- Email: [email protected]
Scattering of sound by an elastic plate with flow
Journal of Sound and Vibration (1983) 89(2), 213-231
SCATTERING BY AN ELASTIC I. D.
OF SOUND
PLATE
WITH FLOW
ABRAHAMS
Department of Mathematics, The University, Manchester Ml3 9PL, England (Received 27 September 1982)
An elastic plate, set in an infinite baffle and immersed subsonic velocity, is excited by an acoustic source. The when fluid-plate coupling is large, and a solution is found expansions. The far field is found to approximate to
in a fluid moving with a uniform scattered sound field is analyzed by the use of matched asymptotic the solution obtained when the
elastic plate is absent. At a plate resonance, however, the outer field must include eigensolutions with singularities at the plate edges, and close to the plate the dominant terms are travelling plate waves. These plate waves are found to have a wavelength independent of the frequency of the source. It is also shown that a plate resonance corresponds to a divergence instability of aerodynamic flutter theory and that the stability results found in this paper are in agreement with those obtained by using modal expansions. The limit as the Mach number goes to zero is found to be singular, suggesting an analysis of the model for small flow velocity. This calculation is performed and the results match smoothly to the respective solutions for a stationary fluid and for a large subsonic flow.
1. INTRODUCTION There has been much attention given in recent years to acoustic problems with finite or infinite flexible surfaces. With finite geometries asymptotic or other approximate methods must be employed to obtain an estimate of the sound field [l, Leppington 1976; 2, Handscomb 19771. The problem of a thin elastic plate set in an infinite baffle has been analyzed in the limits of both small and large fluid loading [l, Leppington 1976; 3, Abrahams 198 1, respectively] and in this paper the latter work is extended by introducing a uniform subsonic flow of the fluid enveloping the plate. The method of matched asymptotic expansions is employed with modification to allow for the presence of plate resonances. It is also expected, due to the addition of flow to the problem, that flutter or divergence instability of the elastic plate may occur. This problem has been analyzed by use of modal expansions to calculate the onset of instability [4, Weaver and Unny 1970; 5, Ellen 19721 and the results will be shown to be in agreement with those found in this paper. In section 2 the problem is analyzed for a Mach number of order unity (M < 1) and, as will be shown, the results do not reduce to that given by Abrahams [3] when the flow velocity goes to zero (i.e., M + 0 is a singular limit). It is therefore necessary to analyze separately the problem when the Mach number is small (section 4) as this result matches to both the M = 0 and M = 0( 1) results. Cartesian co-ordinates (x, y) are chosen in the two-dimensional problem, and for simple harmonic time dependence, with radian frequency o, the velocity potential can be defined as Re 14 (x, Y1 exp ( - ie.41.
(1.1)
213 0022-460X/83/140213+19$03.00/0
@ 1983 Academic Press Inc. (London) Limited
214
I.
D. ABRAHAMS
The time factor is henceforth suppressed for brevity. An elastic plate (lying on y = 0, Ix) < a) is immersed in a compressible inviscid fluid and when there is a uniform mean flow in the x direction the potential satisfies the corresponding convected wave equation ~x,(1-M2)+~,,+k2~+2ik~~,=0,
(1.2)
where M( = V/c) is the Mach number, U is the flow velocity, c is the sound speed in the fluid, and k ( = o/c) is the acoustic wavenumber. For a thin elastic plate, of length 2a, the equation governing small flexural vibrations is (D a4/ax4-&0*)7j(X)
= -p(x, O)]“,
lxI
Y =O,
(1.3)
where D is the plate bending stiffness, M,, its mass per unit area, and n and p denote the plate deflection and fluid pressure respectively. Note that p(x, O)]’ denotes the discontinuity in pressure across the plate. The deflection n is related to the potential by (1.4) where the subscripts denote partial differentiation,
IXI<%
p(x,O)=-p(U&-i&), where p is the mean fluid density. Boundary and these are taken to be B = % = 0, which correspond
and similarly Y =O,
(1.5)
conditions are required at the plate ends,
x=*a,
y =o,
(1.6)
to clamped edges. Equation (1.3) can be rearranged
(a4/ax4-CL4)r1(x)+(ia/20)(d+~~,/k)l+=0,
IxI
to give Y =O,
(1.7)
where cr =2pw2/D, p4- -Mp*/D, and in this paper the heavy loading limit is taken, corresponding to (Y+ co. An additional assumption is made that the plate length, a, is not very small and will in fact satisfy the condition a >>(k2/a)1’3. 2. UNIFORM SUBSONIC FLOW (M=O(l)) 2.1. INNER AND OUTER REGION LENGTH SCALES It is expected that the basic structure of the acoustic potential will be similar to that in the problem with no flow and so the matching scheme in reference [3] can be used. It was shown there that for large fluid loading (a! + co) the outer potential (away from the plate) is to leading order the solution of the boundary value problem with the elastic plate removed. This “outer” approximation becomes invalid near the plate and standing waves are present on the plate (resulting in plate resonances) for particular lengths of the flexible surface. When a resonance occurs, eigensolutions, with singular behaviour at the plate edges, become of the same order as the outer solution. To perform asymptotic matching between the outer and inner regions it is convenient to split the potential into the form 4
=4o+A(~h+B(~k52
(2.1)
in the outer region and @ = &,+A(E)@~ +B(E)&
(2.2)
SOUND
SCATTERING
BY
A PLATE
IN
215
A FLOW
in the inner region, where A(E) and B(o) are functions of the small parameter, E, given subsequently in equation (2.7), and change their order in E as a resonance is approached. Matching is then performed separately between ~$aand @a, 41 and @I, etc. Away from the plate the co-ordinates can be non-dimensionalized by using the suitable length scale k-‘; hence ky = Y,
kx =X,
(2.3)
where (X, Y) are the new outer region co-ordinates. An inner region length scale can be found by examining when the outer region approximation fails. This occurs (from equation (1.7)) when
a47j/ax4 - c~U$,/w 2 and
& - UQ,
(2.4)
or at a distance I from an edge, where the inner length scale, I is given by 1 = (k’/c&“,
h4 = O(l),
upon assuming &, c$, - ~$11,
Hence (X, y), (.-i’,jj), the left- and right-hand inner region co-ordinates, I-‘(a +x) =X,
The matching parameter length scales,
r’y
= y,
l-‘(a -x)
a >>1.
(2.5)
are given by
=Z’.
(2.6)
E, first written in equation (2.1), is the ratio of inner to outer E = kl= (k”/c#‘“c
1,
(2.7)
and in this paper 1CCa. 2.2. MATCHING #JO, @o The outer potential & can be expanded in the form ~O-~Ol+~hJ2+~2~03+*
’
*,
(2.8)
where ~$01is the potential of the problem in the absence of the plate and with external forcing. The potentials &2 and r$03 satisfy the same boundary value problem and are eigensolutions with singular behaviour at the edges of the plate. Scaling on the acoustic wavelength (2.3) and replacing do by the expansion (2.8) gives the boundary value problem satisfied by &r, &, do3; this is defined by (1 - M2)~xx + C& + C$+ 2iit4& = 0
&=O, (++iWW=O,
IXI>ka,
for all X, Y,
Y =0,
IX] < ka,
(2.9) (2.10)
Y = 0,
(2.11)
M >>E, M = O(l), ka >>E. The potential r$ is forced by an incoming wave, and, as with all the following potentials, satisfies the condition that only outgoing waves are present at infinity. This problem cannot be solved exactly but an asymptotic estimate can be found for small or large values of ka [3,6, Abrahams 1981, 19821. Note that the outer problem can have more complex geometries as long as these boundaries are of the order of one acoustic wavelength from the plate. The outer potential near to an edge can therefore be shown, by observing that near an edge the potential behaves as the field in an incompressible semi-infinite problem, to exhibit the form
4l&l -
a01 +
bd
1’2sin $6 + cotR cos 8 + dolR 3’2 sin $I + . . . ,
(2.12)
216
I. R2 =
(1 -M2)_‘(X
D. ABRAHAMS
+kLQ2+ Y2,
8 = tan-’ ((1 -M2)“2Y/(X+ka)}
(2.13)
near to the left-hand edge, and near to the right edge R2=(1-A42)-‘(X-ka)2+
Y2,
~=tan-‘{(1-M2)“2Y/(-X+ka)}.
(2.14)
The coefficients aor, bar, etc., can be taken to be known in principle (found by numerical solution of the outer problem or otherwise) and are dependent on the external geometry and forcing (incident wave or monopole source for instance). The expansions of the potentials (i.e., expression (2.12)) can be split into two parts, one being even in y, the other odd in y. The even function automatically satisfies the plate boundary condition and therefore does not require an inner and outer matching as it is valid for the whole region. Thus it is convenient to subtract off this part of the potential to be included again after the matching has been performed, and then attention is restricted to analyzing the region y > 0. The inner problems are again found by substituting the inner co-ordinates into expressions (1.2)-( 1.7); thus the left-hand inner problem becomes @,-,-(1-M2)+@Pr+E2@+2isM@f=0 XCO,
@,-=o, +j =jii=O, 2c(a4/ai4-
for all i, jj,
f =o,
(2.17)
1 =o,
region. From expression
Go = gr(&)(@or +&C&Z+E2& which, after substituting written as
(2.16)
y=o,
(E~/R)~)* +i(c@ +i.M@~)l’= 0 > Qi? = -iccjj + U+jli 1
and similarly for the right-hand Q. can be expanded as
into expressions
(2.15)-(2.18),
@,-,-(l-M2)+@rV=0
@zzzV-M’@,-
= 0,
f>O,
jj=o,
+ * * a),
(2.19)
defines the problem for Qol, (2.20) (2.21)
jr=o, f>O,
(2.18)
(2.8) it can be expected that
forallf,jr
fS0,
@,-=o,
(2.15)
y=o.
(2.22)
In this matching scheme the form of the outer expansion has been suggested in equation (2.19) and it can easily be shown that E~‘~, cl/2 log E, etc., terms are not present in this expansion [3]. Only a brief sketch of the matching procedure is presented in this paper due to the fact that only the forms of A(E) and B(E) are of interest. Upon changing now to inner co-ordinates, the leading order term in C#J~ (from expression (2.12)) has the form 40-e r2 = (1 -M2)-rf2
1’2bolr1’2 sin $0,
+ y2,
(2.23)
8 = tan-’ {y( 1 - M2)“2/ff},
(2.24)
and this must match with the inner potential Q. giving gr(e) = E”‘. Thus the inner potential Go1 is forced by singular behaviour (~l’~) as r + co, and similarly Go2 behaves as ao2 - dolr3’2 sin $0. The form, as r -, 00, of the potential Go can be written as bol(r1’2 sin $0 fAolr-“2
sin $3 +Bolr-3’2
sin $3 + + - a),
(2.25)
where Aol and Bol are found by solution of the ao1 problem. It is therefore
found from
@Ol
r-m
-
SOUND
SCATTERING
BY A PLATE
217
IN A FLOW
continued matching that -AolbolRe?a
1’2
*
1
an Te,
$
Because of the symmetry of the geometry must have the forms 402
=Aolbor4w
qa-BolbolR-3’2
sit&.
(2.26)
it is clear that the eigenpotentials
+A0160141,2~,
403
= Bolbo143,2L
r$02,&3 (2.27)
+Bolbo143,2L,
where &‘2L denotes a potential satisfying expressions (2.9)-(2.11) with a singularity of O(R-1’2) at the left-hand edge, etc., and the bar denotes a right-hand coefficient (i.e., Fol). The problems for QoI (expressions (2.20)-(2.22)) can be solved by using the WienerHopf technique (see section 3) and the solution is composed of a scattered field plus an outgoing wave on the elastic plate of the form (3.23) (see section 3.1) ibi-_(l--MV/2bY @OL,
= aolbol
e
,
f>O,
(2.28)
y>o,
where sol is a known complex constant, and b = {M2/(1 -M2)1’2}1’3.
