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Acoustic radiation from slender bodies
Wave Motion 18 (1993) 371-381 Elsevier
371
Acoustic radiation from slender bodies M. Tran Van Nhieu Thomson Sintra ASM, 94117 Arcueil, France
Received 17 August 1992, Revised 15 June 1993
A general approach is proposed to compute the acoustic field radiated by axisymmetric slender bodies at medium (Ka -- 1) and low (Ka << 1) frequencies. A multiple-scales technique is developedto determine the sound pressure in the near field. The far field solution is built up as a continuous distribution of sectorial spherical harmonics and their densities are deduced by matching with the near field solution. Some applications are presented to support the analytical results.
1. Introduction Because of their particular geometry, slender bodies have been the subject of numerous studies and several approximate methods have been proposed to calculate the sound pressure field; among these papers we mention in particular Pond [ 1 ], Junger [2], Chertock [3] for radiation problems and Geer [4], Fedoryuk [5], Ahluwalia and Keller [6] for scattering problems. All methods are based upon an asymptotic expansion of the solution with respect to the slenderness ratio e which is the ratio of the maximum body cross-section radius Rma x tO the longitudinal body half-length L. However most of them are restricted to low frequency as they are similar to the classical hydrodynamic slender-body theory which can be rigorously derived from the matched asymptotic expansion method [ 7]. Two key points limit their extension to high frequency: i) in the "inner region" scaled by Rma x the boundary condition is assumed to vary slowly along the longitudinal axis and ii) the "outer solution" that is valid on the lengthscale of order L has been built up as a continuous distribution of monopoles and dipoles along the body axis. In a series of papers, we proposed a formalism based on the matched asymptotic method to deal with scattering by perfectly rigid slender bodies at medium frequencies [ 8-9 ]. The extension to higher frequency has been made possible firstly by considering the inner region as a local region scaled in all directions by Rma x around a current point on the body axis; as a consequence of the change in the spatial structure, the inner solution is then function of two stretched coordinates instead of only the radial boundary-layer variable as in previous methods and secondly by describing the outer solution as a continuous distribution of sectorial spherical harmonics instead of sources and doublets; the choice of such elementary singularities was suggested by the form of the inner solution and by the matching condition. In many applications in acoustics, it is interesting to compute the sound pressure radiated from a vibrating slender shell excited by mechanic forces or incident acoustic waves. In these cases strong fluid-solid interactions can occur and resonance effects could be used to determine the elastical parameters of the shell. Very often the frequency band of interest which leads to acoustic wavelengths of order Rmax is in the medium frequencies ( K R m a x = 1 ) and the vibrations of the shell are described in terms of the in vacuo eigenfunctions with structural wavelengths of different orders of magnitude. In order to address this problem, we have shown that the multi-variable expansions technique is available for treating this kind of singular perturbation problem; the method leads to a solution which 0165-2125/93/$06.00 © 1993- Elsevier SciencePublishers B.V. All rights reserved
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M. Tran Van Nhieu / Acoustic radiation,[}'ool slender bodie,s
is asymptotically equivalent with our previous results in the "near and intermediate" fields for rigid bodies [ 10] and is able to deal with the fluid-solid coupling of a finite cylindrical shell insonified by an incident plane wave Ill]. The purpose of this paper is to extend the work presented in Ref. [ 11] and to propose an unified approach for slender-body acoustics. The development of the theory is presented in Section 2 where the derivation of the governing equations is restricted to axisymmetric slender bodies. In order to illustrate the application of the formulas derived here, several examples concerning radiation and scattering from slender bodies are presented in Sections 3 and 4. It is found that the analytical results of previous works are reproduced in a systematic fashion while numerical computations show good agreement with benchmark cases and T matrix calculations.
2. T h e o r y
Consider a slender body of revolution immersed in an unbounded fluid. The body is placed with its longitudinal axis on the Z axis; let the equation of the body surface in cylindrical coordinates be
R=Rma,,F(Z/L)
(2.1)
where R is the radial coordinate. We denote the body half-length by L and the maximum transverse section radius A sound pressure field is induced by the body surface vibration; we assume that the normal component V of the velocity can be represented by a discrete double Fourier series with respect to the longitudinal variable Z and the azimuth angle q~; for simplicity it has been taken to be a cosine series of q~.
b y R m a x.
V = V o e i~,, ~ n~O
U,,p(Z/L) cosnq~e ix'z
~ p=
(2.2)
--~
where the axial wavenumbers Kp and the complex coefficients v,,i, are given by the Fourier analysis of the body surface vibration. In the above, capital letters denote dimensional variables. It is convenient to introduce dimensionless variables defined as follows
r=R/L;
z=Z/L;
e.=Rmax/L;
k~, =KpL;
k=KL;
t,= V/Vo;
p = P / p , wR . . . .
