the following problem: $~, + ~byy- (1/c2)$Zoro = {A cos [(q +½),ry/h] + B(y + h) sin [(q + ½)Try~h]} cos [(p +½),rx] e -i'°T°
[x]
x=+l,O
0
¢=O, Ix]<~l,y=-h,
(A1) (A2, A3)
Ixl ~< 1, y = 0,
(A4)
where
to = tom = cTrx/[(p + 1 ) 2 + ((q +½) ~r/h)2].
(A5)
It is convenient to m a k e the substitution ~ = 4~ + 2 r r ( q + ½ )
A + 2 ~ . ( q +½)
- ~ - (y + h) cos [(q +½)1ry/h]
sin[(q+½)~ry/h] cos [(p +l)Trx] e -i~T°,
(A6)
so that ~b satisfies the h o m o g e n e o u s wave equation, and the b o u n d a r y condition on y = 0 becomes 1 q~y = [ C f ( x ) - ( h / 2 ) { A - hB/2~r(q +½)} cos ((p +~)Trx)] e - i~ o To,
Ixl
1.
(A7)
This m a y be expressed as the Fourier series thy = C ~
a~ cos [(m +½)~rx] e -i°'ro ,
(A8)
where am =
I
+l
cos [(m +½)Trx]f(x) dx,
m # p,
(A9)
--1
a p = ( - h / 2 C ) { a - h B / 2 z c ( q + ½ ) } + I ~ cos [(p+l)*rx]f(x) dx. 1
(A10)
404
i.D. ABRAHAMS
The solution to the steady state boundary value problem may now easily be found as
4' = C ~ am cos [(m +½)¢rx] sin [bm(y + h)] e-i°'ro-~ h(y + h) .1=o bm cos (bmh) 2~r(q+ 1) x[\~:A+ 27r(q +½)]hB~ sin
[(q+l)~'y/h]-B(y+h)
cos [(q+½)zry/h]}
x cos [(p + ½)¢rx] e-i~'r°,
(A11)
where bm= x/[(to/c) 2 - ((m +½) ¢r)2].
(A12)
However, from equation (A5), b~ = (q + 1)7r/h and so the denominator of the pth term vanishes. Therefore 4' can be bounded only if a, =0,
(A13)
which corresponds to A
hB 2¢r(q+½)
-
2C Ih
cos [(p +½)~rx]f(x) dx.
(A14)
This is the secularity condition which can be used to evaluate the amplitudes of the resonant cavity eigensolutions. Finally, it should be noted that equation (All) is the particular solution of 4' for the steady state boundary value problem. Complementary functions, or eigensolutions, as written in equation (4.11), may be added to this solution.