(2.29)
This non-attenuating wave must obviously match to an outgoing wave (travelling on the plate but far from either edge) which satisfies the full plate and governing equations ((1.2) and (1.7)). With this wave taken to be is(x+a)-yy
4 wave
(2.30)
= e
then, to satisfy all conditions, y = (1 -M2)1’2{s -k/(1
+M)}“‘{S +/c/(1--M)}“‘,
(2.31)
and ~(s4-~4)-C+-MS/k)2=0.
(2.32)
But if (Y+CO, s can be expanded in powers of E to give, from expression replacing x by Z’, etc., 4 wave= e
-ibf’-(1-M2)1/2by[l
+E((2
(2.30) after
_M2Jift
-2(1-M2)1’2(2M2-l)~}/{3M(1-it42)}+O(E2)], b = {M’/(l -IV~)“~}“~.
(2.33)
Similarly for a wave in the opposite direction L,, y1= (1-M2)“*{s~+k/(l
=e
+A4)}“2{sl--k/(l
-is,(x+n)-y,y
(2.34)
I
-M)}1’2, (s’: -~4)yl--Lr(l+Msr/k)2=
0,
(2.35, 2.36) which becomes &.,., = e
-iM-(l-M*)l/zbql
_ E(t2
-2(1 -A42)“2(2M2-
_M21ti
l)Y}/{3M(l -M2)}+ 0(E2)].
(2.37)
Thus, for consistent matching, part of the outgoing wave term in oo2 must add to Qolw given by expression (2.28) to merge into the wave in equation (2.33). This will be shown when solving the inner problems in section 3.
218
I. D. ABRAHAMS
2.3. MATCHING &,
@I
The potentials c5r, & have been introduced into the problem to allow for possible plate resonances (by changing the order in E of A(E) and B(E)). The forcing for the cS1,@r matching can therefore be taken as an un-attenuated plate wave travelling into the left-hand edge, having initially been launched from the right edge. The potential cP1 can be expanded as @r-&r+&&+.
(2.38)
* *,
where Qirr satisfies the boundary conditions in expressions (2.20)-(2.22) with forcing by an incoming wave along the plate of the form @lri = exp { - ibZ - (1 -JM*)~‘*~~J}. The form of @r1 as r + 00 can now be written as -l/2 ?k
-Allr
-3/2 sin $0
.
sin $3 +Bllr
+ ..
.,
(2.39)
as there is no forcing from the outer scattered potential. Thus, upon using the expansion (2.40)
“*~11+E3’*~,*+E5’*~,J,
41-c
where c5r1 and d12 satisfy expressions (2.9)-(2.11),
it is clear that
411 -~rr(R-“*sin$3+a~~~R~‘~sin$I+~
(2.41)
- e),
R+O
and so aI1 = A 11.Continued matching also gives a*-
I’QO
Atlall2r
"*
(2.42)
sin $I,
where G12 satisfies the problem given by ~12~~(l-M2)+~12~~=-2iM~11,a129
=
0,
3s0,
(2.43)
forallZ,f>O,
(2.44)
y=o,
Glziifz -M2@r2,- = -i(@llY,,+M2@rr)/M,
f>O,
y=o.
(2.45)
Note that Q12 is forced by (i) r1’2 behaviour at infinity, (ii) terms in both expressions (2.43) and (2.45), and (iii) aI must also have an incoming wave of the form ib.f-(l-M*)‘&J @l*i
=
e
3M(l -M2)
{2(1 -M2)“2(2M2-
l)y -i(2-M*)i}
(2.46)
to match with the incoming wave in the &I problem (equation (2.37)). As a final point the leading order outer potential for 4r can be rewritten as (2.47)
411 =A1141/2~, where cSl12=is given in equation (2.27) 2.4.
MATCHING
42,
@2
matching procedure for cS2,ep, is analogous to that with the previous potentials (+r, @r) where G2 is the right-hand inner region potential and the forcing is by a right travelling wave along the plate. Again upon expanding the potentials as The
@*-@*r+&&+’
* *,
(2.48)
SOUND
SCATTERING
BY
A PLATE
IN
219
A FLOW
it is found that &I satisfies the boundary value problem defined in expressions (2.20)(2.22) when 2 is replaced by Z’ of equation (2.6) with an incident plate wave providing the forcing. Thus it is clear that @21=
@llW,
(2.49)
9,
and so 22 -AllF-1’2 f2 = (I -&-lx’2
sin $$+sB~~T-~‘~ sin %,
+ y2,
(2.50) (2.51)
e= tan-l {y(l -M2)/Z}.
Matching to leading order gives 42-421
1’2&1h2~
=E
(2.52)
+ O(E~‘~),
where 41’2R is an eigensolution of the problem defined by equations (2.9)-(2.11) has the form 41’2R-R”2 sin ~8+~i~‘~R’/~ sin $x?,
and
R+O
4Rli;R
+
- b1,2R I’2 sin $6,
R2 = (1 -M2)-‘(x
- ka)2+ Y2.
(2.53)
The constants u~‘~, 6;‘2 are found by solving the outer problems for 41’2~ #1/2R, and are again assumed known in principle. The boundary value problem for 422, analogous with ti12, is found to be @22fSPS(1 -M2) + @22gy= 2iM@21x, for all ,?, jj >O,
(2.54) (2.55)
@22yx’ ___ _ -h/f x’X
2
@22x’ ::yi(;09.. Zlyx’x’ _ :‘;O;
2
i’>O,
y=o.
(2.56)
It can also be shown that @22must behave as
$k -A11~1’2F”2
(2.57)
sin it?,
and has an incident plate wave providing the forcing, of the form (from equation (2.33)) -ibf’-(l-M2)ll*by
@22i =
e3M( 1 -M2)
((2 -M2)if’-
2(1 -M2)1’2(2M2-
l)jj}.
(2.58)
As a final observation note that the potential d2, as R + 0, will behave as 42
-E
1/2A ll
R+O
~$1’2~
=
E
1’2A11b1,2R1’2 sin $0 +O(E~‘~)
(2.59)
R--O
which will therefore match with part of the inner left-hand potential, @2L,say, giving @2~
-
(~Ah,2/bod@o~
+ W2).
(2.60)
This can be repeated at the right-hand edge to match the 41 potential. 2.5.
ASYMPTOTIC
SOLUTION
Matching has been accomplished to leading order and the solution may be written as 4 total=+ol +E(Aolbol~l,2=+Aol~ol~l,2~)+O(&~) +A(E){E~,~A~~~~,zL+O(E~‘~)}+B(E){E~’~A~~~~,~R
+O(E~‘~)},
(2.61)
220
I.
D. ABRAHAMS
with inner approximations ~,~,~=E~‘~~~~+O(E~‘~)+A(E){~~~+E~~Z+O(E~)}+B(E){EA~~~~,~~~~/~~~+O(E*)}, (2.62) @right
=
+o(~~‘~)+A(~)(~A11b;,z~ollbol
& 1’2@O~60~/60~
+O(E*)}
+B(E){~2~+&~22+O(&2)}.
(2.63)
Near the plate, the total potential is the sum of the travelling waves on the plate and the potential in equation (2.61), i.e., (2.64)
d,,,,l+A(E)exp{-isl(x+a)-yly}+B(&)exp{is(x-a)-yy},
where y, s, etc., satisfy equations (2.32) and (2.36). It now remains to solve the inner problems &, @ii, etc., so that the coefficients A o1,A 11,etc., can be found. The functions A(E) and B(E) are determined by equating the incoming and outgoing waves at the left-hand edge with those at the right-hand edge and this will give the condition for plate _ resonance (362)-(3.65).
3. THE INNER
3.1.
PROBLEM
THE 011 POTENTIAL
A solution @ii for the inner problem can be found by use of the Wiener-Hopf technique. The potential @ii satisfies the boundary conditions (6XX(1-M2)+~yy=0
forallx,y>O,
xso,
&=O,
where for convenience plate wave
y=o, x>o,
4 xx*y- M2& = 0,
(3.1) (3.2)
y=o,
(3.3)
i, jj is written as x, y and forcing is supplied by an incoming b3 =M*/(l
& =exp{-i6x-(1-M2)“26y},
-M’)l’*.
(3.4)
The usual radiation condition allowing only outgoing waves at infinity is also imposed. Note that the plate equation (3.3) can be integrated to give c$,..,,- M*qS= constant,
x>o
y=o,
(3.5)
but from matching considerations it can be shown that this constant must be zero. Upon subtracting off the incident wave by introducing a new potential (I, where I&- C$- di, the problem becomes I,L(~-M~)+(L,,,=O forallx,y>O, (3.6) (jlY= (1 -M2)“26
ewibx,
&‘xxy - M24 = 0,
x S 0, x >o,
y = 0,
(3.7)
y =o.
(3.8)
The half range Fourier transforms are defined as P(x, y) = ??+(x, y ) + q-(x, y), where 0 P+(s,
y ) =
lam
eisxW,
Y1 dx,
*-(s,
y 1 = _m e%(x, I
Y) dx,
(3.9,3.10)
P+ being analytic in an upper region of the complex s plane and similarly Y_ analytic in the lower half plane. Both P, and ly_ are assumed to have algebraic growth as IsI + Co
SOUND
SCATTERING
BY A PLATE
221
IN A FLOW
in their respective half planes. Transforming the governing equation (3.6) gives the result (3.11)
P(s,y)=A(s)exp{-y[~[(l-M*)“~}, which allows only outgoing wave solutions. Note that
Is1= F; {(s + iq)“2(s -
(3.12)
iq)“*},
where q is a fictitious dissipation term included for mathematical
convenience,
and
Re (s)>O Re
l’l=I -:, (s)
(3.13)
Y+ = !P+(s, 0), etc., will be used and the trans-
= a{P_(s, y)}/J13yl~=~= -(l -M2)“*bi/(s
-6),
N(S)-S2!fY
-M2P+=0, (3.14,3.15)
where N(s) = -&,(O, 0) +is&,(O, 0) and is regular over all s. From equation (3.13) it is known that !P++‘P_=A(s)=-(‘Py:
which, on rearranging functional equation
(3.16)
and using expressions (3.14) and (3.15), gives the Wiener-Hopf
{N(s)/M2jK_(s)
+ W-K_(s) -{(l -M’)i’*bi/(s
= -!?:K+(s)+{(l The Wiener-Hopf
+P~))/{~s~(1-M*)“*},
-M2)1’2bi/(s
-b)}{s*K-(s)
+K+(b)}
-b)}{K+(s)-K+(b)}.
(3.17)
kernel K(s) is given by
K(s) = ((1 -M2)-1’2/ISI}-S2/M2
=K+(s)/K_(s),
K+(s) = l/K-(-s),
(3.18)
where K+(S), K_(s) are regular and non-zero in their respective upper and lower half s-planes. The left-hand side of equation (3.17) is therefore analytic in the lower half plane (the pole at s = b lying just above the line joining the two half planes), and similarly the right-hand side is regular in the region Im (s) > 0. Note that a regular function J(s), say, can equal both sides of equation (3.17) for values of s lying on the line of common analyticity. By analytic continuation J(S) can be extended throughout the whole s plane and the form of J(s) is found by examining the behaviour of equation (3.17) as IsI + co. By using the fact (from equation (A9) of Appendix A) that K+(s)-O(s)
ass+oo,
etc.,
(3.19)
together with the edge condition t&(0,0) = 6(1 -M*)l’*, J(s) behaves as J(S) + 0 as IsI+OO and so by Liouville’s theorem J(s)=O. Thus $1 is now found from equation (3.17), and equation (3.11) can be written as ~(~,y)=~biK+(b)lJsI(s-b)K+(s)~exp~-Isl(l-~~)~’~y},
(3.20)
or finally the solution is K+(b)
exp
{-id-Is1(1-~2)1’2~)ds
K+(s)lsl(s -6)
+,-ibi-(l-M2)1/Zb$!
