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Acoustic scattering by two parallel slightly staggered rigid plates
WAVE MOTION 12 (1990) 281-297 NORTH-HOLLAND
ACOUSTIC SCATrERING RIGID PLATES
281
BY T W O
PARALLEL
SLIGHTLY
STAGGERED
I.D. A B R A H A M S Department of Mathematics and Statistics, The University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK
G.R. W I C K H A M Department of Mathematics, The University of Manchester, Manchester M13 9PL, UK
Received 27 April 1989, Revised 10 July 1989
In a recent paper Abrahams and Wickham [1] showed how to exactly solve the diffraction problem for two parallel semi-infinite plates which are arranged to form a duct. The edges of the plates are misaligned so that the geometry is asymmetric with respect to the central line of the duct. To evaluate the field it is first necessary to solve the integral equations derived in [1], and this can be accomplished analytically when the 'stagger' of the plates is small. In this paper we derive this analytic solution and use it to determine several terms in the asymptotic expansion of the velocity potential. The solution found by the method of matched asymptotic expansions is shown to be identical to this rigorous result.
1. Introduction and boundary value problem This p a p e r is concerned with the scattering of plane sound waves obliquely incident onto two semi-infinite rigid parallel plates. The plate edges are 'staggered' so that an imaginary line joining the two plate edges is not perpendicular to the plane of either plate. The motion is assumed to be purely two-dimensional. This boundary value problem was examined recently in a p a p e r by the authors, Abrahams and Wickham [1] and by Fourier transformation was reposed as a matrix W i e n e r - H o p f functional equation. The 'unstaggered' geometry has attracted a good deal of attention since the work of Heins [2, 3]; and an explicit solution of the matrix W i e n e r - H o p f equation is possible in this case because the system may be reduced to two scalar equations. However, when the plates are staggered, the matrix kernel contains exponential phase factors which, on decomposing in the manner suggested by Heins, give rise to exponentially growing terms. This inhibits the application of Liouville's theorem and renders the equation unusable without modification. One method to overcome this difficulty is to premultiply both sides of the equation by an entire matrix, which must have suitable exponential behaviour at infinity, and this was performed on the specific problem in [1] and on a general class of matrix W i e n e r - H o p f equations in Abrahams and Wickham [4]. The elements of this matrix are all shown to depend on a pair of coupled Fredholm integral equations of the second kind (see equations (2.8), (2.9)) and therefore any numerical evaluation of the scattered sound field (or transmitted duct modes) will necessitate the solution of these equations. The object of the present investigation is to obtain an asymptotic solution of the exact analysis presented in [1], when a physical parameter is taken to be small, and then to compare the field obtained from this solution with that derived by the method o f matched asymptotic expansions. This will not only prove useful in helping to understand the physical relevance of the exact solution, but will enable a check on 0165-8641/90/$03.50 © 1990--Elsevier Science Publishers B.V. (North-Holland)
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LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
the numerical solution of the integral equations (see Abrahams and Wickham [5]). The asymptotic limit chosen in this study is the situation when the 'stagger' is very small, i.e. the ratio of the difference in plate lengths to the plate spacing tends to zero. It will further be assumed that the ratio of the incident acoustic wavelength to the plate spacing is of order unity. The organization of this paper will be as follows; In the remaining part of Section 1 the boundary value problem is stated. In Section 2 an asymptotic expansion for the field is derived from the exact analysis of [1]. The method of matched asymptotic expansions is then employed in Sections 3, 4 and 5. In Section 3 the inner and outer variables are introduced together with the matching principle, and Section 4 contains the derivation of the inner solution (i.e. the asymptotic form of the sound field near to the edge of the top plate). The outer solution is found in Section 5 up to several unknown constant factors, and in Section 6 these constants are determined by matching with the inner solution. In Section 7 this solution is shown to exactly agree with that found in Section 2, and we also make some concluding remarks. The mathematical formulation of the physical problem is expressed in terms of a time-harmonic velocity potential Re[~b(x, y) e -i'~,]
(1.1)
where to is the angular frequency, (x, y) are cartesian coordinates and e -i~'' is henceforth suppressed for brevity. The potential, ~b(x, y), satisfies the reduced wave equation
(32/ax2+O2/ay2+k2) dp =0,
(x,y)~ ~,
(1.2)
and the boundary conditions
a4, - 0 ,
-a
(1.3)
3¢b O, y = - h ±, O < x < o o ,
(1.4)
Oy
y = O ±,
by
where ~ is the whole of the (x, y)-plane, cut along the two semi-infinite lines y = O, - a < x < oo, and y = - h , O< x <0% and the wavenumber
k = to/ c
(1.5)
with c the phase speed in ~. We wish to solve for the potential, ~bs(x,y), scattered by an incident plane wave, ~bi(x, y), i.e. 4~s= ~b- ~i,
(1.6)
4~i.= exp(ikx cos O+iky sin 0)
(1.7)
where 0 is the direction of propagation relative to the positive x-axis (see Figure 1). We require that 4~, represents outgoing waves as (x2+y2)~/2-*oo both inside and outside the duet region, and that 4~s is bounded in the neighbourhood of both plate edges.
2. Asymptotic solution from exact analysis The exact solution of the boundary value problem was determined [1] via the introduction of the entire matrix K* which contained the scalar functions N*, M*, S* and T*. To recap I
1 ~ ~ ( a , y) e -i~x da, ~bs(x, y) = 2--~ fd-oo
(2.1)
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283
Y
(-a,o)
(O,-h) Fig. 1. Physicalconfiguration. is the scattered field where a runs along the real line (if k has slight positive imaginary part), 1 • (a, y) = - ~ )
V(a, y) K-(a) ( K - ( - k cos 0))-' F ( a ) ,
'(e-iaa-yy(a), V(ot, y)
(2.2) y > O,
0),
( _ e _ i , ~ cosh y(a)(y+h) cosh y(a)y~ sinh y ( a ) h 'sinh y(a)h]"
0 > y > -h,
(0, --eY(~)(Y+h)),
-- h > y,
(2.3)
and F(a)=
- k s i n 0 ( e -'k . . . . 0~ k co--s"0--+a \ e-ikh sin 0]"
(2.4)
The square root function y ( a ) is defined as y ( a ) = (a 2 - k2) 1/2
(2.5)
which has the chosen branch with y ( 0 ) = - i k , and K-(a) is
\e-" K-(a)
-e-i"~L-(a)) K*(a)'
(2.6)
where K-(a), L-(a) are scalar factorizations which are analytic in ~ - defined by I m ( a ) < k and are written in the Appendix (A3, A11). Also
( M * ( a ) + e i"" S * ( a ) + 1~ K*(a)=\N*(a)-eia" T*(a)+l]'
(2.7)
where the entire elements M*(a) and N*(a) satisfy the integral equations
N*(,~)- M*(~)+~i I ~ M*(~) \/(--7~-~ N*(ot)+ M*(a)+ ..,o,
\K-(~)
\
•
d~"
1) e -'"(~-'~> ~_,~ =0,
~-ot
(2.8)
LD. Abrahams, G.R. Wickham / Acousticscatteringby plates
284 in which
1 r i,,, { L,(~') ) d~ °*(a)---2~r-'~J~e kK-(~----~ 1, ~ _ ~ ,
(2.10)
and Jr (L,) denotes an integration path from -oo to +oo which passes above (below) all the integrand's singularities. We also find that
S*(ot)=-eia" N * ( - a ) ,
T*(a)=eiaa M*(-ot),
(2.11)
and note that the inverse Fourier transform path passes above (below) the branch cut emanating from - k (k) to oo in the lower (upper) half plane and above the pole at a = - k cos 0. We now wish to obtain an asymptotic solution from this exact solution as ka-> 0 with kh fixed and order unity. In [1] it was shown that the system may be solved by iteration for each value of kh < ~r and a/h sufficiently small, e.g. for kh <2 then a/h must be less than 1.8 (see Theorem 1 and Figure 3 of [1]). The first iteration yields
N*(a) ~ M * ( a ) ~ ½0*(a).
(2.12)
in which we write O*(a) as
1 L.e'~ (L-(,/a)_l) \K-(~/a)
O*(a) = 2~r---i
d,
(2.13)
(~-aa)
where the contour has been deformed into ~ which passes from - ~ to -1 along the real line of ~, from - 1 to + 1 in a semi-circular arc in the lower half plane and then to +oo along the real line. The path is also indented below aa. It is now straightforward to show that as ka = e -~ O, the integrand may be replaced by
v(kh) kh~
L-(~/a)_l_ K-(~/a)
--e+
I~(kh) ~.2
e2+O(e3)
(2.14)
uniformly on ~, where v(kh) is given in (A21) and ~(kh) is some function of kh which will not actually be required to the order of the expansion we examine. Substituting this into (2.13) and closing the integrals in the upper half plane yields O*(a) =
v(kh) ( 1 - e i ~ ) 4 k21~(kh) (e i ~ - 1 - i a a ) + O ( e 3 ) , ha a2
where the first term is uniformly of O(e) for I m ( a ) ~ - l / a , uniformly O(e 2) and O(e 3) respectively. The next iteration gives
1 f
\ K---~)
1 f O*(~)(L-(~) - - -
2,n.i L T
\ KZ(-~-i
1
M ( a ) = - ~ i f,~ 0 " ( , ) ( L + ( , ) _ 4
If.
2~ri
and the second term and remainder are
d~"a O * ( ~ ) { L+(~") - 1 e-ia(g-~) ~--
N *( a ) ~- - 2ar-~l.J~ T
.
(2.15)
\K+(~ ")
O *_(4,_____(~) L- ( , ) \~'('~-
d~" + l o , ( a ) , 1 e -ia(¢-~) ~'1 ) e -'"('-~)
1'~
"
(2.16)
d~r ~-ot
d, +½0*(a),
] e-'"(¢-~) K- a
(2.17)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
285
in which we substitute from (2.15) and evaulate the integrals in the same way as for (2.13) to obtain the asymptotic expansions. These are
v(kh),._ei~. ) [u2(kh) - -
N*(ot) = 2-~--~t--ot(1
+ ~ 4 - ~ ' j 2 [2(1 - e
i,,.-
•
)+laot(l+ei~'a)]
-+k21z(kh---~)2a z (e i~a - 1 -iota)) M*(a) = v(kh)2ha(1-ei~'")+ [iav2(k4h2o h)[ t
(1-eia~)4
+ O(e3),
k21z(kh)2ot2(ei~a-1
(2.18)
-iota)} +O(e 3)
(2.19)
where the terms in parentheses are O(e 2) and the error is uniform for Im(ot)~>-1/a. S*(ot) and T*(a) are obtained by reflection in the origin using equations (2.11). Generally the integral operators in (2.8), (2.9) are O(e), and hence to obtain an asymptotic solution to O(e") one requires n iterations. The asymptotic solution written in (2.18), (2.19) may now be substituted into (2.7), (2.6) to form K*(ot) and then K-(a). These are then employed in 4~(a, y), equation (2.2), to give the scattered field 4~s(x,y) up to and including terms of O(e2). This solution will be shown in the concluding section of this paper to be identical to that obtained in the following sections by the method of matched asymptotic expansions.
3. Inner and outer variables and the matching principle
In Section 1 the boundary value problem was introduced, and after subtraction of the incident wave (1.7), the scattered potential ~bs can be shown to satisfy the governing equation (1.2), boundary conditions (1.3, 1.4) and edge and radiation conditions. If the stagger is very small, i.e.,
ka=e-->O, a/h-->O,
(3.1)
then it is reasonable to suppose that ~bs behaves to leading order (in the small parameter e) like the potential to the unstaggered boundary value problem. Thus we write ~b, = ~bu+ ~bc
(3.2)
where the 'unstaggered' potential ~busatisfies the same governing, boundary, edge and radiation conditions as 4~ except for the condition on the top plate, 4~uy= - i k sin 0 e i~e°s °,
y = 0 ±,
x>~O,
(3.3)
0 being the incident wave angle, and ~b¢ is the 'correction' potential which we assume to be ~bc'~ ~bu.
(3.4)
The scattering of a plane wave by a pair of semi-infinite parallel unstaggered plates has a well-known solution (Heins [2, 3], Noble [6]) and may be obtained simply from [1, equations (6.4), (6.5)], when a is set to zero. This gives
_1
dp~(x,Y)=2~r f--~oO• U(a, y) e --lax dot,
(3.5)
LD. Abrahams,G.R. Wickham/ Acousticscatteringby plates
286 where
--i e v(~)h ¢'u(a, y) -- 4 K + ( a ) L+(a) (,~ - ~)
×
,[L+(a)-K+(a). L+(a) + K+(°t)] (K+(~)+L+(~)] V'(a'Y)\L+(a)+ K+(oO L+(a)-K+(oO,I \K+(¢)-L+(~),I '
V~(a, y) = (sgn(y) e -lyI~), sgn(y + h) e-ly+hl*~)),
(3.6) (3.7)
K+(ot), L + ( a ) are defined in Appendix A, 3,(00 is written in (2.5) and = - k cos 0.
(3.8)
The integration path in (3.5) runs along the real axis and lies above the branch cut emanating from a = - k , and just below the cut from a = k to + m as k is allowed to have a small positive imaginary part. We will now solve the problem for the 'correction' potential ~bc by the method of matched asymptotic expansions. Firstly we note that ~bc satisfies the same governing equation, edge and boundary conditions on the lower plate, and radiation condition as r = (x2+ y2)~/2_~ oo as the total scattered potential ~bs. The difference lies in the condition on y = 0, which is now
=~-iksinOei~°s°-qb~y, ~cy [0,
- a < x < O , y = O ±, X ~ 0, y = 0 ~.
(3.9)
Next, we introduce inner and outer dimensionless variables
X=x/a, ~=xk,
Y=y/a,
(3.10)
37=yk,
(3.11)
respectively so that (X, Y)=(~,37)/e
(3.12)
where e is the small parameter and kh is assumed to be of order unity. The inner region is where (x, y) are of order a, and here the potential is denoed by d,~-~b(X, Y),
(X, Y ) = O ( 1 ) .
(3.13)
External to this tiny neighbourhood of the origin is the outer region, and here the potential will be written as ~b~-- ~b(~, 37), (~, 37) = O(1).
(3.14)
Let us now suppose that in the inner region we can obtain an asymptotic expansion for 0 up to and including terms of order e m, m not necessarily an integer. This is denoted as ~b(X, Y ) ~ ~btm)(X, Y),
(X, Y ) = O(1).
(3.15)
Similarly assume that an expansion may be obtained for the scattered potential in the outer field up to terms of order e" (n not necessarily an integer), written &(~, 37)~ ~bc")(~, 37).
(3.16)
The inner and outer potential expansions are related by the matching principle (see e.g. Van Dyke [7], Crighton and Leppington [8]) ~O~"'") ~ q~"");
(3.17)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
287
here ¢("'") is obtained by rewriting ~btin) in terms of outer variables, expanding in powers of e (holding the outer variables fixed) and truncating after terms of O(e"). The outer potential ~b~' '~) is ~bt") re-expressed in inner coordinates, re-expanded in e, and truncated after terms of O(em). Note that we are only really interested in the outer solution for this problem and so, as will be seen, not all coefficients in the inner problem need be evaluated.
4. The inner solution
The inner region near the u p p e r plate edge is very far (on the inner lengthscale) from the lower plate, i.e. the lower plate is outside the inner region. Thus the effect of the lower plate on the field in this region is negligible and so the problem may be written as:
~lxx "[- ~lyy + E2¢ = O, Sv=_iesin0e
Cy=O,
(4.1)
i. . . . o x _ 4 ~ . v '
(4.2)
Y=0 ±,-I
(4.3)
Y=0 ±,X~>0,
together with the edge and radiation condtions. Note that $(X, 0) is required to be continuous across X = 0. The forcing term on the fight-hand side of (4.2) can be evaluated asymptotically as follows. The solution to the unstaggered problem (3.5) is written in inner variables, differentiated with respect to Y and then evaluated on Y = 0. This gives
& . y ( X , o ) = l I ~ cP~(a,O)e-~X da,
(4.4)
where
C~Uy(a,O)
- i a e v(°h
(~-~)
(K-(a) K+(~) - L-(a)
L+(~)),
(4.5)
and the pole at a = ~"= - k cos 0 lies below the contour of integration• We now deform the path to run from -oo to -1/a along the real line, from -1/a to 1/a in a semi-circular path lying in the lower half plane, and from 1/a to 0o along the real line. The pole at a = ~ is picked up in the deformation and we replace a by 6t/a to give
4,~(x,
0 ) = : -~--1 f 2"rra J~
qb~g(rt/a,0 ) e -i~x
d & - i e sin 0 e i. . . . 0x,
(4.6)
where
~y(~i / a, O)
_ i a 2 eV(Oh
(a -~a) ( K-( ~/ a) K+(~)- L-( ~/ a) L+(~)),
(4.7)
in which ca, as in (2.13), is the straight line contour from -00 to co indented into a semi-circle of centre 0 and radius 1 in the lower half plane. The path of ~i now lies entirely in ~ - , [ a ] > ~ ! , ' a n d as e ~ 0 the • f asymptotic forms (for large argument) of K-(~/a), L-(~/a) may be employed u m f o r m l y on ~. These are written in the Appendix (A16, A17), and finally after insertion into (4.7) we find th:~boundary condition ;I1
o n 0 as
A e l/2
~ y = ]XII/2 ~-Bea/21xJl/2+O(eS/2), Y = 0 , - 1 < X < 0 ,
(4.8)
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LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
where
e-ikh sin 0 e-i~r/2 A=
( K + ( ~ ) - L+(~')) k -~/2,
(2~)1/2
(4.9)
--2 e-ikh sin 0
(2.rr)~/2 {(cos 0+/32) K + ( ~ ) - ( c o s 0+/31) L+(~)} k -1/2,
B=
(4.10)
and ill, 132 are constants (dependent on kh only) given in (A19) and (A18) respectively. From the forcing (4.8) it is clear that the inner expansion will contain terms of order e l/E, e 3/2 and e 5/2 etc., but there may be other terms in the expansion at different orders. We therefore write L
q,_ q,<3/2)=
e~/~¢,l/~+ ~/~¢,~/~+ y~ GI(~) ~7
(4.11)
/=o
where it is assumed that
G,(e) ~, G,+l(e),
Go(e) <~1,
GL(e) >I e a/2.
(4.12)
Substituting the expansion (4.11) into the boundary value problem (4.1)- (4.3) and collecting like terms in e gives a sequence of problems for each ¢. Thus ~bl/2, ~b3/2both satisfy
~ x x + ¢ v v = O , all X, Y,
(4.13)
,~y = 0 ,
(4.14)
Y=0,
X~>0,
and in particular
~,l/2,y---Alxl -'/2,
Y=0,-I
(4.15)
Y=0,-l
(4.16)
and
,3/2,y=BlXl All the ee potentials satisfy
e x~x + ~yy-O,* $~.=0,
all X, Y,
(4.17)
Y=0, X~>-I,
(4.18)
and are therefore eigensolutions of the boundary value problem. The edge condition restricts these potentials to be bounded at the edge Y = 0, X = -1, and so it is easily shown that only solutions of the form
~b*= CpRp/2cos(pA/2),
p = 0 , 1,2,3 . . . . ,
(4.19)
R 2 = (X+ 1)2+ y2,
(4.20)
A = tan-l( Y / { X + 1}),
(4.21)
are possible. The presence or absence of such term, and therefore the value of the integer p and the constant Cp for each ~bT, will be determined by matching with the outer potential. For convenience, however, we now take only two terms ( p = 0 , 1) as these will be shown later to be the only required potentials (up t o O ( E 3 / 2 ) ) . The forced terms q~l/2, ¢3/2 can be solved in a straightforward manner using the Wiener-Hopf technique. Their solutions are respectively e3i~/4A foo e-iX,~(1 _ e-i,~) ~bl/2 = ~ lqimj_~ ~-~a+Tq'-~ e-"~2+q2)'/ZlYIdasgn(Y)" (4.22) _ e i~/4 j
.
foC~ e-iX,~(1--ia
~b3/2- 4x/~ li~moL ~
e -i'~ - - e - i a ) e-(a2+q2)l/2lYI
a2---~a+i-'~ ~
d a sgn(Y)
(4.23)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
289
where q is positive real constant which is set to zero after performing the integrations, and the square root functions are defined such that (or q- iq) 1/2 -->ei=/4q 1/2,
(or -- iq) 1/2 "-> e-i'#/4q 1/2
(4.24)
as ot --) 0. With these solutions we find the inner expansion, up to and including g3/2 terms, as ~1(3/2) =
el/2~bl/2_l_ e3/2~b3/2_F Go(e) Co+ Gt(e) el R1/2 cos(A/2).
(4.25)
NOW the inner variables are written in outer coordinates defined in ~= (X2+372)1/2 = e(X2+ y2)1/2,
(4.26)
O = tan-~07/~) = t a n - l ( Y / X ) .