(3.21)
222
I. D. ABRAHAMS
integral can be estimated for large r hence giving the coefficients AI1 and BII of expression (2.39) and the pole in the integrand at s = -b (in K+(s)) gives the outgoing plate wave. After much algebra the outgoing wave (from equation (AM)) is determined as @rlw = e5i7r/4 eibf-gb(l-M2)1/*, (3.22)
The
The above procedure
could be repeated for aoI which would determine the coefficients (2.25), and the coefficient (~0~multiplying the outgoing wave: i.e., the outgoing plate wave term in QzoIis (from equation (2.28))
AoI and B,-,r of expression
GOlo
=
crolbol exp {ibf - (1 -M*)“*bjf},
(3.23)
where (yo1 is assumed known in this analysis, 3.2. THE aL2 POTENTIAL The boundary value problem for G12 is written in equations (2.43)-(2.45) and forcing is supplied by an incoming wave along the plate, given in equation (2.46). As in section 3.1 a new potential, 4, can be defined which is given (from equation (2.46)) by (3.24)
4 =@l*-@**i
and, after substitution into the relevant equations, 4 is found to satisfy the problem &, (1 -M*) + c%,,+ 2iM& = 0 &, = -epib”(Ax + b), -M*cL + (ilM)($,,,, 4 YXXX
for all x, y > 0,
x =S0,
(3.25)
y = 0,
x>o,
+M*tj) = 0,
(3.26) y=o.
(3.27)
The potential $ is the inverse Fourier transform of equation (3.11), and A = (2 -Mz)ib/3M(l
Transforming
B = 2(2M2-
-M*)l’*,
1)/3M(l
-M2)1’2.
(3.28)
the governing equation gives
@(s, y) =D(s) e-‘S’y’1--MZ”‘2+yMibK+(b) e-‘s’y(1-M2)“2/s(1 -M*)l’*(s
- b)K+(s),
(3.29)
where @(s, y) is the Fourier transform of C&(X,y), and @+, @I, etc., are half range transforms as previously defined in equations (3.9) and (3.10) and can be shown to have the same regions of analyticity as the P+ and P_ functions. The problem now reduces to determining D(s) for the given boundary conditions. The plate equation transforms to give P(s)+is3@: P(s)={-c%:,, the superscript
+iM*s@++(i/M)(-s*PL +isf$$+s*cSy
+M*P+)=O,
(3.31)
+M290-(iIM)~~,-(slM)9~),
denoting x = y = 0, and 4: = -B
and 4; = b(1 -M*)“*.
@J: +@F =-~s~(l-~2)“2~(s)-{M/S(1-M2)}(ly: @++@_=D(s)
(3.30)
+YL),
The identities (3.32) (3.33)
can be used, together with the left-hand boundary condition, to give the Wiener-Hopf
SOUND
SCATTERING
BY A PLATE
223
IN A FLOW
equation. After much algebra this equation is found to be P(s)K_(s)+is’K-(s)(
A-&]
-M2s@-K-(s)
+iM2 ( A[sK+(b)+b(s-b)K(b)l(S_b)2
f++(b)]
M3bK+(b) 1+RJs) R+(b) -(s _b)(l-My I T+MZ I +2(1-Mz)“26 M(s -6)
[S2K_(S) -K+(b){R-(s)
+R+(b)H
-iM’(&&K+(s)-X+(b)-b(s-b)K:(Wl
= isM2K+(s)dJ>
+26(1 -M2)“2K+(b)
[R+(s) -R+(b)19
M(s-b)
(3.34)
where K+(s) and K_(s) are defined in equations (3.18), K’(b) = dK(s)/dsI,+,,
R+(s) +R-(s)
= S2/K(S),
(3.35)
R+(s) and R_(s) being regular in the upper and lower half planes, respectively. The left-hand side of equation (3.34) is regular in the lower half plane, and the right-hand side is regular in the upper half plane. After allowing both sides of equation (3.34) to be equal to a function regular over all s, and examining the behaviour of each side as 1sI+ co, one finds from the extended form of Liouville’s theorem that this function must be a constant, C, say. Note that C is determined by matching to the outer region and therefore from equation (2.42) must be proportional to A11a1,2. It is now assumed to be a known constant and so the function D(s) can be written as AK+(b) AbK: (b) Bi6K+(6) (s-b)2K+(s)+(s-b)sK+(s)-(s-b)sK+(s)
D(s) =
ib(2 -M2)K+(b> -M3(1 -M2)*‘2(s -b)sK+(s)
rR+(s)-R+(b)l+M2;g (J + (3.36)
+*
The solution can finally be written as jMibK+(b) s(s -b)(l -M2)“2K+(s)
D(s)+
e-~s)~(l-M*)l/*-isf
ds +@rzi.
(3.37)
The outgoing wave can again be found by picking up the pole at s = -6, and, from equation (BlO) of Appendix B, after considerable algebra, @120
=
i=/4 (2 -M’)Il + 2( 1 -M2)“2(2M2 - l)F 3M(l -M2) 3M( 1 -M2) I
I [ e
-
cei”‘86(2/3)“2 M3
I
e
ibz-b(l-M2)“2g
(3.38)
224
I. D. ABRAHAMS
The analogous outgoing wave @220 can be written down after noting that the forcing in equations (2.54), (2.53) and (2.58) is the negative of that in the & problem. Thus it is found that @22@ =
sin/4 (2 - M2)if ’ +2(1 -M2)“2(2M23M(l -M2) 3M(l -M2)
l)jj
( [ e
- Dei”‘8b(2/3)“2 M3
1
e
I
ibf’-b(l-M2)11*_
(3.39)
YI
where D is found from matching with the outer field and can be considered known constant.
to be a
3.3. DETERMINING A(E), B(E) It is clear that the outgoing wave from the left-hand edge must be equal to the incoming wave into the right-hand region as the waves do not decay with distance along the plate. This must also be true for waves emanating from the right-hand edge and travelling into the left-hand edge. The outgoing wave from the left-hand edge is the sum of all waves generated by the potentials in equation (2.62). Thus, upon using expressions (3.22), (3.23), (3.38) and (2.60), the total outgoing wave is @lcfto
=
[E 1’2ao~bo~+O(~3’2)+B(~)(~A~~b~,2~~~+O(~2)} +A(~)(eSi+r/4{1-~(2-~2i~+2(1-M2)1’2(2M2-l)~)/(3M(1-M2))} -&f-2CeiW’86(2/3)1’2+O(~2)]exp{ib~-6(1-M2)”2~},
(3.40)
or, in outer co-ordinates, (9 lefto
--[
E 1’2~01601+O(~3’2)+B(~){~AIlb~,2~~1+O(~2)}
+A(E)(eSin/4_
E~-2~
eir/8b(2/3)l/2+
0(E2))] eir(x+a)-vy,
(3.41)
where y and s satisfy equations (2.31) and (2.32). This wave must be equal to the incident wave into the right edge which, from equation (2.64), gives @lefto =B(E)e
is(x-a)-vy
(3.42)
.
Similarly the left going wave condition gives Q,right- = [&1’2cy,,&,~+O(E~‘~)+A(E){EA*~~;,~(Y~~ +~(~){~5id4_ =A(&)
e
._M-2D
eid8b (2/3)1/2 + 0(E2)}]
e-is,(x-a)-vly
-is,(x+aky,y
with y1 and s1 satisfying equations equations finally gives the results A(E)
+O(E~)}
= E 1’2ao~{b~~esirr’4+&
(3.43) (2.35) and (2.36). Solving the above simultaneous
exp (-2ika[b/~
-(2-M2)/(3M(1-M2))])+O(e)}/E(~), (3.44)
B(E) = ~“~a~~{b~~ exp (-2ika[b/E + iDleSir/ + O(E))/&),
+(2-M2)/(3M(1
-M’))]) (3.45)
SOUND
SCATTERING
E(E) = exp (-4iakb/&)
BY A PLATE
225
IN A FLOW
-exp (5ir/2)
+e{[2iak2/M2(1-M2)*][14M4-49M2+38]exp(-4iukb/e) -Am01
e
-2iakb’e[bti2 exp (- 2ika (2 -k4*)/(3M(l
+6;,* exp (2ika(2 -M*)/(3M(l
-M*)))
-M*)))]
+(D +C)(b/A4*)(2/3)“*}+0(~*).
(3.46)
A resonance is defined to occur when A(E) and B(E) change from O(E “*) to O(E -?, and this condition occurs when e-4iakb/e
=
e5in/*
(3.47)
9
or
(3.48)
2pU2u3/(1 -M2)1’2D = {(3~/8)+ (n~r/2)}~, II = 0, 1,2, . . . .
This resonance condition can be altered by changing the plate edge conditions. instance, if the plate were simply supported at the edges, i.e., I)=%X=
x=*u,
0,
y=o,
For (3.49)
then repeating the previous analysis would yield the resonance condition 2pU2u3/(1 -M*)“*D
n = 0. 1,2,. . . .
={(~/24)+(nr/2)}~,
(3.50)
Note that if fluid were on only one side of the plate (with a vacuum on the other) then the resonance condition (3.50) would become n=0,1,2
pU2u3/(1 -M2)1’2D ={(*/24)+(n~/2)}~,
)...
.
(3.51)
4. LOW MACH NUMBER FLOW
The resonance condition obtained in section 2 is valid for a Mach number of order unity, and does not reduce to the zero flow resonance condition [3, Abrahams 19811 CY
l/5
a
-1 -*?T+$m,
n=0,1,2
,...,
(4.1)
as the Mach number is reduced. The reason for this is breakdown of the leading order expansion of the equations (2.15)-(2.19) written in expressions (2.20)-(2.22). This is therefore a singular limit suggesting that an examination be made of the problem when the Mach number is of the order of the other small parameter in the problem. As suggested in the zero flow problem the small parameter is taken as E =
k/c?,
(4.2)
and so the Mach number is now defined to be where m = O(1).
M = me,
The plate length, a, is kept in the same range as previously: co-ordinates can now be written as Z=P(u+x),
y = (y“SY,
(4.3) i.e., ku >>E. The inner
f’ = cPS(u -x),
for the left and right inner regions, respectively, and the outer co-ordinates acoustic wavelength, so that X=kx,
Y = ky.
(4.4) scale on the (4.5)
226
I.
D. ABRAHAMS
The leading order outer problem is found after resealing on the outer co-ordinates, is defined by ~&+c&,+~#J=O 4Y =O,
forallx,
Y,
C$I”=o,
IXI>ka,
and (4.6)
IXl
(4.7,4.8)
Similarly the inner problems can be written as @,-,-+@,-,-+~~(@++irn@,-m2@,-,-)=0 @,=O,
forallf,y,
*=*,-=0,
f
~(a~/f!IZ~+~~e~)ji +(i@ -m0,)/2clT @? = --Ec(ijj -m?j,-),
(4.9)
i=O,
= 0,
(4.10,4.11)
y=o,
y=o,
x>o ,
x>o.
(4.12) (4.13)
The matching scheme for this low Mach number limit is no different from that described in section 2, and will therefore not be analyzed. To calculate the condition for resonance it is only necessary to obtain the reflection coefficient of a plate wave travelling into an edge. Thus the leading order inner problem is found to be (after noting that Qi is odd in Y) @,,,
@oy=O,
x
+ aoee =
f=O,
for all Z,y > 0,
0
jjo=?jo~=o,
c?~+~o/c~X~ + (l/c)(i@% - m@&) = 0, @Ok= -c (iii0 - mfjoi), Tj =no+&nl+.
ff=o, x>o,
x>o,
(4.14) jj=o, y=o,
(4.16)
y=o,
(4.17)
@=,@,+2@,+~~-.