(4.27)
Holding these variables fixed, expanding the various terms in powers of e, and truncating after terms of O(e 2) gives ~k(3/2"2)= - A e f - l / 2 cos(O/2) + ~Ae2~ -3/2 cos(3 0 / 2 ) - ~Be2~ -1/2 c0s(19/2) + CoGo(e)+ CIG1(e) e-1/2~ 1/2 cos(O/2)+½CIGI(e) el/27 -1/2 cos(19/2).
(4.28)
The outer solution will be fully determined in Section 6 by employing (4.28) in the matching principle.
5. The outer solution
The outer solution for the correction potential (3.14), q~, can easily be shown to satisfy the following conditions: ~b~+&~y+~b =0, 4,y = 0,
all ~,)7,
)~=0 and - k h , ~>~0,
(5.1) (5.2)
together with a radiation condition at infinity. As the lower plate in the original problem starts at ~ = 0 then ~b must be bounded at this edge. Thus & can only be non-zero if it contains eigensolutions of the above boundary value problem with singular behaviour near the edge of the upper plate. In particular, the outer potentials must match with the behaviour of the inner problems, and this behaviour can be seen in eq. (4.28). Thus we write _ ~(2) = ~ 4', + e 2 ~2
(5.3)
where ~bl will be assumed to behave as ~1 ~ O(r-1/2),
r--h0
(5.4)
~-~0.
(5.5)
and similarly ~ 2 - O(~-3/2),
Their solutions can again be obtained by application of the Wiener-Hopf technique. Although in both cases a homogeneous functional equation is obtained, the solutions are non-zero because both sides of the rearranged equations are shown, after an application of the extended form of Liouville's theorem, to
290
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
be equal to polynomials in the complex variable a, say. For 4), the singular behaviour at the upper edge means that this polynomial is a constant whereas for 4)2 the polynomial is linear in a. The solutions are
4),(& y) = -~-~1 I_°c b e_i~~ ~(a, y) {O(a)/~'(oO} do~,
,£
4)2(& y) = -}--£~ -2--£
( ¢ a + d ) e -i"*
(5.6)
V(a, y) {Q(~)/~(~)} da
c(/3,-/32) L-(ka) e -i"~ da,
(5.7)
where b, c and d are constants which must be determined by matching, l~(a, ~) is the row vector
f(e -~(")~, 0), ¢(a, y)
_-
/\
y > 0,
sinh(~(t~)kh)
[(0, - e -~(")(p kh)),
' sinh(~(a)kh)/'
0 > f i > -kh,
(5.8)
- k h > fi,
(~(a) is the column vector
( K - ( k a ) + L-(koO~ Q(o~) = \ K - ( k a ) - L-(ko~)/'
(5.9)
~(o~) = k-' ~,(kot) = (a 2 - l) '/2
(5.10)
and whose branch is chosen such that ~(0) = - i .
(5.11)
As before, the integration path runs along the real axis, just above the left hand branch cut and below the right hand cut. For matching purposes we now introduce the inner variables via the transformation (3.12), and holding these fixed we expand 4),, 4)2 in powers of e. After considerable algebra and some manipulation it is found that 4)(2.a/2) = e,/2{cc,R-3/2 cos(3 O/2) + bb~R -'/2 cos(O/2)}
+ ebb2+ e3/2{(db, + cc2)R-'/2+ bb3 R'/2} cos(O/2),
(5.12)
where
c, = - ( k / ( 2 ~ ) ) '/2, b, = 2ic,,
c2 = -2ic,/32
ba=-2c,(/3,+/32),
(5.13, 5.14) (5.15,5.16)
b2 is a constant which has not been determined as it is not required in the final solution, and/32 and/3, are found in the expansions for K - , L-, (A18), (A19) respectively.
6. Matched solution To obtain the asymptotic solution for small stagger all that remains is to determine the constants b, c and d in the outer solution. This is achieved by the application of the matching principle (3.17) with
291
I.D. Abrahams, G.R. Wickham / Acoustic scattering by plates
n = 2, m = 3/2. Thus (4.28) and (5.12) are equated:
e 1/2{1AR-3/2 cos(3 0 / 2 ) - A R - 1/2 cos(O/2)} + CoGo(e) + C1 G1 (e)R 1/2 c o s ( 0 / 2 ) + ½C1Gi ( e)R-1/2 c o s ( 0 / 2 ) - ¼Be3/2R-1/2 c o s ( 0 / 2 )
-- e 1/2{cc~R-3/2 cos(3 0 / 2 ) + bbtR-I/2 cos(O/2)} + ebb2 + e3/2{(db~ + cc2)R-~/2+ bb3R ~/2} c o s ( 0 / 2 )
(6.1)
which immediately reveals that the eigensolutions must have the orders
Go(e) =- e,
Gl(e) =- e 3/2.
(6.2, 6.3)
Equating the various powers in R gives
¼A = cct,
A = -bb~,
(6.4, 6.5)
Co = bb2,
CI = bb3,
(6.6, 6.7)
½C~ - ~ B = db~ + cc2,
(6.8)
which are rearranged to yield the following constants:
c =-(2,n/k)~/2A/4,
b =-i(2.tr/k)~/2A/2,
(6.9, 6.10)
d = (2~r/k)l/2[2Afl~-iB]/8, Co =-ib2(2xt/k)~/EA/2,
and
(6.11) Cl=-iA(13~+fl2),
(6.12, 6.13)
where A and B are written in (4.9) and (4.10) respectively. The first two terms of the correction potential, ~bc, have now been obtained and so finally the scattered solution is given by da d p s ( x , y ) ~ d p . ( x , y ) - - ~1 I_ °~o( e b + e 2 d + e 2 c o t / k ) e -i'~x Vo(a,y) Q ( a ) y(a'----~
2~r
c(fl~-/32) L - ( a ) e -i"x Vo(a,y)
_
y(a)'
k(x2+y2) ~/2 = 0(1),
(6.14)
where ~b, is written in (3.5)-(3.7), Vo(a, y) is given as V(a, y) in (2.3) with a -- O, or alternatively
V o ( a , y ) = ~'(a/k, ky),
and
Q(a) = Q(a/k).
(6.15, 6.16)
7. Equivalence of exact and matched asymptotic solutions, and concluding remarks Suppose we are given an inverse Fourier transform of the form ~b(x, y) = a ~
e -i'x-~(~)y O ( a a ) dot,
(7.1)
where y ( a ) and the integration path are defined as in Section 2. We wish to determine an asymptotic expansion of this expression in the outer region, defined by (~,)7)= O(1) where ~ )7 are given by (3.11), as ka = e-~ 0. Changing variables to (~,)7) and letting a = ,~/a gives ~b = [ ~ e x p { - i t ~ / e -(of 2- e2)l/2)7/e} tlb(,-~) d~7.
(7.2)
LD. Abrahams,G.R. Wickham/ Acousticscatteringby plates
292
When e is small it is reasonable to assume that the major contribution comes from a region near the origin on the integration path. Thus we write
I ~.1/2 4' = J - ' / 2 e x p { - i 6 ~ / e +
if%f / L J-~
j
l/2j
- (6 2- e2)~/2y/e}
@(6) d~
e x p { - i 6 ~ / e - ( a 2 - e2)'/zy/e} @(a) da,
(7.3)
where the limits of integration in the first term have been chosen to lie within the intervals (-1, - e ) and (e, 1) respectively. For ~ # 0 it is clear that the contributions from the infinite integrals are exponentially small and for the first term, since 6 is uniformly small on the range of integration, we may replace @(~) by its asymptotic expansion @(~) (say). Thus, in the outer region, (~, ~) = 0(1), (7.1) is approximated by A
i
~1/2
4~~ J-~'/~ exp{-ia~/e - (a 2-
e2)'/2y/e} ~(~i) d 6
(7.4)
to the same order in e as ~ ( a ) approximates @. It now follows that if we wish to show that (7.1) is asymptotically equivalent to some alternative Fourier transform, then we merely have to verify that the corresponding @(aa) are asymptotically identical at the origin. That is, if
d~j(x,y)=a f f exp{-icex-y(ce)y} @~(aa)doe, j= l,2,
(7.5)
where @~(ota) are slowly varying in the neighbourhood of the origin, then ~b,(x, y) ~ ~b2(x, y), as e-~O and
(7.6)
(~,~)=(x,y)/k=O(1),
@l(aa) ~
@2(aa)
if
as a ~ 0.
(7.7)
In what follows we shall use this principle to compare (2.1) with the outer solution obtained in Section 6. However, note that (2.2) and (2.3) are more complicated than the model of (7.1) in that our @(a, y) is actually a function of a/k as well as aa. Since a/k is not uniformly small on a e [ - e l/z, e~/2], we must expand q~ for small aa, uniformly in a/k on the range of integration. Returning to the asymptotic solutions of the integral equations (2.18), (2.19), derived from the exact analysis of [1], we obtain as e ~ 0
N*(a)= M*(a) =
-
iu(kh) e+e--'4~'(kh)(a/k) 2k-----~
iu(kh)2k_..._~e +
[
kh
I~(kh)
}
+O(e3),
(7.8)
~,(kh)(ot/k)~kh--k2h2 ~(kh) +O(e3),
(7.9)
and from (2.11),
S*(a) =i~'(kh)2khe+ e___~{ ~,(kh)(a/k)+l.~(kh)}+O(ea) ' k h T*(oQ=
iv(kh) 2kh e+
~'(kh)(a/k) t k2h----T ~(kh) kh
(7.10) +O(e3),
(7.11)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
293
where note that it has only been necessary to expand e i~ about the origin. Hence, if we write ~ = - k cos 0 again, then i e v(Oh 2(,-a){(:-i)-2-'~\-1
(K-(,))-'F(a)
iev ( 1 1 l)+ie('lk)(~ ~)
o
-2(Uk) 2
+(~)]
+ O(e3~l(-K " J \ L+(~) ]" This may be premultiplied by (0 -'a~
(7.12)
K - ( a ) to yield
~)K-(a)(K-(~))-'F(a)
L-(=)~r=/l 7)+le(....~)(l 11) -L-(o0/L \0
ie "r~Oh (K-(a)
.
~(7-~) \K-(o,)
e2(~-a]2(1
~-a
1
e2(~-a] v(kh) -T\---~]
\ L+(~") ]
(7.13)
or rearranging gives the final result i e v(¢)h y(a)
q~(a,y)=--
iE
+~~
f L-(,,) Vo(,',, y ) / ( ~ - '~x_l/g-(~) ~g-(,~) -L-(o,)]\ Z+(~) / I.
Q(a) (-K+(~')+
e 2 [v(kh)
+4-~ t - - - ~
2
L+(sr))+-~2
g+(~') - sr(L+(~') -
Q(a) (L+(~')- K+(sr))
K+(~')))
O(a)
e 2 -v(kh) - L-(a) (L+(~)- K+(~)) (_I) +O(e3) } , 4k kh where
Vo(a,y) is V(a, y) (eq.