. *)
(4.15)
(4.18,4.19)
Forcing is due to a travelling plate wave into the inner region and will have the form e where d satisfies the plate equation region. Thus d must be a solution of
-idf-dy
(4.20)
,
and kinematic
boundary
condition
ds=(1+md)2. The inner problem which yields
(4.14)-(4.17)
in the inner (4.21)
can be solved by using the Wiener-Hopf
technique
(4.22) ’
(4.23)
K(s)=K+(~)/K_(s)=(l+sm)~/lsI-s4,
f being a constant found by matching or by edge conditions. The non-attenuating outgoing plate wave can be found in the limit m + 0, by using the results d=l+O(m),
(4.24)
f=O+O(m);
thus @owave =
e
3im/4 eif-V+~(mj.
(4.25)
Equating the waves at either edge of the plate gives the resonance condition r/=Ll= $?r + $l?r, CY
n=0,1,2
,...,
(4.26)
SOUND
SCATTERING
BY A PLATE
IN A FLOW
227
which is identical to that in the zero flow problem. Alternatively when the factor m in equation (4.27) becomes very large it is found that, from equation (4.21), d = m2’3,
K(s)-(m2/lsI-s2)s2+U(m-1).
(4.27)
The constant, f, is found to be (4.28)
f = -X+(d),
which gives the required behaviour at infinity, and so the outgoing wave is found, by picking up the pole at s = -d, to be otrn-l). 0 O,.“, = eSiw/4 eidf-dT+ ’ (4.29) The resonance condition therefore becomes da”‘a
12= 0, 1,2,. . . )
= (37/g) + (nr/2),
(4.30)
or, upon using equation (4.3) to give d = (M/E)~‘~, 2pU2a3/0 This M + It flow
= {(3~/8) + (n7r/2)}3,
n=0,1,2
,....
(4.31)
result must be equivalent to the resonance condition found in section 3 in the limit 0, and by expanding equation (3.48) in powers of M this is found to be correct. has now been shown that the potential of the problem with small Mach number smoothly matches the zero flow solution to that of the M = O(1) problem. 5. DISCUSSION
It has been shown that in the heavy loading limit the outer potential behaves to leading order as if the plate were absent. Near a resonance, however, eigensolutions of the outer boundary value problem with singularities at the plate edges also become present at this leading order. Close to the plate it is found that near resonance the leading order terms are standing plate waves (of order E-1’2) and these have a wavelength of the order of the size of the inner region. These results are similar in form to those in the zero flow problem except that for M = O(1) the wavelength of the plate waves is given, from equation (2.5) by, I= (k2/ap3
= (D/pc2y3.
(5.1)
This result is interesting because it shows that the wavelength (or frequency) of the un-attenuated waves on the flexible plate is independent of the frequency of the acoustic source supplying the forcing. Thus for any frequency of forcing it can be expected that the frequency of plate waves will be a constant dependent only on D, p, cand U, subject of course to the heavy loading approximation E =klc
1.
(5.2)
As was discussed briefly in the introduction, it was hoped that comparison could be made between this analysis and flutter analysis by using modal expansions. Much work has been performed on calculating the onset of flutter of finite or infinite elastic plates [4, Weaver and Unny 1970; 7, Dowel1 1966; 8, Bohon and Dixon 19641. The usual method for calculating instability is to replace the unknown plate deflection by a modal expansion (where each term satisfies the plate edge conditions) and the problem is solved by using a Galerkin approximation. The validity of Galerkin’s method is difficult to justify in a rigorous sense and it is therefore helpful to solve the problem by an alternative technique (in this case by matched asymptotic expansions) to compare the predictions
228
I.
D. ABRAHAMS
made. The two-dimensional problem of flow of a compressible fluid over one side of a simply supported elastic plate was analyzed by Ellen [5] (again using a truncated modal expansion) and his results predicted divergence instability (when the complex frequency crosses the real axis at w = 0) when pU’a~/D(l
-&f2)“2 = r5n4/2{n7r Si (n7r) - 1+ (- l)“},
aE = 2a,
Si(z)=
I0
’sin x dx ___ x
n=1,2,...,
subject to the restriction kaE c 1. ’
(5.3) (5.4)
In the analysis presented in this paper a time harmonic solution was assumed. If, instead, a Fourier transform in time had been taken, i.e., @p(x_, o) = jrn 4(x_, t) eiwt dt, --oo where4(x,t)=O
(5.5)
fort
al
@(x_,w) e-‘“’ do,
(5.6)
where the poles of @(x_,w) must lie below the real axis. Thus to satisfy causality the result in equation (2.61) must have poles (of complex o) lying in the lower half plane. This condition leads to the result (from equation (3.63)) that the frequency vanishes (i.e., a divergence instability occurs) at a plate resonance. Therefore divergence instability is predicted here for a simply supported plate with flow over one side when (from equation (3.51)) pU2a~/D(1
-M2)1’2 ={(~/12)+(nr)}~,
n=0,1,2
,...)
(5.7)
with the conditions kl c 1,
l/a
(5.8)
Note that the latter constraint implies that equation (5.7) is strictly valid only for large n. It can be shown that the values of the dimensionless parameter pU2a~/[D(1 -M2)1’2] predicted by both methods are within 2% of each other for n 2 1 and both tend to the same value (n7r3) as n + 00. The n = 0 term in equation (5.7) violates the latter constraint in expressions (5.8) and does not satisfy the condition that kaE <<1 in equations (5.4) and can therefore be rejected; it is reassuring, however, to find that the results are in very good agreement, even for low n (when neither approximation is strictly valid). As a final point it can be noted that an advantage of the present method over that of Ellen is the ability to predict resonance for any plate edge condition whereas the modal expansion technique has been applied only to simply supported panels. ACKNOWLEDGMENTS
The author would like to thank Dr F. G. Leppington for much advice and assistance, and the S.E.R.C. and A.U.W.E. Portland for the financial support of a C.A.S.E. studentship. REFERENCES
1976 Quarterly Journal of Mechanics and Applied Mathematics Scattering of sound by finite membranes and plates near resonance.
1. F. G. LEPPINGTON 527-546,
2. D. C. HANDSCOMB 183-189. Vibrations
1977 Journal
of the Institute of Mathematics
of a submerged window.
and its Applications
29, 20,
229
SOUND SCATTERING BY A PLATE IN A FLOW
3. I. D. ABRAHAMS 1981 Proceedings of the Royal Society London A378, 89-117. Scattering of sound by a heavily loaded finite elastic plate. 4. D. S. WEAVER and T. E. UNNY 1970 American Society of Mechanical Engineers, Journal of Applied Mechanics 37, 823-827. The hydroelastic stability of a flat plate. 5. C. H. ELLEN 1973 American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 68-72. The stability of simply supported rectangular surfaces in uniform subsonic flow. 6. 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. 7. E. H. DOWELL 1966 American Institute of Aeronautics and Astronautics Journal 4, 1370-1377. Flutter of infinitely long plates and shells-part 1: Plate. Some recent developments 8. H. L. BOHON and S. C. DIXON 1964 JournalofAircraft 1,280-288. in flutter of flat panels.
APPENDIX From equation
A: FACTORIZATION
OF THE WIENER-HOPF
(3.18) K(s) = {Isj(l -M2)1’2}-1
which can be rewritten
(AlI
-s2/M2,
as K_(s) = MSY2 ei?r’4Q-(s),
K+(s) = (eiT’4/MS:‘2) Q+(s), where Si”
KERNEL
(A21
= lim,,e (s +iq)1’2, etc., and
c?+(s)/Q-(s)
Q+(s) = Q-(-s).
= V--s21sl),
(A3)
and Q- are regular and non-zero in the same regions as K, and K-, respectively, and it can be shown that
Q,
In [Q+(s)] = (2ri)-r
lim
where 1,s > I,& Note that the integration
(A4)
path in equation (A4) passes below the point
5 = b and so
ln[Q+ts)l=~,
say,
=& Iom In (b3-13) tc2_s2)
After much algebra, with careful interpretation it is found that
1
(A3
of the branch of the logarithm chosen,
1
___-1
(s+bc,)+3(s+bl)
dS.
3(s+b_l)
(A61
I’
where 6, = b e2nni’3 . To estimate the order of K+(s), as s + co, the expansion of expression
(A6) in powers of l/s is dI/ds -(3+/s)+O(l/s*),
s+oO,
(A7)
which, after integrating, gives Q+(s)=ZJs3’2+O(s”2),
s+Q),
W9
where P is a constant, and so from equation (A2)
K+(s) = O(s)
when s + ~0.
(A9)
230
I.
D. ABRAHAMS
It is now required to find K+(6) so that the coefficient of the outgoing wave in equation (3.21) can be found. First, by integrating by parts, equation (A5) can be rewritten as
and so I(0) = 37ri In 6,
(All)
as the integral in equation (AlO) is zero when s = 0. The expression in equation (A6) can now be integrated and, upon letting s = 6, the equation simplifies to
I(6) -I(O) 2ri
(A12)
=ln(~}=-&[om&dv+t$)ln6-~.
The integral can be shown to have the value rr*/8 and so substituting equation (A12) into equation (A2) finally gives K+(6) = 6 e’““&/M. The pole contribution can be written as =
6
lim
(A13)
at s = -6 in equation (3.21) gives the outgoing wave for @ir, and
K+(6) e-isi-‘s’8(1--M2)“*= --K: (6) 26 K+(s)lsl(s -6)
ibX-bJ(l-Mz)l/2 .
(A14)
The coefficient in front of the exponential in equation (A14) is found -Kt (6)M*/66*, and upon using equation (A13) the solution becomes
to be
@ll”
s+-b
5irr/4 @I~,
=
e
e
e
ibl-bf(l-.W*)l/*
APPENDIX B: OUTGOING WAVE Q&, The expression for or2 is written in equation (3.37) and the outgoing wave is again found by closing the integral in the lower half plane thus picking up the pole at s = -6. Upon noting that the term in D(s), from equation (3.36), containing R+(s) has a pole of order two at s = -6, the residue of the pole can be written as residue (at s = -6) 1 = 2~6*(1 -M*)l’* iC -y+ M
AK+(b)+AKi(b) 46 2
MiK+(6) 2(1-M*)r/*+
_
Bi6K+(6)+iK+(6)(:!-M2)R+(6) 26 2M3(1 -M*)l’*
yMiK+(6)6 2 e R+(~)(~ +6)* eiSf-lSl~(l--M2)“2 sK+(s)lsl(s -6)
’
(Bl)
By using the identities R+(s) =s*/K(s)-R_(s),
R-(-b)=R+(b),
(I329B3) (B4)
SOUND
SCATTERING
BY
A PLATE
IN A FLOW
231
the last term in equation (Bl) can be written as it2
_MZJK+(~J
eibf-bg(l-M2)“2
-
(jj(l-A42)“2
47rM3( 1 -iW)
-Z)K+(b)-K:(b)
It is easily shown that !;mb($(zr}=O,
(B6-B8)
and these values are substituted into equations (Bl) and (B6). By using equations (3.28), (B5) and (Bl) the outgoing wave of Cpi2, after considerable cancellation, is determined: @12”
=
=
-2ri{residue
at s = -b}
(2kf2-l)~K: (b)b+i(2-M2)K:
(b)bz cK+(b) 18M(1-M2)“2 -x
9M
ibf_b~(l_MZ)1/2 e (B9)
or, upon using equation (A13),
C -
ei+b(2/3)l'*
M3
I
e
ibf-bg(l_&f2)1/2
(BlO)
SCATTERING BY AN ELASTIC I. D.