(2.3)) with a = O, and
(K-(a)+L-(a)] Q(a)=\K-(a)_L-(a)]"
(7.14)
Q(a) is (7.15)
It is a simple matter to show that (7.14) in (2.1) is identical to the expression derived via matching, (6.14) correct to O(e2). We could further prove that the inner solution is in agreement with ~b~ from Section 2 if we did not expand the e ~a~ terms in (2.18), (2.19). As this region is of little physical interest this calculation is not performed here.
294
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
We can conclude that both solutions show to leading order that the field scattered by the staggered plates is identical to the field scattered by the unstaggered plates. The effect of the stagger first manifests itself in the outer field expansion at O(ka), and this correction potential is an eigensolution of the unstaggered plates problem with singular behaviour at the edge of the upper plate. This potential behaves like r -~/2 as r = (x2+y2)~/2~ 0. The next order correction potential ( O ( k a ) 2) is similarly an eigensolution of the unstaggered problem but it has r -3/2 behaviour as r-~ 0. As a final exercise it is useful to compare the asymptotic solution with that obtained from the full numerical analysis of the integral equations (2.8-2.10) performed by the authors elsewhere [5]. In particular it is important to establish just how small the dimensionless length ka has to be in order to give an accurate estimate of the sound field both inside and outside the duct. In Figures 2a and 2b the magnitude of the scattered acoustic field far from both plate edges (kr = 30.0) is shown plotted as a function of the observation angle O. Note that the curves are discontinuous because the specular field has been subtracted from the total scattered potential, i.e. the reflected wave is removed in 2~ > O > 2 ~ - 0 and the incident wave is added in the shadow zone (0 < O < 0). The value O = 0 ° corresponds to a point on top of the u p p e r plate and O = 360 ° is a point on the underside of the lower plate. The incident wave makes an angle 0 = "#/4 with the horizontal and the plate spacing is chosen as kh = ¢ / 2 . In Figure 2a the stagger is chosen as ka = 0.1 whereas in Figure 2b this is increased to ka = 2.0. The unbroken line corresponds to the numerical results of [5], and the large dashed line is that of the far field predicted here by both the matched asymptotic analysis and solution using the approximation given in (7.8-7.11). In Figure 2a the two curves are indistinguishable but as the stagger increases the two results have deviated quite markedly from each other for certain angles O. The small dashed line in Figure 2b is the asymptotic solution employing the integral equation approximation given in (2.18, 2.19) and (2.11). This result appears to give slightly better agreement with the exact result over the range 0 < O < "rr. Figure 3 shows the variation of the sound field magnitude (at kr = 30.0) as a function of ka at a particular observation angle O = 3~r/2. The incident wave angle is chosen as 0 = 3~r/4 and the plate spacing is again ~/2. The asymptotic result (dashed line) is in very good agreement with the exact result (unbroken curve) for stagger lengths ka less than approximately 1.3 but then deviates rapidly as ka continues to increase. It may be noted that the
3.0 ~ (a)
4.0
(b)
2.5
-•3.0 .~2.0
D_ E
"~ 2.o
-"
,
b~
~ t.O
t~-0.5 o.o
..... ""~I '""='~68 ...... ~ ..... Observation Angle Theta
0.0
k i
,,,y v= "
6 ........ i~ ........ iN ............ Observation Angle Theta
Fig. 2. Magnitude o f scattered sound field (with specular terms omitted) against observation angle O for kr = 30.0, 0 = ~/4, kh = ~r/2 and (a) ka =0.1, (b) kh =2.0. Unbroken line is exact solution, large dashed line is asymptotic result from (7.8-7.11) and small dashed line is approximation using (2.18, 2.19, 2.11).
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
295
2o !
LO.5
-~
\
3
o.o
~.o 2.0 3.0 Dimensionless length ka
~.o
Fig. 3. Magnitude of sound field against dimensionless length ka for kr = 30.0, 0 = 3Tt/4, O = 3 ~ / 2 and kh = ~r/2. Unbroken line
is exact solution and dashed line is asymptotic result.
approximation employing N * , M* etc. from equations (2.18, 2.19) rather than (7.8-7.11) in fact leads to worse results as/ca is increased. This is partly due to an artificial resonance built into this solution near k a = "rr. F i g u r e s 4 a a n d 4 b c o n f i r m t h a t t h e a s y m p t o t i c
r e s u l t is v e r y g o o d
f o r k a <~ 1.3, a n d s h o w t h e
amplitude of the propagating duct mode as a function of ka. In these figures kh is again ~ / 2 and in 4a, 0 = xr/2 whereas 0 = 3"tr/2 in 4b. In conclusion the method of matched asymptotic expansions is appealing and powerful in its relatiohship with the physics of complicated problems such as that considered here and, analytically, it can be simpler to apply than exact techniques (e.g. as given in [1] and [5]). Given that there is not as yet a general and rigorous justification of the method, it is helpful to check, where exact solutions are known, that it does indeed produce correct asymptotic expansions. This paper has not only given the asymptotic solution of
25.0
2.5
(a)
(b)
" 0 20.0
Q) " 0 2.0 , "-I .--
I I |
O.
',
~.~ 15.0
iII /\\
,'/
-
I
~D . ~ I0•0
\
\
~
1.0,
~
0.5.
"'........
\ ~
5.O
°'°o.o. . . . . . . . 'i.b ";.b ........ Lb ........ Dimensionless length ko .
.
.
.
.
.
.
.
~,:o
0.0
0•0
1.0
2.0
3.0
g I
4.0
Dimensionless length ka
Fig. 4. Magnitude of propagating duct mode against dimensionless length/ca for kh = ~ / 2 and (a) 0 = ~r/2, (b) 0 = 31r/2. Unbroken line is exact solution and dashed line is asymptotic result•
296
LD. Abrahams, G.IL Wickham / Acoustic scattering by plates
the field scattered by slightly staggered plates, and validated the matching principle employed herein, it has also shown the wide range of ka for which the solution is valid.
Appendix A The following scalar factorizations were introduced in [1] and are repeated here for convenience. Let K ( a ) = -½y(a)(1 + e -~(~)h)
(A1)
where y ( a ) is written in (2.5), and introduce the product decomposition
(A2)
K ( o ) = K+(a) K - ( a )
where the + ( - ) denotes a function which is analytic in the upper (lower) half of the complex a plane (both half planes are assumed to contain the real line if k has slightly positive imaginary part). It is found that {(1 - k 2d._l/2) 2 1/2;:.ladn_l/2} exp{+
K=~(a)=(a+k)l/2exp{i½~r;:fl(a)-T±(a)}fi
iadn-l/2},
n=l
(A3) where T+(a) _ h~r_7,a.____.~arccos(a/k), ,
2~r
f,(a)
ixr + ah 4
T-(a) = T+(-a),
. a__h_h {1 _ C +in( ~r ) }
1 2"#
-'~
(A4, 5)
h
d.-,/2 - (2n - 1),rr'
,
(A6, 7)
and C = 0.5772... is Euler's constant. Also we take
(A8)
L ( - ) = -½y(a)(1 - e -~(~)h) = L+(a) L - ( a ) ,
(A9, 10)
L+(a) = L - ( - a ) ,
in which case L±(a) = (½h)~/2(a + k)
expIi½"rrwf2(a)-
H {(1 - k2d2.)~/2~:iad.} exp/+iaa.},
T~(a)}
(All)
where
A(a)
2¢ri + ah 4
i a~h { 1 - C+ln(4~-~)}
d.-
'
h 2nqv"
(A12, 13)
It can be shown that the following asymptotic properties are held by the various factors:
K+(a) =---~ e~"/'{1 + kOJ,, + O(1/~)} al/2
L+(,,) = - ~ - ei~/'{1 + k#~/~ + O0/,,~)}
(A14)
[al-~ oo, 0~
LD. Abrahams, G.R. Wickham I Acoustic scattering by plates
ot 1/2 e3i~/4{1 K-(,',) =--~-k/32/a+O(llot2)}
297 (A16)
l a l + ~ o , -~r <~a r g ( a ) <~ 0, L-(ot) -- ~ - ~3i~r/4~f 1 ,.-kfltla+O(lla2)}
(A17)
where 1 ikh /32 = ~ + ~ - ( 1 - 2 C )
I
+i ~ .{(1-k n=l
kh
8
ikh l n ( _ ~ ) i~r 4¢r +l-~h
2 2
dn-1/2) 1/2 - 1 } + ~ k1 kd,,-l/2
2 2 dn-t/2] J
(A18)
and /31=l+ikh kh ikh (4~r) ~ - (1 - 2 C ) - - ~ - + ~ - ~ - In ~
+i ~ .=!
[
i~r 6kh
122]
((1-k2d2)'/2-1}+-2k
d.
kd.
(A19)
"
We can also write the ratio L + ( a ) / K + ( a ) as
L+(a)
K+(a)-l+~hh)+o(1/t~2),
(h20)
lal -~co, 0~
where v ( k h ) = k h ( f l l - / 3 2 ) =¼(2kh-i,rr)+~ri
~ {2nf,(kh)-(2n
-1)f,-t/2(kh)},
(A21)
and f , ( k h ) = (1 - k2d2) ~/2-1.
(A22)
References [1] I.D. Abrahams and G.R. Wickham, "On the scattering of sound by two semi-infinite parallel staggered plates I. Explicit matrix Wiener-Hopf factorization", Proc. Roy. Soc. London A 420, 131-156 (1988). [2] A.E. Heins, "The radiation and transmission properties of a pair of semi-infinite parallel plates-I", Q. Appl. Math. 6, 157:-166 (1948). [3] A.E. Heins, "The radiation and transmission properties of a pair of parallel plates-IF', Q. Appl. Math. 6, 215-220 (1948). [4] I.D. Abrahams and G.R. Wickham, "General Wiener-Hopf factorization of matrix kernels with exponential phase factors", SIAM J. Appl. Math. A 50 (3), to appear (1990). [5] I.D. Abrahams and G.R. Wickham, "On the scattering of sound by two semi-infinite parallel staggered plates II. Evaluation of the velocity potential for an incident plane wave and incident duct mode", Proc. Roy. Soc. London A 427, 139-171 (1990). [6] B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon Press, London (1958). [7] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford (1964). [8] D.G. Crighton and F.G. Leppington, "Singular perturbation methods in acoustics: Diffraction by a plate of finite thickness", Proc. Roy. Soc. London A 335, 313-339 (1973).
ACOUSTIC SCATrERING RIGID PLATES
281
BY T W O
PARALLEL
SLIGHTLY
STAGGERED
I.D. A B R A H A M S Department of Mathematics and Statistics, The University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK
G.R. W I C K H A M Department of Mathematics, The University of Manchester, Manchester M13 9PL, UK
Received 27 April 1989, Revised 10 July 1989
In a recent paper Abrahams and Wickham [1] showed how to exactly solve the diffraction problem for two parallel semi-infinite plates which are arranged to form a duct. The edges of the plates are misaligned so that the geometry is asymmetric with respect to the central line of the duct. To evaluate the field it is first necessary to solve the integral equations derived in [1], and this can be accomplished analytically when the 'stagger' of the plates is small. In this paper we derive this analytic solution and use it to determine several terms in the asymptotic expansion of the velocity potential. The solution found by the method of matched asymptotic expansions is shown to be identical to this rigorous result.
1. Introduction and boundary value problem This p a p e r is concerned with the scattering of plane sound waves obliquely incident onto two semi-infinite rigid parallel plates. The plate edges are 'staggered' so that an imaginary line joining the two plate edges is not perpendicular to the plane of either plate. The motion is assumed to be purely two-dimensional. This boundary value problem was examined recently in a p a p e r by the authors, Abrahams and Wickham [1] and by Fourier transformation was reposed as a matrix W i e n e r - H o p f functional equation. The 'unstaggered' geometry has attracted a good deal of attention since the work of Heins [2, 3]; and an explicit solution of the matrix W i e n e r - H o p f equation is possible in this case because the system may be reduced to two scalar equations. However, when the plates are staggered, the matrix kernel contains exponential phase factors which, on decomposing in the manner suggested by Heins, give rise to exponentially growing terms. This inhibits the application of Liouville's theorem and renders the equation unusable without modification. One method to overcome this difficulty is to premultiply both sides of the equation by an entire matrix, which must have suitable exponential behaviour at infinity, and this was performed on the specific problem in [1] and on a general class of matrix W i e n e r - H o p f equations in Abrahams and Wickham [4]. The elements of this matrix are all shown to depend on a pair of coupled Fredholm integral equations of the second kind (see equations (2.8), (2.9)) and therefore any numerical evaluation of the scattered sound field (or transmitted duct modes) will necessitate the solution of these equations. The object of the present investigation is to obtain an asymptotic solution of the exact analysis presented in [1], when a physical parameter is taken to be small, and then to compare the field obtained from this solution with that derived by the method o f matched asymptotic expansions. This will not only prove useful in helping to understand the physical relevance of the exact solution, but will enable a check on 0165-8641/90/$03.50 © 1990--Elsevier Science Publishers B.V. (North-Holland)
282
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
the numerical solution of the integral equations (see Abrahams and Wickham [5]). The asymptotic limit chosen in this study is the situation when the 'stagger' is very small, i.e. the ratio of the difference in plate lengths to the plate spacing tends to zero. It will further be assumed that the ratio of the incident acoustic wavelength to the plate spacing is of order unity. The organization of this paper will be as follows; In the remaining part of Section 1 the boundary value problem is stated. In Section 2 an asymptotic expansion for the field is derived from the exact analysis of [1]. The method of matched asymptotic expansions is then employed in Sections 3, 4 and 5. In Section 3 the inner and outer variables are introduced together with the matching principle, and Section 4 contains the derivation of the inner solution (i.e. the asymptotic form of the sound field near to the edge of the top plate). The outer solution is found in Section 5 up to several unknown constant factors, and in Section 6 these constants are determined by matching with the inner solution. In Section 7 this solution is shown to exactly agree with that found in Section 2, and we also make some concluding remarks. The mathematical formulation of the physical problem is expressed in terms of a time-harmonic velocity potential Re[~b(x, y) e -i'~,]
(1.1)
where to is the angular frequency, (x, y) are cartesian coordinates and e -i~'' is henceforth suppressed for brevity. The potential, ~b(x, y), satisfies the reduced wave equation
(32/ax2+O2/ay2+k2) dp =0,
(x,y)~ ~,
(1.2)
and the boundary conditions
a4, - 0 ,
-a
(1.3)
3¢b O, y = - h ±, O < x < o o ,
(1.4)
Oy
y = O ±,
by
where ~ is the whole of the (x, y)-plane, cut along the two semi-infinite lines y = O, - a < x < oo, and y = - h , O< x <0% and the wavenumber
k = to/ c
(1.5)
with c the phase speed in ~. We wish to solve for the potential, ~bs(x,y), scattered by an incident plane wave, ~bi(x, y), i.e. 4~s= ~b- ~i,
(1.6)
4~i.= exp(ikx cos O+iky sin 0)
(1.7)
where 0 is the direction of propagation relative to the positive x-axis (see Figure 1). We require that 4~, represents outgoing waves as (x2+y2)~/2-*oo both inside and outside the duet region, and that 4~s is bounded in the neighbourhood of both plate edges.
2. Asymptotic solution from exact analysis The exact solution of the boundary value problem was determined [1] via the introduction of the entire matrix K* which contained the scalar functions N*, M*, S* and T*. To recap I
1 ~ ~ ( a , y) e -i~x da, ~bs(x, y) = 2--~ fd-oo
(2.1)
LD. Abrahams,G.R. Wickham/ Acousticscatteringby plates
283
Y
(-a,o)
(O,-h) Fig. 1. Physicalconfiguration. is the scattered field where a runs along the real line (if k has slight positive imaginary part), 1 • (a, y) = - ~ )
V(a, y) K-(a) ( K - ( - k cos 0))-' F ( a ) ,
'(e-iaa-yy(a), V(ot, y)
(2.2) y > O,
0),
( _ e _ i , ~ cosh y(a)(y+h) cosh y(a)y~ sinh y ( a ) h 'sinh y(a)h]"
0 > y > -h,
(0, --eY(~)(Y+h)),
-- h > y,
(2.3)
and F(a)=
- k s i n 0 ( e -'k . . . . 0~ k co--s"0--+a \ e-ikh sin 0]"
(2.4)
The square root function y ( a ) is defined as y ( a ) = (a 2 - k2) 1/2
(2.5)
which has the chosen branch with y ( 0 ) = - i k , and K-(a) is
\e-" K-(a)
-e-i"~L-(a)) K*(a)'
(2.6)
where K-(a), L-(a) are scalar factorizations which are analytic in ~ - defined by I m ( a ) < k and are written in the Appendix (A3, A11). Also
( M * ( a ) + e i"" S * ( a ) + 1~ K*(a)=\N*(a)-eia" T*(a)+l]'
(2.7)
where the entire elements M*(a) and N*(a) satisfy the integral equations
N*(,~)- M*(~)+~i I ~ M*(~) \/(--7~-~ N*(ot)+ M*(a)+ ..,o,
\K-(~)
\
•
d~"
1) e -'"(~-'~> ~_,~ =0,
~-ot
(2.8)
LD. Abrahams, G.R. Wickham / Acousticscatteringby plates
284 in which
1 r i,,, { L,(~') ) d~ °*(a)---2~r-'~J~e kK-(~----~ 1, ~ _ ~ ,
(2.10)
and Jr (L,) denotes an integration path from -oo to +oo which passes above (below) all the integrand's singularities. We also find that
S*(ot)=-eia" N * ( - a ) ,
T*(a)=eiaa M*(-ot),
(2.11)
and note that the inverse Fourier transform path passes above (below) the branch cut emanating from - k (k) to oo in the lower (upper) half plane and above the pole at a = - k cos 0. We now wish to obtain an asymptotic solution from this exact solution as ka-> 0 with kh fixed and order unity. In [1] it was shown that the system may be solved by iteration for each value of kh < ~r and a/h sufficiently small, e.g. for kh <2 then a/h must be less than 1.8 (see Theorem 1 and Figure 3 of [1]). The first iteration yields
N*(a) ~ M * ( a ) ~ ½0*(a).
(2.12)
in which we write O*(a) as
1 L.e'~ (L-(,/a)_l) \K-(~/a)
O*(a) = 2~r---i
d,
(2.13)
(~-aa)
where the contour has been deformed into ~ which passes from - ~ to -1 along the real line of ~, from - 1 to + 1 in a semi-circular arc in the lower half plane and then to +oo along the real line. The path is also indented below aa. It is now straightforward to show that as ka = e -~ O, the integrand may be replaced by
v(kh) kh~
L-(~/a)_l_ K-(~/a)
--e+
I~(kh) ~.2
e2+O(e3)
(2.14)
uniformly on ~, where v(kh) is given in (A21) and ~(kh) is some function of kh which will not actually be required to the order of the expansion we examine. Substituting this into (2.13) and closing the integrals in the upper half plane yields O*(a) =
v(kh) ( 1 - e i ~ ) 4 k21~(kh) (e i ~ - 1 - i a a ) + O ( e 3 ) , ha a2
where the first term is uniformly of O(e) for I m ( a ) ~ - l / a , uniformly O(e 2) and O(e 3) respectively. The next iteration gives
1 f
\ K---~)
1 f O*(~)(L-(~) - - -
2,n.i L T
\ KZ(-~-i
1
M ( a ) = - ~ i f,~ 0 " ( , ) ( L + ( , ) _ 4
If.
2~ri
and the second term and remainder are
d~"a O * ( ~ ) { L+(~") - 1 e-ia(g-~) ~--
N *( a ) ~- - 2ar-~l.J~ T
.
(2.15)
\K+(~ ")
O *_(4,_____(~) L- ( , ) \~'('~-
d~" + l o , ( a ) , 1 e -ia(¢-~) ~'1 ) e -'"('-~)
1'~
"
(2.16)
d~r ~-ot
d, +½0*(a),
] e-'"(¢-~) K- a
(2.17)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
285
in which we substitute from (2.15) and evaulate the integrals in the same way as for (2.13) to obtain the asymptotic expansions. These are
v(kh),._ei~. ) [u2(kh) - -
N*(ot) = 2-~--~t--ot(1
+ ~ 4 - ~ ' j 2 [2(1 - e
i,,.-
•
)+laot(l+ei~'a)]
-+k21z(kh---~)2a z (e i~a - 1 -iota)) M*(a) = v(kh)2ha(1-ei~'")+ [iav2(k4h2o h)[ t
(1-eia~)4
+ O(e3),
k21z(kh)2ot2(ei~a-1
(2.18)
-iota)} +O(e 3)
(2.19)
where the terms in parentheses are O(e 2) and the error is uniform for Im(ot)~>-1/a. S*(ot) and T*(a) are obtained by reflection in the origin using equations (2.11). Generally the integral operators in (2.8), (2.9) are O(e), and hence to obtain an asymptotic solution to O(e") one requires n iterations. The asymptotic solution written in (2.18), (2.19) may now be substituted into (2.7), (2.6) to form K*(ot) and then K-(a). These are then employed in 4~(a, y), equation (2.2), to give the scattered field 4~s(x,y) up to and including terms of O(e2). This solution will be shown in the concluding section of this paper to be identical to that obtained in the following sections by the method of matched asymptotic expansions.