OF SOUND
PLATE
WITH FLOW
ABRAHAMS
Department of Mathematics, The University, Manchester Ml3 9PL, England (Received 27 September 1982)
An elastic plate, set in an infinite baffle and immersed subsonic velocity, is excited by an acoustic source. The when fluid-plate coupling is large, and a solution is found expansions. The far field is found to approximate to
in a fluid moving with a uniform scattered sound field is analyzed by the use of matched asymptotic the solution obtained when the
elastic plate is absent. At a plate resonance, however, the outer field must include eigensolutions with singularities at the plate edges, and close to the plate the dominant terms are travelling plate waves. These plate waves are found to have a wavelength independent of the frequency of the source. It is also shown that a plate resonance corresponds to a divergence instability of aerodynamic flutter theory and that the stability results found in this paper are in agreement with those obtained by using modal expansions. The limit as the Mach number goes to zero is found to be singular, suggesting an analysis of the model for small flow velocity. This calculation is performed and the results match smoothly to the respective solutions for a stationary fluid and for a large subsonic flow.
1. INTRODUCTION There has been much attention given in recent years to acoustic problems with finite or infinite flexible surfaces. With finite geometries asymptotic or other approximate methods must be employed to obtain an estimate of the sound field [l, Leppington 1976; 2, Handscomb 19771. The problem of a thin elastic plate set in an infinite baffle has been analyzed in the limits of both small and large fluid loading [l, Leppington 1976; 3, Abrahams 198 1, respectively] and in this paper the latter work is extended by introducing a uniform subsonic flow of the fluid enveloping the plate. The method of matched asymptotic expansions is employed with modification to allow for the presence of plate resonances. It is also expected, due to the addition of flow to the problem, that flutter or divergence instability of the elastic plate may occur. This problem has been analyzed by use of modal expansions to calculate the onset of instability [4, Weaver and Unny 1970; 5, Ellen 19721 and the results will be shown to be in agreement with those found in this paper. In section 2 the problem is analyzed for a Mach number of order unity (M < 1) and, as will be shown, the results do not reduce to that given by Abrahams [3] when the flow velocity goes to zero (i.e., M + 0 is a singular limit). It is therefore necessary to analyze separately the problem when the Mach number is small (section 4) as this result matches to both the M = 0 and M = 0( 1) results. Cartesian co-ordinates (x, y) are chosen in the two-dimensional problem, and for simple harmonic time dependence, with radian frequency o, the velocity potential can be defined as Re 14 (x, Y1 exp ( - ie.41.
(1.1)
213 0022-460X/83/140213+19$03.00/0
@ 1983 Academic Press Inc. (London) Limited
214
I.
D. ABRAHAMS
The time factor is henceforth suppressed for brevity. An elastic plate (lying on y = 0, Ix) < a) is immersed in a compressible inviscid fluid and when there is a uniform mean flow in the x direction the potential satisfies the corresponding convected wave equation ~x,(1-M2)+~,,+k2~+2ik~~,=0,
(1.2)
where M( = V/c) is the Mach number, U is the flow velocity, c is the sound speed in the fluid, and k ( = o/c) is the acoustic wavenumber. For a thin elastic plate, of length 2a, the equation governing small flexural vibrations is (D a4/ax4-&0*)7j(X)
= -p(x, O)]“,
lxI
Y =O,
(1.3)
where D is the plate bending stiffness, M,, its mass per unit area, and n and p denote the plate deflection and fluid pressure respectively. Note that p(x, O)]’ denotes the discontinuity in pressure across the plate. The deflection n is related to the potential by (1.4) where the subscripts denote partial differentiation,
IXI<%
p(x,O)=-p(U&-i&), where p is the mean fluid density. Boundary and these are taken to be B = % = 0, which correspond
and similarly Y =O,
(1.5)
conditions are required at the plate ends,
x=*a,
y =o,
(1.6)
to clamped edges. Equation (1.3) can be rearranged
(a4/ax4-CL4)r1(x)+(ia/20)(d+~~,/k)l+=0,
IxI
to give Y =O,
(1.7)
where cr =2pw2/D, p4- -Mp*/D, and in this paper the heavy loading limit is taken, corresponding to (Y+ co. An additional assumption is made that the plate length, a, is not very small and will in fact satisfy the condition a >>(k2/a)1’3. 2. UNIFORM SUBSONIC FLOW (M=O(l)) 2.1. INNER AND OUTER REGION LENGTH SCALES It is expected that the basic structure of the acoustic potential will be similar to that in the problem with no flow and so the matching scheme in reference [3] can be used. It was shown there that for large fluid loading (a! + co) the outer potential (away from the plate) is to leading order the solution of the boundary value problem with the elastic plate removed. This “outer” approximation becomes invalid near the plate and standing waves are present on the plate (resulting in plate resonances) for particular lengths of the flexible surface. When a resonance occurs, eigensolutions, with singular behaviour at the plate edges, become of the same order as the outer solution. To perform asymptotic matching between the outer and inner regions it is convenient to split the potential into the form 4
=4o+A(~h+B(~k52
(2.1)
in the outer region and @ = &,+A(E)@~ +B(E)&
(2.2)
SOUND
SCATTERING
BY
A PLATE
IN
215
A FLOW
in the inner region, where A(E) and B(o) are functions of the small parameter, E, given subsequently in equation (2.7), and change their order in E as a resonance is approached. Matching is then performed separately between ~$aand @a, 41 and @I, etc. Away from the plate the co-ordinates can be non-dimensionalized by using the suitable length scale k-‘; hence ky = Y,
kx =X,
(2.3)
where (X, Y) are the new outer region co-ordinates. An inner region length scale can be found by examining when the outer region approximation fails. This occurs (from equation (1.7)) when
a47j/ax4 - c~U$,/w 2 and
& - UQ,
(2.4)
or at a distance I from an edge, where the inner length scale, I is given by 1 = (k’/c&“,
h4 = O(l),
upon assuming &, c$, - ~$11,
Hence (X, y), (.-i’,jj), the left- and right-hand inner region co-ordinates, I-‘(a +x) =X,
The matching parameter length scales,
r’y
= y,
l-‘(a -x)
a >>1.
(2.5)
are given by
=Z’.
(2.6)
E, first written in equation (2.1), is the ratio of inner to outer E = kl= (k”/c#‘“c
1,
(2.7)
and in this paper 1CCa. 2.2. MATCHING #JO, @o The outer potential & can be expanded in the form ~O-~Ol+~hJ2+~2~03+*
’
*,
(2.8)
where ~$01is the potential of the problem in the absence of the plate and with external forcing. The potentials &2 and r$03 satisfy the same boundary value problem and are eigensolutions with singular behaviour at the edges of the plate. Scaling on the acoustic wavelength (2.3) and replacing do by the expansion (2.8) gives the boundary value problem satisfied by &r, &, do3; this is defined by (1 - M2)~xx + C& + C$+ 2iit4& = 0
&=O, (++iWW=O,
IXI>ka,
for all X, Y,
Y =0,
IX] < ka,
(2.9) (2.10)
Y = 0,
(2.11)
M >>E, M = O(l), ka >>E. The potential r$ is forced by an incoming wave, and, as with all the following potentials, satisfies the condition that only outgoing waves are present at infinity. This problem cannot be solved exactly but an asymptotic estimate can be found for small or large values of ka [3,6, Abrahams 1981, 19821. Note that the outer problem can have more complex geometries as long as these boundaries are of the order of one acoustic wavelength from the plate. The outer potential near to an edge can therefore be shown, by observing that near an edge the potential behaves as the field in an incompressible semi-infinite problem, to exhibit the form
4l&l -
a01 +
bd
1’2sin $6 + cotR cos 8 + dolR 3’2 sin $I + . . . ,
(2.12)
216
I. R2 =
(1 -M2)_‘(X
D. ABRAHAMS
+kLQ2+ Y2,
8 = tan-’ ((1 -M2)“2Y/(X+ka)}
(2.13)
near to the left-hand edge, and near to the right edge R2=(1-A42)-‘(X-ka)2+
Y2,
~=tan-‘{(1-M2)“2Y/(-X+ka)}.
(2.14)
The coefficients aor, bar, etc., can be taken to be known in principle (found by numerical solution of the outer problem or otherwise) and are dependent on the external geometry and forcing (incident wave or monopole source for instance). The expansions of the potentials (i.e., expression (2.12)) can be split into two parts, one being even in y, the other odd in y. The even function automatically satisfies the plate boundary condition and therefore does not require an inner and outer matching as it is valid for the whole region. Thus it is convenient to subtract off this part of the potential to be included again after the matching has been performed, and then attention is restricted to analyzing the region y > 0. The inner problems are again found by substituting the inner co-ordinates into expressions (1.2)-( 1.7); thus the left-hand inner problem becomes @,-,-(1-M2)+@Pr+E2@+2isM@f=0 XCO,
@,-=o, +j =jii=O, 2c(a4/ai4-
for all i, jj,
f =o,
(2.17)
1 =o,
region. From expression
Go = gr(&)(@or +&C&Z+E2& which, after substituting written as
(2.16)
y=o,
(E~/R)~)* +i(c@ +i.M@~)l’= 0 > Qi? = -iccjj + U+jli 1
and similarly for the right-hand Q. can be expanded as
into expressions
(2.15)-(2.18),
@,-,-(l-M2)+@rV=0
@zzzV-M’@,-
= 0,
f>O,
jj=o,
+ * * a),
(2.19)
defines the problem for Qol, (2.20) (2.21)
jr=o, f>O,
(2.18)
(2.8) it can be expected that
forallf,jr
fS0,
@,-=o,
(2.15)
y=o.
(2.22)
In this matching scheme the form of the outer expansion has been suggested in equation (2.19) and it can easily be shown that E~‘~, cl/2 log E, etc., terms are not present in this expansion [3]. Only a brief sketch of the matching procedure is presented in this paper due to the fact that only the forms of A(E) and B(E) are of interest. Upon changing now to inner co-ordinates, the leading order term in C#J~ (from expression (2.12)) has the form 40-e r2 = (1 -M2)-rf2
1’2bolr1’2 sin $0,
+ y2,
(2.23)
8 = tan-’ {y( 1 - M2)“2/ff},
(2.24)
and this must match with the inner potential Q. giving gr(e) = E”‘. Thus the inner potential Go1 is forced by singular behaviour (~l’~) as r + co, and similarly Go2 behaves as ao2 - dolr3’2 sin $0. The form, as r -, 00, of the potential Go can be written as bol(r1’2 sin $0 fAolr-“2
sin $3 +Bolr-3’2
sin $3 + + - a),
(2.25)
where Aol and Bol are found by solution of the ao1 problem. It is therefore
found from
@Ol
r-m
-
SOUND
SCATTERING
BY A PLATE
217
IN A FLOW
continued matching that -AolbolRe?a
1’2
*
1
an Te,
$
Because of the symmetry of the geometry must have the forms 402
=Aolbor4w
qa-BolbolR-3’2
sit&.
(2.26)
it is clear that the eigenpotentials
+A0160141,2~,
403
= Bolbo143,2L
r$02,&3 (2.27)
+Bolbo143,2L,
where &‘2L denotes a potential satisfying expressions (2.9)-(2.11) with a singularity of O(R-1’2) at the left-hand edge, etc., and the bar denotes a right-hand coefficient (i.e., Fol). The problems for QoI (expressions (2.20)-(2.22)) can be solved by using the WienerHopf technique (see section 3) and the solution is composed of a scattered field plus an outgoing wave on the elastic plate of the form (3.23) (see section 3.1) ibi-_(l--MV/2bY @OL,
= aolbol
e
,
f>O,
(2.28)
y>o,
where sol is a known complex constant, and b = {M2/(1 -M2)1’2}1’3.