3. Inner and outer variables and the matching principle
In Section 1 the boundary value problem was introduced, and after subtraction of the incident wave (1.7), the scattered potential ~bs can be shown to satisfy the governing equation (1.2), boundary conditions (1.3, 1.4) and edge and radiation conditions. If the stagger is very small, i.e.,
ka=e-->O, a/h-->O,
(3.1)
then it is reasonable to suppose that ~bs behaves to leading order (in the small parameter e) like the potential to the unstaggered boundary value problem. Thus we write ~b, = ~bu+ ~bc
(3.2)
where the 'unstaggered' potential ~busatisfies the same governing, boundary, edge and radiation conditions as 4~ except for the condition on the top plate, 4~uy= - i k sin 0 e i~e°s °,
y = 0 ±,
x>~O,
(3.3)
0 being the incident wave angle, and ~b¢ is the 'correction' potential which we assume to be ~bc'~ ~bu.
(3.4)
The scattering of a plane wave by a pair of semi-infinite parallel unstaggered plates has a well-known solution (Heins [2, 3], Noble [6]) and may be obtained simply from [1, equations (6.4), (6.5)], when a is set to zero. This gives
_1
dp~(x,Y)=2~r f--~oO• U(a, y) e --lax dot,
(3.5)
LD. Abrahams,G.R. Wickham/ Acousticscatteringby plates
286 where
--i e v(~)h ¢'u(a, y) -- 4 K + ( a ) L+(a) (,~ - ~)
×
,[L+(a)-K+(a). L+(a) + K+(°t)] (K+(~)+L+(~)] V'(a'Y)\L+(a)+ K+(oO L+(a)-K+(oO,I \K+(¢)-L+(~),I '
V~(a, y) = (sgn(y) e -lyI~), sgn(y + h) e-ly+hl*~)),
(3.6) (3.7)
K+(ot), L + ( a ) are defined in Appendix A, 3,(00 is written in (2.5) and = - k cos 0.
(3.8)
The integration path in (3.5) runs along the real axis and lies above the branch cut emanating from a = - k , and just below the cut from a = k to + m as k is allowed to have a small positive imaginary part. We will now solve the problem for the 'correction' potential ~bc by the method of matched asymptotic expansions. Firstly we note that ~bc satisfies the same governing equation, edge and boundary conditions on the lower plate, and radiation condition as r = (x2+ y2)~/2_~ oo as the total scattered potential ~bs. The difference lies in the condition on y = 0, which is now
=~-iksinOei~°s°-qb~y, ~cy [0,
- a < x < O , y = O ±, X ~ 0, y = 0 ~.
(3.9)
Next, we introduce inner and outer dimensionless variables
X=x/a, ~=xk,
Y=y/a,
(3.10)
37=yk,
(3.11)
respectively so that (X, Y)=(~,37)/e
(3.12)
where e is the small parameter and kh is assumed to be of order unity. The inner region is where (x, y) are of order a, and here the potential is denoed by d,~-~b(X, Y),
(X, Y ) = O ( 1 ) .
(3.13)
External to this tiny neighbourhood of the origin is the outer region, and here the potential will be written as ~b~-- ~b(~, 37), (~, 37) = O(1).
(3.14)
Let us now suppose that in the inner region we can obtain an asymptotic expansion for 0 up to and including terms of order e m, m not necessarily an integer. This is denoted as ~b(X, Y ) ~ ~btm)(X, Y),
(X, Y ) = O(1).
(3.15)
Similarly assume that an expansion may be obtained for the scattered potential in the outer field up to terms of order e" (n not necessarily an integer), written &(~, 37)~ ~bc")(~, 37).
(3.16)
The inner and outer potential expansions are related by the matching principle (see e.g. Van Dyke [7], Crighton and Leppington [8]) ~O~"'") ~ q~"");
(3.17)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
287
here ¢("'") is obtained by rewriting ~btin) in terms of outer variables, expanding in powers of e (holding the outer variables fixed) and truncating after terms of O(e"). The outer potential ~b~' '~) is ~bt") re-expressed in inner coordinates, re-expanded in e, and truncated after terms of O(em). Note that we are only really interested in the outer solution for this problem and so, as will be seen, not all coefficients in the inner problem need be evaluated.
4. The inner solution
The inner region near the u p p e r plate edge is very far (on the inner lengthscale) from the lower plate, i.e. the lower plate is outside the inner region. Thus the effect of the lower plate on the field in this region is negligible and so the problem may be written as:
~lxx "[- ~lyy + E2¢ = O, Sv=_iesin0e
Cy=O,
(4.1)
i. . . . o x _ 4 ~ . v '
(4.2)
Y=0 ±,-I
(4.3)
Y=0 ±,X~>0,
together with the edge and radiation condtions. Note that $(X, 0) is required to be continuous across X = 0. The forcing term on the fight-hand side of (4.2) can be evaluated asymptotically as follows. The solution to the unstaggered problem (3.5) is written in inner variables, differentiated with respect to Y and then evaluated on Y = 0. This gives
& . y ( X , o ) = l I ~ cP~(a,O)e-~X da,
(4.4)
where
C~Uy(a,O)
- i a e v(°h
(~-~)
(K-(a) K+(~) - L-(a)
L+(~)),
(4.5)
and the pole at a = ~"= - k cos 0 lies below the contour of integration• We now deform the path to run from -oo to -1/a along the real line, from -1/a to 1/a in a semi-circular path lying in the lower half plane, and from 1/a to 0o along the real line. The pole at a = ~ is picked up in the deformation and we replace a by 6t/a to give
4,~(x,
0 ) = : -~--1 f 2"rra J~
qb~g(rt/a,0 ) e -i~x
d & - i e sin 0 e i. . . . 0x,
(4.6)
where
~y(~i / a, O)
_ i a 2 eV(Oh
(a -~a) ( K-( ~/ a) K+(~)- L-( ~/ a) L+(~)),
(4.7)
in which ca, as in (2.13), is the straight line contour from -00 to co indented into a semi-circle of centre 0 and radius 1 in the lower half plane. The path of ~i now lies entirely in ~ - , [ a ] > ~ ! , ' a n d as e ~ 0 the • f asymptotic forms (for large argument) of K-(~/a), L-(~/a) may be employed u m f o r m l y on ~. These are written in the Appendix (A16, A17), and finally after insertion into (4.7) we find th:~boundary condition ;I1
o n 0 as
A e l/2
~ y = ]XII/2 ~-Bea/21xJl/2+O(eS/2), Y = 0 , - 1 < X < 0 ,
(4.8)
288
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
where
e-ikh sin 0 e-i~r/2 A=
( K + ( ~ ) - L+(~')) k -~/2,
(2~)1/2
(4.9)
--2 e-ikh sin 0
(2.rr)~/2 {(cos 0+/32) K + ( ~ ) - ( c o s 0+/31) L+(~)} k -1/2,
B=
(4.10)
and ill, 132 are constants (dependent on kh only) given in (A19) and (A18) respectively. From the forcing (4.8) it is clear that the inner expansion will contain terms of order e l/E, e 3/2 and e 5/2 etc., but there may be other terms in the expansion at different orders. We therefore write L
q,_ q,<3/2)=
e~/~¢,l/~+ ~/~¢,~/~+ y~ GI(~) ~7
(4.11)
/=o
where it is assumed that
G,(e) ~, G,+l(e),
Go(e) <~1,
GL(e) >I e a/2.
(4.12)
Substituting the expansion (4.11) into the boundary value problem (4.1)- (4.3) and collecting like terms in e gives a sequence of problems for each ¢. Thus ~bl/2, ~b3/2both satisfy
~ x x + ¢ v v = O , all X, Y,
(4.13)
,~y = 0 ,
(4.14)
Y=0,
X~>0,
and in particular
~,l/2,y---Alxl -'/2,
Y=0,-I
(4.15)
Y=0,-l
(4.16)
and
,3/2,y=BlXl All the ee potentials satisfy
e x~x + ~yy-O,* $~.=0,
all X, Y,
(4.17)
Y=0, X~>-I,
(4.18)
and are therefore eigensolutions of the boundary value problem. The edge condition restricts these potentials to be bounded at the edge Y = 0, X = -1, and so it is easily shown that only solutions of the form
~b*= CpRp/2cos(pA/2),
p = 0 , 1,2,3 . . . . ,
(4.19)
R 2 = (X+ 1)2+ y2,
(4.20)
A = tan-l( Y / { X + 1}),
(4.21)
are possible. The presence or absence of such term, and therefore the value of the integer p and the constant Cp for each ~bT, will be determined by matching with the outer potential. For convenience, however, we now take only two terms ( p = 0 , 1) as these will be shown later to be the only required potentials (up t o O ( E 3 / 2 ) ) . The forced terms q~l/2, ¢3/2 can be solved in a straightforward manner using the Wiener-Hopf technique. Their solutions are respectively e3i~/4A foo e-iX,~(1 _ e-i,~) ~bl/2 = ~ lqimj_~ ~-~a+Tq'-~ e-"~2+q2)'/ZlYIdasgn(Y)" (4.22) _ e i~/4 j
.
foC~ e-iX,~(1--ia
~b3/2- 4x/~ li~moL ~
e -i'~ - - e - i a ) e-(a2+q2)l/2lYI
a2---~a+i-'~ ~
d a sgn(Y)
(4.23)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
289
where q is positive real constant which is set to zero after performing the integrations, and the square root functions are defined such that (or q- iq) 1/2 -->ei=/4q 1/2,
(or -- iq) 1/2 "-> e-i'#/4q 1/2
(4.24)
as ot --) 0. With these solutions we find the inner expansion, up to and including g3/2 terms, as ~1(3/2) =
el/2~bl/2_l_ e3/2~b3/2_F Go(e) Co+ Gt(e) el R1/2 cos(A/2).
(4.25)
NOW the inner variables are written in outer coordinates defined in ~= (X2+372)1/2 = e(X2+ y2)1/2,
(4.26)
O = tan-~07/~) = t a n - l ( Y / X ) .
(4.27)
Holding these variables fixed, expanding the various terms in powers of e, and truncating after terms of O(e 2) gives ~k(3/2"2)= - A e f - l / 2 cos(O/2) + ~Ae2~ -3/2 cos(3 0 / 2 ) - ~Be2~ -1/2 c0s(19/2) + CoGo(e)+ CIG1(e) e-1/2~ 1/2 cos(O/2)+½CIGI(e) el/27 -1/2 cos(19/2).