(2.29)
This non-attenuating wave must obviously match to an outgoing wave (travelling on the plate but far from either edge) which satisfies the full plate and governing equations ((1.2) and (1.7)). With this wave taken to be is(x+a)-yy
4 wave
(2.30)
= e
then, to satisfy all conditions, y = (1 -M2)1’2{s -k/(1
+M)}“‘{S +/c/(1--M)}“‘,
(2.31)
and ~(s4-~4)-C+-MS/k)2=0.
(2.32)
But if (Y+CO, s can be expanded in powers of E to give, from expression replacing x by Z’, etc., 4 wave= e
-ibf’-(1-M2)1/2by[l
+E((2
(2.30) after
_M2Jift
-2(1-M2)1’2(2M2-l)~}/{3M(1-it42)}+O(E2)], b = {M’/(l -IV~)“~}“~.
(2.33)
Similarly for a wave in the opposite direction L,, y1= (1-M2)“*{s~+k/(l
=e
+A4)}“2{sl--k/(l
-is,(x+n)-y,y
(2.34)
I
-M)}1’2, (s’: -~4)yl--Lr(l+Msr/k)2=
0,
(2.35, 2.36) which becomes &.,., = e
-iM-(l-M*)l/zbql
_ E(t2
-2(1 -A42)“2(2M2-
_M21ti
l)Y}/{3M(l -M2)}+ 0(E2)].
(2.37)
Thus, for consistent matching, part of the outgoing wave term in oo2 must add to Qolw given by expression (2.28) to merge into the wave in equation (2.33). This will be shown when solving the inner problems in section 3.
218
I. D. ABRAHAMS
2.3. MATCHING &,
@I
The potentials c5r, & have been introduced into the problem to allow for possible plate resonances (by changing the order in E of A(E) and B(E)). The forcing for the cS1,@r matching can therefore be taken as an un-attenuated plate wave travelling into the left-hand edge, having initially been launched from the right edge. The potential cP1 can be expanded as @r-&r+&&+.
(2.38)
* *,
where Qirr satisfies the boundary conditions in expressions (2.20)-(2.22) with forcing by an incoming wave along the plate of the form @lri = exp { - ibZ - (1 -JM*)~‘*~~J}. The form of @r1 as r + 00 can now be written as -l/2 ?k
-Allr
-3/2 sin $0
.
sin $3 +Bllr
+ ..
.,
(2.39)
as there is no forcing from the outer scattered potential. Thus, upon using the expansion (2.40)
“*~11+E3’*~,*+E5’*~,J,
41-c
where c5r1 and d12 satisfy expressions (2.9)-(2.11),
it is clear that
411 -~rr(R-“*sin$3+a~~~R~‘~sin$I+~
(2.41)
- e),
R+O
and so aI1 = A 11.Continued matching also gives a*-
I’QO
Atlall2r
"*
(2.42)
sin $I,
where G12 satisfies the problem given by ~12~~(l-M2)+~12~~=-2iM~11,a129
=
0,
3s0,
(2.43)
forallZ,f>O,
(2.44)
y=o,
Glziifz -M2@r2,- = -i(@llY,,+M2@rr)/M,
f>O,
y=o.
(2.45)
Note that Q12 is forced by (i) r1’2 behaviour at infinity, (ii) terms in both expressions (2.43) and (2.45), and (iii) aI must also have an incoming wave of the form ib.f-(l-M*)‘&J @l*i
=
e
3M(l -M2)
{2(1 -M2)“2(2M2-
l)y -i(2-M*)i}
(2.46)
to match with the incoming wave in the &I problem (equation (2.37)). As a final point the leading order outer potential for 4r can be rewritten as (2.47)
411 =A1141/2~, where cSl12=is given in equation (2.27) 2.4.
MATCHING
42,
@2
matching procedure for cS2,ep, is analogous to that with the previous potentials (+r, @r) where G2 is the right-hand inner region potential and the forcing is by a right travelling wave along the plate. Again upon expanding the potentials as The
@*-@*r+&&+’
* *,
(2.48)
SOUND
SCATTERING
BY
A PLATE
IN
219
A FLOW
it is found that &I satisfies the boundary value problem defined in expressions (2.20)(2.22) when 2 is replaced by Z’ of equation (2.6) with an incident plate wave providing the forcing. Thus it is clear that @21=
@llW,
(2.49)
9,
and so 22 -AllF-1’2 f2 = (I -&-lx’2
sin $$+sB~~T-~‘~ sin %,
+ y2,
(2.50) (2.51)
e= tan-l {y(l -M2)/Z}.
Matching to leading order gives 42-421
1’2&1h2~
=E
(2.52)
+ O(E~‘~),
where 41’2R is an eigensolution of the problem defined by equations (2.9)-(2.11) has the form 41’2R-R”2 sin ~8+~i~‘~R’/~ sin $x?,
and
R+O
4Rli;R
+
- b1,2R I’2 sin $6,
R2 = (1 -M2)-‘(x
- ka)2+ Y2.
(2.53)
The constants u~‘~, 6;‘2 are found by solving the outer problems for 41’2~ #1/2R, and are again assumed known in principle. The boundary value problem for 422, analogous with ti12, is found to be @22fSPS(1 -M2) + @22gy= 2iM@21x, for all ,?, jj >O,
(2.54) (2.55)
@22yx’ ___ _ -h/f x’X
2
@22x’ ::yi(;09.. Zlyx’x’ _ :‘;O;
2
i’>O,
y=o.
(2.56)
It can also be shown that @22must behave as
$k -A11~1’2F”2
(2.57)
sin it?,
and has an incident plate wave providing the forcing, of the form (from equation (2.33)) -ibf’-(l-M2)ll*by
@22i =
e3M( 1 -M2)
((2 -M2)if’-
2(1 -M2)1’2(2M2-
l)jj}.
(2.58)
As a final observation note that the potential d2, as R + 0, will behave as 42
-E
1/2A ll
R+O
~$1’2~
=
E
1’2A11b1,2R1’2 sin $0 +O(E~‘~)
(2.59)
R--O
which will therefore match with part of the inner left-hand potential, @2L,say, giving @2~
-
(~Ah,2/bod@o~
+ W2).
(2.60)
This can be repeated at the right-hand edge to match the 41 potential. 2.5.
ASYMPTOTIC
SOLUTION
Matching has been accomplished to leading order and the solution may be written as 4 total=+ol +E(Aolbol~l,2=+Aol~ol~l,2~)+O(&~) +A(E){E~,~A~~~~,zL+O(E~‘~)}+B(E){E~’~A~~~~,~R
+O(E~‘~)},
(2.61)
220
I.
D. ABRAHAMS
with inner approximations ~,~,~=E~‘~~~~+O(E~‘~)+A(E){~~~+E~~Z+O(E~)}+B(E){EA~~~~,~~~~/~~~+O(E*)}, (2.62) @right
=
+o(~~‘~)+A(~)(~A11b;,z~ollbol
& 1’2@O~60~/60~
+O(E*)}
+B(E){~2~+&~22+O(&2)}.
(2.63)
Near the plate, the total potential is the sum of the travelling waves on the plate and the potential in equation (2.61), i.e., (2.64)
d,,,,l+A(E)exp{-isl(x+a)-yly}+B(&)exp{is(x-a)-yy},
where y, s, etc., satisfy equations (2.32) and (2.36). It now remains to solve the inner problems &, @ii, etc., so that the coefficients A o1,A 11,etc., can be found. The functions A(E) and B(E) are determined by equating the incoming and outgoing waves at the left-hand edge with those at the right-hand edge and this will give the condition for plate _ resonance (362)-(3.65).
3. THE INNER
3.1.
PROBLEM
THE 011 POTENTIAL
A solution @ii for the inner problem can be found by use of the Wiener-Hopf technique. The potential @ii satisfies the boundary conditions (6XX(1-M2)+~yy=0
forallx,y>O,
xso,
&=O,
where for convenience plate wave
y=o, x>o,
4 xx*y- M2& = 0,
(3.1) (3.2)
y=o,
(3.3)
i, jj is written as x, y and forcing is supplied by an incoming b3 =M*/(l
& =exp{-i6x-(1-M2)“26y},
-M’)l’*.
(3.4)
The usual radiation condition allowing only outgoing waves at infinity is also imposed. Note that the plate equation (3.3) can be integrated to give c$,..,,- M*qS= constant,
x>o
y=o,
(3.5)
but from matching considerations it can be shown that this constant must be zero. Upon subtracting off the incident wave by introducing a new potential (I, where I&- C$- di, the problem becomes I,L(~-M~)+(L,,,=O forallx,y>O, (3.6) (jlY= (1 -M2)“26
ewibx,
&‘xxy - M24 = 0,
x S 0, x >o,
y = 0,
(3.7)
y =o.
(3.8)
The half range Fourier transforms are defined as P(x, y) = ??+(x, y ) + q-(x, y), where 0 P+(s,
y ) =
lam
eisxW,
Y1 dx,
*-(s,
y 1 = _m e%(x, I
Y) dx,
(3.9,3.10)
P+ being analytic in an upper region of the complex s plane and similarly Y_ analytic in the lower half plane. Both P, and ly_ are assumed to have algebraic growth as IsI + Co
SOUND
SCATTERING
BY A PLATE
221
IN A FLOW
in their respective half planes. Transforming the governing equation (3.6) gives the result (3.11)
P(s,y)=A(s)exp{-y[~[(l-M*)“~}, which allows only outgoing wave solutions. Note that
Is1= F; {(s + iq)“2(s -
(3.12)
iq)“*},
where q is a fictitious dissipation term included for mathematical
convenience,
and
Re (s)>O Re
l’l=I -:, (s)
(3.13)
Y+ = !P+(s, 0), etc., will be used and the trans-
= a{P_(s, y)}/J13yl~=~= -(l -M2)“*bi/(s
-6),
N(S)-S2!fY
-M2P+=0, (3.14,3.15)
where N(s) = -&,(O, 0) +is&,(O, 0) and is regular over all s. From equation (3.13) it is known that !P++‘P_=A(s)=-(‘Py:
which, on rearranging functional equation
(3.16)
and using expressions (3.14) and (3.15), gives the Wiener-Hopf
{N(s)/M2jK_(s)
+ W-K_(s) -{(l -M’)i’*bi/(s
= -!?:K+(s)+{(l The Wiener-Hopf
+P~))/{~s~(1-M*)“*},
-M2)1’2bi/(s
-b)}{s*K-(s)
+K+(b)}
-b)}{K+(s)-K+(b)}.
(3.17)
kernel K(s) is given by
K(s) = ((1 -M2)-1’2/ISI}-S2/M2
=K+(s)/K_(s),
K+(s) = l/K-(-s),
(3.18)
where K+(S), K_(s) are regular and non-zero in their respective upper and lower half s-planes. The left-hand side of equation (3.17) is therefore analytic in the lower half plane (the pole at s = b lying just above the line joining the two half planes), and similarly the right-hand side is regular in the region Im (s) > 0. Note that a regular function J(s), say, can equal both sides of equation (3.17) for values of s lying on the line of common analyticity. By analytic continuation J(S) can be extended throughout the whole s plane and the form of J(s) is found by examining the behaviour of equation (3.17) as IsI + co. By using the fact (from equation (A9) of Appendix A) that K+(s)-O(s)
ass+oo,
etc.,
(3.19)
together with the edge condition t&(0,0) = 6(1 -M*)l’*, J(s) behaves as J(S) + 0 as IsI+OO and so by Liouville’s theorem J(s)=O. Thus $1 is now found from equation (3.17), and equation (3.11) can be written as ~(~,y)=~biK+(b)lJsI(s-b)K+(s)~exp~-Isl(l-~~)~’~y},
(3.20)
or finally the solution is K+(b)
exp
{-id-Is1(1-~2)1’2~)ds
K+(s)lsl(s -6)
+,-ibi-(l-M2)1/Zb$!