(4.28)
The outer solution will be fully determined in Section 6 by employing (4.28) in the matching principle.
5. The outer solution
The outer solution for the correction potential (3.14), q~, can easily be shown to satisfy the following conditions: ~b~+&~y+~b =0, 4,y = 0,
all ~,)7,
)~=0 and - k h , ~>~0,
(5.1) (5.2)
together with a radiation condition at infinity. As the lower plate in the original problem starts at ~ = 0 then ~b must be bounded at this edge. Thus & can only be non-zero if it contains eigensolutions of the above boundary value problem with singular behaviour near the edge of the upper plate. In particular, the outer potentials must match with the behaviour of the inner problems, and this behaviour can be seen in eq. (4.28). Thus we write _ ~(2) = ~ 4', + e 2 ~2
(5.3)
where ~bl will be assumed to behave as ~1 ~ O(r-1/2),
r--h0
(5.4)
~-~0.
(5.5)
and similarly ~ 2 - O(~-3/2),
Their solutions can again be obtained by application of the Wiener-Hopf technique. Although in both cases a homogeneous functional equation is obtained, the solutions are non-zero because both sides of the rearranged equations are shown, after an application of the extended form of Liouville's theorem, to
290
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
be equal to polynomials in the complex variable a, say. For 4), the singular behaviour at the upper edge means that this polynomial is a constant whereas for 4)2 the polynomial is linear in a. The solutions are
4),(& y) = -~-~1 I_°c b e_i~~ ~(a, y) {O(a)/~'(oO} do~,
,£
4)2(& y) = -}--£~ -2--£
( ¢ a + d ) e -i"*
(5.6)
V(a, y) {Q(~)/~(~)} da
c(/3,-/32) L-(ka) e -i"~ da,
(5.7)
where b, c and d are constants which must be determined by matching, l~(a, ~) is the row vector
f(e -~(")~, 0), ¢(a, y)
_-
/\
y > 0,
sinh(~(t~)kh)
[(0, - e -~(")(p kh)),
' sinh(~(a)kh)/'
0 > f i > -kh,
(5.8)
- k h > fi,
(~(a) is the column vector
( K - ( k a ) + L-(koO~ Q(o~) = \ K - ( k a ) - L-(ko~)/'
(5.9)
~(o~) = k-' ~,(kot) = (a 2 - l) '/2
(5.10)
and whose branch is chosen such that ~(0) = - i .
(5.11)
As before, the integration path runs along the real axis, just above the left hand branch cut and below the right hand cut. For matching purposes we now introduce the inner variables via the transformation (3.12), and holding these fixed we expand 4),, 4)2 in powers of e. After considerable algebra and some manipulation it is found that 4)(2.a/2) = e,/2{cc,R-3/2 cos(3 O/2) + bb~R -'/2 cos(O/2)}
+ ebb2+ e3/2{(db, + cc2)R-'/2+ bb3 R'/2} cos(O/2),
(5.12)
where
c, = - ( k / ( 2 ~ ) ) '/2, b, = 2ic,,
c2 = -2ic,/32
ba=-2c,(/3,+/32),
(5.13, 5.14) (5.15,5.16)
b2 is a constant which has not been determined as it is not required in the final solution, and/32 and/3, are found in the expansions for K - , L-, (A18), (A19) respectively.
6. Matched solution To obtain the asymptotic solution for small stagger all that remains is to determine the constants b, c and d in the outer solution. This is achieved by the application of the matching principle (3.17) with
291
I.D. Abrahams, G.R. Wickham / Acoustic scattering by plates
n = 2, m = 3/2. Thus (4.28) and (5.12) are equated:
e 1/2{1AR-3/2 cos(3 0 / 2 ) - A R - 1/2 cos(O/2)} + CoGo(e) + C1 G1 (e)R 1/2 c o s ( 0 / 2 ) + ½C1Gi ( e)R-1/2 c o s ( 0 / 2 ) - ¼Be3/2R-1/2 c o s ( 0 / 2 )
-- e 1/2{cc~R-3/2 cos(3 0 / 2 ) + bbtR-I/2 cos(O/2)} + ebb2 + e3/2{(db~ + cc2)R-~/2+ bb3R ~/2} c o s ( 0 / 2 )
(6.1)
which immediately reveals that the eigensolutions must have the orders
Go(e) =- e,
Gl(e) =- e 3/2.
(6.2, 6.3)
Equating the various powers in R gives
¼A = cct,
A = -bb~,
(6.4, 6.5)
Co = bb2,
CI = bb3,
(6.6, 6.7)
½C~ - ~ B = db~ + cc2,
(6.8)
which are rearranged to yield the following constants:
c =-(2,n/k)~/2A/4,
b =-i(2.tr/k)~/2A/2,
(6.9, 6.10)
d = (2~r/k)l/2[2Afl~-iB]/8, Co =-ib2(2xt/k)~/EA/2,
and
(6.11) Cl=-iA(13~+fl2),
(6.12, 6.13)
where A and B are written in (4.9) and (4.10) respectively. The first two terms of the correction potential, ~bc, have now been obtained and so finally the scattered solution is given by da d p s ( x , y ) ~ d p . ( x , y ) - - ~1 I_ °~o( e b + e 2 d + e 2 c o t / k ) e -i'~x Vo(a,y) Q ( a ) y(a'----~
2~r
c(fl~-/32) L - ( a ) e -i"x Vo(a,y)
_
y(a)'
k(x2+y2) ~/2 = 0(1),
(6.14)
where ~b, is written in (3.5)-(3.7), Vo(a, y) is given as V(a, y) in (2.3) with a -- O, or alternatively
V o ( a , y ) = ~'(a/k, ky),
and
Q(a) = Q(a/k).
(6.15, 6.16)
7. Equivalence of exact and matched asymptotic solutions, and concluding remarks Suppose we are given an inverse Fourier transform of the form ~b(x, y) = a ~
e -i'x-~(~)y O ( a a ) dot,
(7.1)
where y ( a ) and the integration path are defined as in Section 2. We wish to determine an asymptotic expansion of this expression in the outer region, defined by (~,)7)= O(1) where ~ )7 are given by (3.11), as ka = e-~ 0. Changing variables to (~,)7) and letting a = ,~/a gives ~b = [ ~ e x p { - i t ~ / e -(of 2- e2)l/2)7/e} tlb(,-~) d~7.
(7.2)
LD. Abrahams,G.R. Wickham/ Acousticscatteringby plates
292
When e is small it is reasonable to assume that the major contribution comes from a region near the origin on the integration path. Thus we write
I ~.1/2 4' = J - ' / 2 e x p { - i 6 ~ / e +
if%f / L J-~
j
l/2j
- (6 2- e2)~/2y/e}
@(6) d~
e x p { - i 6 ~ / e - ( a 2 - e2)'/zy/e} @(a) da,
(7.3)
where the limits of integration in the first term have been chosen to lie within the intervals (-1, - e ) and (e, 1) respectively. For ~ # 0 it is clear that the contributions from the infinite integrals are exponentially small and for the first term, since 6 is uniformly small on the range of integration, we may replace @(~) by its asymptotic expansion @(~) (say). Thus, in the outer region, (~, ~) = 0(1), (7.1) is approximated by A
i
~1/2
4~~ J-~'/~ exp{-ia~/e - (a 2-
e2)'/2y/e} ~(~i) d 6
(7.4)
to the same order in e as ~ ( a ) approximates @. It now follows that if we wish to show that (7.1) is asymptotically equivalent to some alternative Fourier transform, then we merely have to verify that the corresponding @(aa) are asymptotically identical at the origin. That is, if
d~j(x,y)=a f f exp{-icex-y(ce)y} @~(aa)doe, j= l,2,
(7.5)
where @~(ota) are slowly varying in the neighbourhood of the origin, then ~b,(x, y) ~ ~b2(x, y), as e-~O and
(7.6)
(~,~)=(x,y)/k=O(1),
@l(aa) ~
@2(aa)
if
as a ~ 0.
(7.7)
In what follows we shall use this principle to compare (2.1) with the outer solution obtained in Section 6. However, note that (2.2) and (2.3) are more complicated than the model of (7.1) in that our @(a, y) is actually a function of a/k as well as aa. Since a/k is not uniformly small on a e [ - e l/z, e~/2], we must expand q~ for small aa, uniformly in a/k on the range of integration. Returning to the asymptotic solutions of the integral equations (2.18), (2.19), derived from the exact analysis of [1], we obtain as e ~ 0
N*(a)= M*(a) =
-
iu(kh) e+e--'4~'(kh)(a/k) 2k-----~
iu(kh)2k_..._~e +
[
kh
I~(kh)
}
+O(e3),
(7.8)
~,(kh)(ot/k)~kh--k2h2 ~(kh) +O(e3),
(7.9)
and from (2.11),
S*(a) =i~'(kh)2khe+ e___~{ ~,(kh)(a/k)+l.~(kh)}+O(ea) ' k h T*(oQ=
iv(kh) 2kh e+
~'(kh)(a/k) t k2h----T ~(kh) kh
(7.10) +O(e3),
(7.11)
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
293
where note that it has only been necessary to expand e i~ about the origin. Hence, if we write ~ = - k cos 0 again, then i e v(Oh 2(,-a){(:-i)-2-'~\-1
(K-(,))-'F(a)
iev ( 1 1 l)+ie('lk)(~ ~)
o
-2(Uk) 2
+(~)]
+ O(e3~l(-K " J \ L+(~) ]" This may be premultiplied by (0 -'a~
(7.12)
K - ( a ) to yield
~)K-(a)(K-(~))-'F(a)
L-(=)~r=/l 7)+le(....~)(l 11) -L-(o0/L \0
ie "r~Oh (K-(a)
.
~(7-~) \K-(o,)
e2(~-a]2(1
~-a
1
e2(~-a] v(kh) -T\---~]
\ L+(~") ]
(7.13)
or rearranging gives the final result i e v(¢)h y(a)
q~(a,y)=--
iE
+~~
f L-(,,) Vo(,',, y ) / ( ~ - '~x_l/g-(~) ~g-(,~) -L-(o,)]\ Z+(~) / I.
Q(a) (-K+(~')+
e 2 [v(kh)
+4-~ t - - - ~
2
L+(sr))+-~2
g+(~') - sr(L+(~') -
Q(a) (L+(~')- K+(sr))
K+(~')))
O(a)
e 2 -v(kh) - L-(a) (L+(~)- K+(~)) (_I) +O(e3) } , 4k kh where
Vo(a,y) is V(a, y) (eq.