(3.21)
222
I. D. ABRAHAMS
integral can be estimated for large r hence giving the coefficients AI1 and BII of expression (2.39) and the pole in the integrand at s = -b (in K+(s)) gives the outgoing plate wave. After much algebra the outgoing wave (from equation (AM)) is determined as @rlw = e5i7r/4 eibf-gb(l-M2)1/*, (3.22)
The
The above procedure
could be repeated for aoI which would determine the coefficients (2.25), and the coefficient (~0~multiplying the outgoing wave: i.e., the outgoing plate wave term in QzoIis (from equation (2.28))
AoI and B,-,r of expression
GOlo
=
crolbol exp {ibf - (1 -M*)“*bjf},
(3.23)
where (yo1 is assumed known in this analysis, 3.2. THE aL2 POTENTIAL The boundary value problem for G12 is written in equations (2.43)-(2.45) and forcing is supplied by an incoming wave along the plate, given in equation (2.46). As in section 3.1 a new potential, 4, can be defined which is given (from equation (2.46)) by (3.24)
4 =@l*-@**i
and, after substitution into the relevant equations, 4 is found to satisfy the problem &, (1 -M*) + c%,,+ 2iM& = 0 &, = -epib”(Ax + b), -M*cL + (ilM)($,,,, 4 YXXX
for all x, y > 0,
x =S0,
(3.25)
y = 0,
x>o,
+M*tj) = 0,
(3.26) y=o.
(3.27)
The potential $ is the inverse Fourier transform of equation (3.11), and A = (2 -Mz)ib/3M(l
Transforming
B = 2(2M2-
-M*)l’*,
1)/3M(l
-M2)1’2.
(3.28)
the governing equation gives
@(s, y) =D(s) e-‘S’y’1--MZ”‘2+yMibK+(b) e-‘s’y(1-M2)“2/s(1 -M*)l’*(s
- b)K+(s),
(3.29)
where @(s, y) is the Fourier transform of C&(X,y), and @+, @I, etc., are half range transforms as previously defined in equations (3.9) and (3.10) and can be shown to have the same regions of analyticity as the P+ and P_ functions. The problem now reduces to determining D(s) for the given boundary conditions. The plate equation transforms to give P(s)+is3@: P(s)={-c%:,, the superscript
+iM*s@++(i/M)(-s*PL +isf$$+s*cSy
+M*P+)=O,
(3.31)
+M290-(iIM)~~,-(slM)9~),
denoting x = y = 0, and 4: = -B
and 4; = b(1 -M*)“*.
@J: +@F =-~s~(l-~2)“2~(s)-{M/S(1-M2)}(ly: @++@_=D(s)
(3.30)
+YL),
The identities (3.32) (3.33)
can be used, together with the left-hand boundary condition, to give the Wiener-Hopf
SOUND
SCATTERING
BY A PLATE
223
IN A FLOW
equation. After much algebra this equation is found to be P(s)K_(s)+is’K-(s)(
A-&]
-M2s@-K-(s)
+iM2 ( A[sK+(b)+b(s-b)K(b)l(S_b)2
f++(b)]
M3bK+(b) 1+RJs) R+(b) -(s _b)(l-My I T+MZ I +2(1-Mz)“26 M(s -6)
[S2K_(S) -K+(b){R-(s)
+R+(b)H
-iM’(&&K+(s)-X+(b)-b(s-b)K:(Wl
= isM2K+(s)dJ>
+26(1 -M2)“2K+(b)
[R+(s) -R+(b)19
M(s-b)
(3.34)
where K+(s) and K_(s) are defined in equations (3.18), K’(b) = dK(s)/dsI,+,,
R+(s) +R-(s)
= S2/K(S),
(3.35)
R+(s) and R_(s) being regular in the upper and lower half planes, respectively. The left-hand side of equation (3.34) is regular in the lower half plane, and the right-hand side is regular in the upper half plane. After allowing both sides of equation (3.34) to be equal to a function regular over all s, and examining the behaviour of each side as 1sI+ co, one finds from the extended form of Liouville’s theorem that this function must be a constant, C, say. Note that C is determined by matching to the outer region and therefore from equation (2.42) must be proportional to A11a1,2. It is now assumed to be a known constant and so the function D(s) can be written as AK+(b) AbK: (b) Bi6K+(6) (s-b)2K+(s)+(s-b)sK+(s)-(s-b)sK+(s)
D(s) =
ib(2 -M2)K+(b> -M3(1 -M2)*‘2(s -b)sK+(s)
rR+(s)-R+(b)l+M2;g (J + (3.36)
+*
The solution can finally be written as jMibK+(b) s(s -b)(l -M2)“2K+(s)
D(s)+
e-~s)~(l-M*)l/*-isf
ds +@rzi.
(3.37)
The outgoing wave can again be found by picking up the pole at s = -6, and, from equation (BlO) of Appendix B, after considerable algebra, @120
=
i=/4 (2 -M’)Il + 2( 1 -M2)“2(2M2 - l)F 3M(l -M2) 3M( 1 -M2) I
I [ e
-
cei”‘86(2/3)“2 M3
I
e
ibz-b(l-M2)“2g
(3.38)
224
I. D. ABRAHAMS
The analogous outgoing wave @220 can be written down after noting that the forcing in equations (2.54), (2.53) and (2.58) is the negative of that in the & problem. Thus it is found that @22@ =
sin/4 (2 - M2)if ’ +2(1 -M2)“2(2M23M(l -M2) 3M(l -M2)
l)jj
( [ e
- Dei”‘8b(2/3)“2 M3
1
e
I
ibf’-b(l-M2)11*_
(3.39)
YI
where D is found from matching with the outer field and can be considered known constant.
to be a
3.3. DETERMINING A(E), B(E) It is clear that the outgoing wave from the left-hand edge must be equal to the incoming wave into the right-hand region as the waves do not decay with distance along the plate. This must also be true for waves emanating from the right-hand edge and travelling into the left-hand edge. The outgoing wave from the left-hand edge is the sum of all waves generated by the potentials in equation (2.62). Thus, upon using expressions (3.22), (3.23), (3.38) and (2.60), the total outgoing wave is @lcfto
=
[E 1’2ao~bo~+O(~3’2)+B(~)(~A~~b~,2~~~+O(~2)} +A(~)(eSi+r/4{1-~(2-~2i~+2(1-M2)1’2(2M2-l)~)/(3M(1-M2))} -&f-2CeiW’86(2/3)1’2+O(~2)]exp{ib~-6(1-M2)”2~},
(3.40)
or, in outer co-ordinates, (9 lefto
--[
E 1’2~01601+O(~3’2)+B(~){~AIlb~,2~~1+O(~2)}
+A(E)(eSin/4_
E~-2~
eir/8b(2/3)l/2+
0(E2))] eir(x+a)-vy,
(3.41)
where y and s satisfy equations (2.31) and (2.32). This wave must be equal to the incident wave into the right edge which, from equation (2.64), gives @lefto =B(E)e
is(x-a)-vy
(3.42)
.
Similarly the left going wave condition gives Q,right- = [&1’2cy,,&,~+O(E~‘~)+A(E){EA*~~;,~(Y~~ +~(~){~5id4_ =A(&)
e
._M-2D
eid8b (2/3)1/2 + 0(E2)}]
e-is,(x-a)-vly
-is,(x+aky,y
with y1 and s1 satisfying equations equations finally gives the results A(E)
+O(E~)}
= E 1’2ao~{b~~esirr’4+&
(3.43) (2.35) and (2.36). Solving the above simultaneous
exp (-2ika[b/~
-(2-M2)/(3M(1-M2))])+O(e)}/E(~), (3.44)
B(E) = ~“~a~~{b~~ exp (-2ika[b/E + iDleSir/ + O(E))/&),
+(2-M2)/(3M(1
-M’))]) (3.45)
SOUND
SCATTERING
E(E) = exp (-4iakb/&)
BY A PLATE
225
IN A FLOW
-exp (5ir/2)
+e{[2iak2/M2(1-M2)*][14M4-49M2+38]exp(-4iukb/e) -Am01
e
-2iakb’e[bti2 exp (- 2ika (2 -k4*)/(3M(l
+6;,* exp (2ika(2 -M*)/(3M(l
-M*)))
-M*)))]
+(D +C)(b/A4*)(2/3)“*}+0(~*).
(3.46)
A resonance is defined to occur when A(E) and B(E) change from O(E “*) to O(E -?, and this condition occurs when e-4iakb/e
=
e5in/*
(3.47)
9
or
(3.48)
2pU2u3/(1 -M2)1’2D = {(3~/8)+ (n~r/2)}~, II = 0, 1,2, . . . .
This resonance condition can be altered by changing the plate edge conditions. instance, if the plate were simply supported at the edges, i.e., I)=%X=
x=*u,
0,
y=o,
For (3.49)
then repeating the previous analysis would yield the resonance condition 2pU2u3/(1 -M*)“*D
n = 0. 1,2,. . . .
={(~/24)+(nr/2)}~,
(3.50)
Note that if fluid were on only one side of the plate (with a vacuum on the other) then the resonance condition (3.50) would become n=0,1,2
pU2u3/(1 -M2)1’2D ={(*/24)+(n~/2)}~,
)...
.
(3.51)
4. LOW MACH NUMBER FLOW
The resonance condition obtained in section 2 is valid for a Mach number of order unity, and does not reduce to the zero flow resonance condition [3, Abrahams 19811 CY
l/5
a
-1 -*?T+$m,
n=0,1,2
,...,
(4.1)
as the Mach number is reduced. The reason for this is breakdown of the leading order expansion of the equations (2.15)-(2.19) written in expressions (2.20)-(2.22). This is therefore a singular limit suggesting that an examination be made of the problem when the Mach number is of the order of the other small parameter in the problem. As suggested in the zero flow problem the small parameter is taken as E =
k/c?,
(4.2)
and so the Mach number is now defined to be where m = O(1).
M = me,
The plate length, a, is kept in the same range as previously: co-ordinates can now be written as Z=P(u+x),
y = (y“SY,
(4.3) i.e., ku >>E. The inner
f’ = cPS(u -x),
for the left and right inner regions, respectively, and the outer co-ordinates acoustic wavelength, so that X=kx,
Y = ky.
(4.4) scale on the (4.5)
226
I.
D. ABRAHAMS
The leading order outer problem is found after resealing on the outer co-ordinates, is defined by ~&+c&,+~#J=O 4Y =O,
forallx,
Y,
C$I”=o,
IXI>ka,
and (4.6)
IXl
(4.7,4.8)
Similarly the inner problems can be written as @,-,-+@,-,-+~~(@++irn@,-m2@,-,-)=0 @,=O,
forallf,y,
*=*,-=0,
f
~(a~/f!IZ~+~~e~)ji +(i@ -m0,)/2clT @? = --Ec(ijj -m?j,-),
(4.9)
i=O,
= 0,
(4.10,4.11)
y=o,
y=o,
x>o ,
x>o.
(4.12) (4.13)
The matching scheme for this low Mach number limit is no different from that described in section 2, and will therefore not be analyzed. To calculate the condition for resonance it is only necessary to obtain the reflection coefficient of a plate wave travelling into an edge. Thus the leading order inner problem is found to be (after noting that Qi is odd in Y) @,,,
@oy=O,
x
+ aoee =
f=O,
for all Z,y > 0,
0
jjo=?jo~=o,
c?~+~o/c~X~ + (l/c)(i@% - m@&) = 0, @Ok= -c (iii0 - mfjoi), Tj =no+&nl+.
ff=o, x>o,
x>o,
(4.14) jj=o, y=o,
(4.16)
y=o,
(4.17)
@=,@,+2@,+~~-.