(2.3)) with a = O, and
(K-(a)+L-(a)] Q(a)=\K-(a)_L-(a)]"
(7.14)
Q(a) is (7.15)
It is a simple matter to show that (7.14) in (2.1) is identical to the expression derived via matching, (6.14) correct to O(e2). We could further prove that the inner solution is in agreement with ~b~ from Section 2 if we did not expand the e ~a~ terms in (2.18), (2.19). As this region is of little physical interest this calculation is not performed here.
294
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
We can conclude that both solutions show to leading order that the field scattered by the staggered plates is identical to the field scattered by the unstaggered plates. The effect of the stagger first manifests itself in the outer field expansion at O(ka), and this correction potential is an eigensolution of the unstaggered plates problem with singular behaviour at the edge of the upper plate. This potential behaves like r -~/2 as r = (x2+y2)~/2~ 0. The next order correction potential ( O ( k a ) 2) is similarly an eigensolution of the unstaggered problem but it has r -3/2 behaviour as r-~ 0. As a final exercise it is useful to compare the asymptotic solution with that obtained from the full numerical analysis of the integral equations (2.8-2.10) performed by the authors elsewhere [5]. In particular it is important to establish just how small the dimensionless length ka has to be in order to give an accurate estimate of the sound field both inside and outside the duct. In Figures 2a and 2b the magnitude of the scattered acoustic field far from both plate edges (kr = 30.0) is shown plotted as a function of the observation angle O. Note that the curves are discontinuous because the specular field has been subtracted from the total scattered potential, i.e. the reflected wave is removed in 2~ > O > 2 ~ - 0 and the incident wave is added in the shadow zone (0 < O < 0). The value O = 0 ° corresponds to a point on top of the u p p e r plate and O = 360 ° is a point on the underside of the lower plate. The incident wave makes an angle 0 = "#/4 with the horizontal and the plate spacing is chosen as kh = ¢ / 2 . In Figure 2a the stagger is chosen as ka = 0.1 whereas in Figure 2b this is increased to ka = 2.0. The unbroken line corresponds to the numerical results of [5], and the large dashed line is that of the far field predicted here by both the matched asymptotic analysis and solution using the approximation given in (7.8-7.11). In Figure 2a the two curves are indistinguishable but as the stagger increases the two results have deviated quite markedly from each other for certain angles O. The small dashed line in Figure 2b is the asymptotic solution employing the integral equation approximation given in (2.18, 2.19) and (2.11). This result appears to give slightly better agreement with the exact result over the range 0 < O < "rr. Figure 3 shows the variation of the sound field magnitude (at kr = 30.0) as a function of ka at a particular observation angle O = 3~r/2. The incident wave angle is chosen as 0 = 3~r/4 and the plate spacing is again ~/2. The asymptotic result (dashed line) is in very good agreement with the exact result (unbroken curve) for stagger lengths ka less than approximately 1.3 but then deviates rapidly as ka continues to increase. It may be noted that the
3.0 ~ (a)
4.0
(b)
2.5
-•3.0 .~2.0
D_ E
"~ 2.o
-"
,
b~
~ t.O
t~-0.5 o.o
..... ""~I '""='~68 ...... ~ ..... Observation Angle Theta
0.0
k i
,,,y v= "
6 ........ i~ ........ iN ............ Observation Angle Theta
Fig. 2. Magnitude o f scattered sound field (with specular terms omitted) against observation angle O for kr = 30.0, 0 = ~/4, kh = ~r/2 and (a) ka =0.1, (b) kh =2.0. Unbroken line is exact solution, large dashed line is asymptotic result from (7.8-7.11) and small dashed line is approximation using (2.18, 2.19, 2.11).
LD. Abrahams, G.R. Wickham / Acoustic scattering by plates
295
2o !
LO.5
-~
\
3
o.o
~.o 2.0 3.0 Dimensionless length ka
~.o
Fig. 3. Magnitude of sound field against dimensionless length ka for kr = 30.0, 0 = 3Tt/4, O = 3 ~ / 2 and kh = ~r/2. Unbroken line
is exact solution and dashed line is asymptotic result.
approximation employing N * , M* etc. from equations (2.18, 2.19) rather than (7.8-7.11) in fact leads to worse results as/ca is increased. This is partly due to an artificial resonance built into this solution near k a = "rr. F i g u r e s 4 a a n d 4 b c o n f i r m t h a t t h e a s y m p t o t i c
r e s u l t is v e r y g o o d
f o r k a <~ 1.3, a n d s h o w t h e
amplitude of the propagating duct mode as a function of ka. In these figures kh is again ~ / 2 and in 4a, 0 = xr/2 whereas 0 = 3"tr/2 in 4b. In conclusion the method of matched asymptotic expansions is appealing and powerful in its relatiohship with the physics of complicated problems such as that considered here and, analytically, it can be simpler to apply than exact techniques (e.g. as given in [1] and [5]). Given that there is not as yet a general and rigorous justification of the method, it is helpful to check, where exact solutions are known, that it does indeed produce correct asymptotic expansions. This paper has not only given the asymptotic solution of
25.0
2.5
(a)
(b)
" 0 20.0
Q) " 0 2.0 , "-I .--
I I |
O.
',
~.~ 15.0
iII /\\
,'/
-
I
~D . ~ I0•0
\
\
~
1.0,
~
0.5.
"'........
\ ~
5.O
°'°o.o. . . . . . . . 'i.b ";.b ........ Lb ........ Dimensionless length ko .
.
.
.
.
.
.
.
~,:o
0.0
0•0
1.0
2.0
3.0
g I
4.0
Dimensionless length ka
Fig. 4. Magnitude of propagating duct mode against dimensionless length/ca for kh = ~ / 2 and (a) 0 = ~r/2, (b) 0 = 31r/2. Unbroken line is exact solution and dashed line is asymptotic result•
296
LD. Abrahams, G.IL Wickham / Acoustic scattering by plates
the field scattered by slightly staggered plates, and validated the matching principle employed herein, it has also shown the wide range of ka for which the solution is valid.
Appendix A The following scalar factorizations were introduced in [1] and are repeated here for convenience. Let K ( a ) = -½y(a)(1 + e -~(~)h)
(A1)
where y ( a ) is written in (2.5), and introduce the product decomposition
(A2)
K ( o ) = K+(a) K - ( a )
where the + ( - ) denotes a function which is analytic in the upper (lower) half of the complex a plane (both half planes are assumed to contain the real line if k has slightly positive imaginary part). It is found that {(1 - k 2d._l/2) 2 1/2;:.ladn_l/2} exp{+
K=~(a)=(a+k)l/2exp{i½~r;:fl(a)-T±(a)}fi
iadn-l/2},
n=l
(A3) where T+(a) _ h~r_7,a.____.~arccos(a/k), ,
2~r
f,(a)
ixr + ah 4
T-(a) = T+(-a),
. a__h_h {1 _ C +in( ~r ) }
1 2"#
-'~
(A4, 5)
h
d.-,/2 - (2n - 1),rr'
,
(A6, 7)
and C = 0.5772... is Euler's constant. Also we take
(A8)
L ( - ) = -½y(a)(1 - e -~(~)h) = L+(a) L - ( a ) ,
(A9, 10)
L+(a) = L - ( - a ) ,
in which case L±(a) = (½h)~/2(a + k)
expIi½"rrwf2(a)-
H {(1 - k2d2.)~/2~:iad.} exp/+iaa.},
T~(a)}
(All)
where
A(a)
2¢ri + ah 4
i a~h { 1 - C+ln(4~-~)}
d.-
'
h 2nqv"
(A12, 13)
It can be shown that the following asymptotic properties are held by the various factors:
K+(a) =---~ e~"/'{1 + kOJ,, + O(1/~)} al/2
L+(,,) = - ~ - ei~/'{1 + k#~/~ + O0/,,~)}
(A14)
[al-~ oo, 0~
LD. Abrahams, G.R. Wickham I Acoustic scattering by plates
ot 1/2 e3i~/4{1 K-(,',) =--~-k/32/a+O(llot2)}
297 (A16)
l a l + ~ o , -~r <~a r g ( a ) <~ 0, L-(ot) -- ~ - ~3i~r/4~f 1 ,.-kfltla+O(lla2)}
(A17)
where 1 ikh /32 = ~ + ~ - ( 1 - 2 C )
I
+i ~ .{(1-k n=l
kh
8
ikh l n ( _ ~ ) i~r 4¢r +l-~h
2 2
dn-1/2) 1/2 - 1 } + ~ k1 kd,,-l/2
2 2 dn-t/2] J
(A18)
and /31=l+ikh kh ikh (4~r) ~ - (1 - 2 C ) - - ~ - + ~ - ~ - In ~
+i ~ .=!
[
i~r 6kh
122]
((1-k2d2)'/2-1}+-2k
d.
kd.
(A19)
"
We can also write the ratio L + ( a ) / K + ( a ) as
L+(a)
K+(a)-l+~hh)+o(1/t~2),
(h20)
lal -~co, 0~
where v ( k h ) = k h ( f l l - / 3 2 ) =¼(2kh-i,rr)+~ri
~ {2nf,(kh)-(2n
-1)f,-t/2(kh)},
(A21)
and f , ( k h ) = (1 - k2d2) ~/2-1.
(A22)
References [1] I.D. Abrahams and G.R. Wickham, "On the scattering of sound by two semi-infinite parallel staggered plates I. Explicit matrix Wiener-Hopf factorization", Proc. Roy. Soc. London A 420, 131-156 (1988). [2] A.E. Heins, "The radiation and transmission properties of a pair of semi-infinite parallel plates-I", Q. Appl. Math. 6, 157:-166 (1948). [3] A.E. Heins, "The radiation and transmission properties of a pair of parallel plates-IF', Q. Appl. Math. 6, 215-220 (1948). [4] I.D. Abrahams and G.R. Wickham, "General Wiener-Hopf factorization of matrix kernels with exponential phase factors", SIAM J. Appl. Math. A 50 (3), to appear (1990). [5] I.D. Abrahams and G.R. Wickham, "On the scattering of sound by two semi-infinite parallel staggered plates II. Evaluation of the velocity potential for an incident plane wave and incident duct mode", Proc. Roy. Soc. London A 427, 139-171 (1990). [6] B. Noble, Methods Based on the Wiener-Hopf Technique, Pergamon Press, London (1958). [7] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford (1964). [8] D.G. Crighton and F.G. Leppington, "Singular perturbation methods in acoustics: Diffraction by a plate of finite thickness", Proc. Roy. Soc. London A 335, 313-339 (1973).