. *)
(4.15)
(4.18,4.19)
Forcing is due to a travelling plate wave into the inner region and will have the form e where d satisfies the plate equation region. Thus d must be a solution of
-idf-dy
(4.20)
,
and kinematic
boundary
condition
ds=(1+md)2. The inner problem which yields
(4.14)-(4.17)
in the inner (4.21)
can be solved by using the Wiener-Hopf
technique
(4.22) ’
(4.23)
K(s)=K+(~)/K_(s)=(l+sm)~/lsI-s4,
f being a constant found by matching or by edge conditions. The non-attenuating outgoing plate wave can be found in the limit m + 0, by using the results d=l+O(m),
(4.24)
f=O+O(m);
thus @owave =
e
3im/4 eif-V+~(mj.
(4.25)
Equating the waves at either edge of the plate gives the resonance condition r/=Ll= $?r + $l?r, CY
n=0,1,2
,...,
(4.26)
SOUND
SCATTERING
BY A PLATE
IN A FLOW
227
which is identical to that in the zero flow problem. Alternatively when the factor m in equation (4.27) becomes very large it is found that, from equation (4.21), d = m2’3,
K(s)-(m2/lsI-s2)s2+U(m-1).
(4.27)
The constant, f, is found to be (4.28)
f = -X+(d),
which gives the required behaviour at infinity, and so the outgoing wave is found, by picking up the pole at s = -d, to be otrn-l). 0 O,.“, = eSiw/4 eidf-dT+ ’ (4.29) The resonance condition therefore becomes da”‘a
12= 0, 1,2,. . . )
= (37/g) + (nr/2),
(4.30)
or, upon using equation (4.3) to give d = (M/E)~‘~, 2pU2a3/0 This M + It flow
= {(3~/8) + (n7r/2)}3,
n=0,1,2
,....
(4.31)
result must be equivalent to the resonance condition found in section 3 in the limit 0, and by expanding equation (3.48) in powers of M this is found to be correct. has now been shown that the potential of the problem with small Mach number smoothly matches the zero flow solution to that of the M = O(1) problem. 5. DISCUSSION
It has been shown that in the heavy loading limit the outer potential behaves to leading order as if the plate were absent. Near a resonance, however, eigensolutions of the outer boundary value problem with singularities at the plate edges also become present at this leading order. Close to the plate it is found that near resonance the leading order terms are standing plate waves (of order E-1’2) and these have a wavelength of the order of the size of the inner region. These results are similar in form to those in the zero flow problem except that for M = O(1) the wavelength of the plate waves is given, from equation (2.5) by, I= (k2/ap3
= (D/pc2y3.
(5.1)
This result is interesting because it shows that the wavelength (or frequency) of the un-attenuated waves on the flexible plate is independent of the frequency of the acoustic source supplying the forcing. Thus for any frequency of forcing it can be expected that the frequency of plate waves will be a constant dependent only on D, p, cand U, subject of course to the heavy loading approximation E =klc
1.
(5.2)
As was discussed briefly in the introduction, it was hoped that comparison could be made between this analysis and flutter analysis by using modal expansions. Much work has been performed on calculating the onset of flutter of finite or infinite elastic plates [4, Weaver and Unny 1970; 7, Dowel1 1966; 8, Bohon and Dixon 19641. The usual method for calculating instability is to replace the unknown plate deflection by a modal expansion (where each term satisfies the plate edge conditions) and the problem is solved by using a Galerkin approximation. The validity of Galerkin’s method is difficult to justify in a rigorous sense and it is therefore helpful to solve the problem by an alternative technique (in this case by matched asymptotic expansions) to compare the predictions
228
I.
D. ABRAHAMS
made. The two-dimensional problem of flow of a compressible fluid over one side of a simply supported elastic plate was analyzed by Ellen [5] (again using a truncated modal expansion) and his results predicted divergence instability (when the complex frequency crosses the real axis at w = 0) when pU’a~/D(l
-&f2)“2 = r5n4/2{n7r Si (n7r) - 1+ (- l)“},
aE = 2a,
Si(z)=
I0
’sin x dx ___ x
n=1,2,...,
subject to the restriction kaE c 1. ’
(5.3) (5.4)
In the analysis presented in this paper a time harmonic solution was assumed. If, instead, a Fourier transform in time had been taken, i.e., @p(x_, o) = jrn 4(x_, t) eiwt dt, --oo where4(x,t)=O
(5.5)
fort
al
@(x_,w) e-‘“’ do,
(5.6)
where the poles of @(x_,w) must lie below the real axis. Thus to satisfy causality the result in equation (2.61) must have poles (of complex o) lying in the lower half plane. This condition leads to the result (from equation (3.63)) that the frequency vanishes (i.e., a divergence instability occurs) at a plate resonance. Therefore divergence instability is predicted here for a simply supported plate with flow over one side when (from equation (3.51)) pU2a~/D(1
-M2)1’2 ={(~/12)+(nr)}~,
n=0,1,2
,...)
(5.7)
with the conditions kl c 1,
l/a
(5.8)
Note that the latter constraint implies that equation (5.7) is strictly valid only for large n. It can be shown that the values of the dimensionless parameter pU2a~/[D(1 -M2)1’2] predicted by both methods are within 2% of each other for n 2 1 and both tend to the same value (n7r3) as n + 00. The n = 0 term in equation (5.7) violates the latter constraint in expressions (5.8) and does not satisfy the condition that kaE <<1 in equations (5.4) and can therefore be rejected; it is reassuring, however, to find that the results are in very good agreement, even for low n (when neither approximation is strictly valid). As a final point it can be noted that an advantage of the present method over that of Ellen is the ability to predict resonance for any plate edge condition whereas the modal expansion technique has been applied only to simply supported panels. ACKNOWLEDGMENTS
The author would like to thank Dr F. G. Leppington for much advice and assistance, and the S.E.R.C. and A.U.W.E. Portland for the financial support of a C.A.S.E. studentship. REFERENCES
1976 Quarterly Journal of Mechanics and Applied Mathematics Scattering of sound by finite membranes and plates near resonance.
1. F. G. LEPPINGTON 527-546,
2. D. C. HANDSCOMB 183-189. Vibrations
1977 Journal
of the Institute of Mathematics
of a submerged window.
and its Applications
29, 20,
229
SOUND SCATTERING BY A PLATE IN A FLOW
3. I. D. ABRAHAMS 1981 Proceedings of the Royal Society London A378, 89-117. Scattering of sound by a heavily loaded finite elastic plate. 4. D. S. WEAVER and T. E. UNNY 1970 American Society of Mechanical Engineers, Journal of Applied Mechanics 37, 823-827. The hydroelastic stability of a flat plate. 5. C. H. ELLEN 1973 American Society of Mechanical Engineers, Journal of Applied Mechanics 40, 68-72. The stability of simply supported rectangular surfaces in uniform subsonic flow. 6. 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. 7. E. H. DOWELL 1966 American Institute of Aeronautics and Astronautics Journal 4, 1370-1377. Flutter of infinitely long plates and shells-part 1: Plate. Some recent developments 8. H. L. BOHON and S. C. DIXON 1964 JournalofAircraft 1,280-288. in flutter of flat panels.
APPENDIX From equation
A: FACTORIZATION
OF THE WIENER-HOPF
(3.18) K(s) = {Isj(l -M2)1’2}-1
which can be rewritten
(AlI
-s2/M2,
as K_(s) = MSY2 ei?r’4Q-(s),
K+(s) = (eiT’4/MS:‘2) Q+(s), where Si”
KERNEL
(A21
= lim,,e (s +iq)1’2, etc., and
c?+(s)/Q-(s)
Q+(s) = Q-(-s).
= V--s21sl),
(A3)
and Q- are regular and non-zero in the same regions as K, and K-, respectively, and it can be shown that
Q,
In [Q+(s)] = (2ri)-r
lim
where 1,s > I,& Note that the integration
(A4)
path in equation (A4) passes below the point
5 = b and so
ln[Q+ts)l=~,
say,
=& Iom In (b3-13) tc2_s2)
After much algebra, with careful interpretation it is found that
1
(A3
of the branch of the logarithm chosen,
1
___-1
(s+bc,)+3(s+bl)
dS.
3(s+b_l)
(A61
I’
where 6, = b e2nni’3 . To estimate the order of K+(s), as s + co, the expansion of expression
(A6) in powers of l/s is dI/ds -(3+/s)+O(l/s*),
s+oO,
(A7)
which, after integrating, gives Q+(s)=ZJs3’2+O(s”2),
s+Q),
W9
where P is a constant, and so from equation (A2)
K+(s) = O(s)
when s + ~0.
(A9)
230
I.
D. ABRAHAMS
It is now required to find K+(6) so that the coefficient of the outgoing wave in equation (3.21) can be found. First, by integrating by parts, equation (A5) can be rewritten as
and so I(0) = 37ri In 6,
(All)
as the integral in equation (AlO) is zero when s = 0. The expression in equation (A6) can now be integrated and, upon letting s = 6, the equation simplifies to
I(6) -I(O) 2ri
(A12)
=ln(~}=-&[om&dv+t$)ln6-~.
The integral can be shown to have the value rr*/8 and so substituting equation (A12) into equation (A2) finally gives K+(6) = 6 e’““&/M. The pole contribution can be written as =
6
lim
(A13)
at s = -6 in equation (3.21) gives the outgoing wave for @ir, and
K+(6) e-isi-‘s’8(1--M2)“*= --K: (6) 26 K+(s)lsl(s -6)
ibX-bJ(l-Mz)l/2 .
(A14)
The coefficient in front of the exponential in equation (A14) is found -Kt (6)M*/66*, and upon using equation (A13) the solution becomes
to be
@ll”
s+-b
5irr/4 @I~,
=
e
e
e
ibl-bf(l-.W*)l/*
APPENDIX B: OUTGOING WAVE Q&, The expression for or2 is written in equation (3.37) and the outgoing wave is again found by closing the integral in the lower half plane thus picking up the pole at s = -6. Upon noting that the term in D(s), from equation (3.36), containing R+(s) has a pole of order two at s = -6, the residue of the pole can be written as residue (at s = -6) 1 = 2~6*(1 -M*)l’* iC -y+ M
AK+(b)+AKi(b) 46 2
MiK+(6) 2(1-M*)r/*+
_
Bi6K+(6)+iK+(6)(:!-M2)R+(6) 26 2M3(1 -M*)l’*
yMiK+(6)6 2 e R+(~)(~ +6)* eiSf-lSl~(l--M2)“2 sK+(s)lsl(s -6)
’
(Bl)
By using the identities R+(s) =s*/K(s)-R_(s),
R-(-b)=R+(b),
(I329B3) (B4)
SOUND
SCATTERING
BY
A PLATE
IN A FLOW
231
the last term in equation (Bl) can be written as it2
_MZJK+(~J
eibf-bg(l-M2)“2
-
(jj(l-A42)“2
47rM3( 1 -iW)
-Z)K+(b)-K:(b)
It is easily shown that !;mb($(zr}=O,
(B6-B8)
and these values are substituted into equations (Bl) and (B6). By using equations (3.28), (B5) and (Bl) the outgoing wave of Cpi2, after considerable cancellation, is determined: @12”
=
=
-2ri{residue
at s = -b}
(2kf2-l)~K: (b)b+i(2-M2)K:
(b)bz cK+(b) 18M(1-M2)“2 -x
9M
ibf_b~(l_MZ)1/2 e (B9)
or, upon using equation (A13),
C -
ei+b(2/3)l'*
M3
I
e
ibf-bg(l_&f2)1/2
(BlO)