Vo,
(2.3) where e is the slenderness of the body, Pa is the fluid density, K = w/c is the acoustic wavenumber, Vo is a measure of the velocity amplitude and P is the radiated sound pressure. Then the Helmholtz equation becomes 1 Op 1 OZp O2p 82P + - - - + - - + -Or 2 r Or r 2 0(~ 2 OZ 2
__
(2.4)
+kZp=0
and the boundary condition on the surface reduces to
r=EF(z);
Izl <1:
OP - ¢ F ' ( z ) Oz Op = i , - I E E v,,,,(z) cos nq~e 'k'~ 0--7 n
(2.5)
p
where the symbol ' denotes the derivative with respect to the argument; the e - i ' ° t time dependence and the double summation limits are suppressed in the above equation as in the remainder of this paper. The present analysis is made under the following assumptions: (i) the body is slender: E << 1; F and its derivatives are regular functions of z and are of order unity. (ii) the n, p component of the normal velocity is the product of a regular slowly varying function t'np with a rapidly
M. Tran Van Nhieu / Acoustic radiation from slender bodies
373
oscillating phase function along the z axis: vnp are supposed to be of order unity and kp of order 6 - ~ with 8<<1. (iii) the normalized acoustic wavenumber k is large k >> 1.
2.1. Multi-variable expansion procedures In the following, we propose a general formalism based on the multiple scale method to deal with the acoustic radiation from slender bodies. In addition to the geometrical variables (r, z, ~), we introduce the boundary-layer variable r and the short longitudinal scale g defined as
F=rlE,
£=Z/6.
(2.6)
According to the multiple scale method [ 12 ], the radiated pressure p is considered as a function of the independent variables previously introduced. We replace the r and z derivatives by 0 Or
--
--*
8 Or
--
+
l0 . . e OF'
.
0 3z
.
*--
0 18 + - - Oz ~ O~
(2.7)
and using (2.7), we find the following equation from eq. (2.4) __
_ _
LOOp
OZp + 1F Op + F2 O ~ OF 2 OF
2
+(e/6)z
OzOf]
+2EL0r0F
0£ 2
+(kE)Z p
L Orz + r O---r+ 0z21 = 0
(2.8)
while the boundary condition (2.5) becomes
F=F(z);
Izl < 1 :
0p + E l 0~rp - ( e / 6 ) F ' ( z ) . ~ ] O-r
Op = i E E Unp(Z) COS n t l ~ e i(/p')z-. -- E 2 F ' ( z ) -~Z n p
(2.9)
To proceed further, we regard 8, k and kp as unknown functions of e and their relative orders of magnitude are chosen by applying the principle of least degeneracy (in French "principe de moindre drgrnrrescence" ). It can be easily shown that the richest equation which retains the largest possible number of terms occurs if 6 = O(E) and k - 1= O(E). So we take 3-- E and introduce the e-independent parameters/~ and/~p defined by (2.10)
=,kp = 6i..
Now we will construct an approximate solution of (2.8)-(2.9) in the form oo
p(r, z, q~, F, ~, e) = y ' ~ ~ e" elg"Z-cosn*P'~p(r, Y, z) m=O n
(2.11)
p
where Pn'p and its derivatives are bounded functions of (r, f, z). Substitution of the expansion (2.11 ) into (2.8) gives Y'. E cos nCI)ei~pe[L~oP~,p+ e(L~oP~,p + L~IP~p) +" "'] = 0 n
p
where the operators LP~oand L~l are defined by (2.13)
(2.12)
M. Tran Van Nhieu /Acoustic radiation from slender bodies
374
a2
13
n2
(0 2
U"°= --OF 2 + ---F OF +)~2_ ?~'
O)
L~, =2 ~ - ~ +i/~, ~
(2.13)
and
= f ( k 2 - k ~ ) ~/2 Xp
/.i(k 2 _ k 2),/2
for Ik, I < k for Ik~,l > k
(2.14)
with .~p = eXp. Since the terms in brackets in the left member of (2.12) are independent of • and ~, we have at the first two orders L,,nOuo -- nO = 0 ,
(2.15)
p 1 p 0 L,oP,r = --L,lP,,p •
(2.16)
We seek a solution of (2.15 ) in the form
P~',, =anp( r, z)H~"t)~p?)
(2.17)
and substituting this expression into the right member of (2.16) yields p 1 L.oP~p =g~p(t7)
(2.18)
where gnp(r-) = --2
Oanp
Oanp
~,--~r H~'~'+ifcp ~
H'",
)
.
(2.19)
The determination of anp is obtained by requiring that the right-hand side of (2.18) contains no secular terms in the limit of f ~ ~; in order to analyze the asymptotic behaviour of the solution of (2.18) for large ~, let we set
F--+~: P],p= ~,p/X/~.
(2.20)
Inserting (2.20) into (2.18) and expanding the different terms in 1 / ~ in the two members, we find the following differential equation for ~np at the first approximation
dzaglnp ( XP--~r Oa,p +fcP 0a"p]eiX/ dr 2 +X2aIe"p=-2(-i)"I/(2i/TrXP) OZ ]
(2.21)
where )~2 are real positive or negative numbers. It is readily seen that the right member of (2.21 ) gives rise to a term in the solution that is proportional to g. Now according to the classical multi-variable expansion procedures, the expansion (2. l 1 ) is supposed to be uniformly valid in the variable ~ and therefore this term must vanish to keep (2.11 ) well-ordered with respect to e when r-~ ~; it follows that ~)anp + f~p ~)anp
-ff-r
=0.
(2.22)
Thus the functions anp verify a first partial derivative equation in the geometrical variables (r, z) ; the integration of (2.22) is straightforward and yields
a,p( r, Z)=bnp(Z-rkn/X p)
(2.23)
M. Tran Van Nhieu / Acoustic radiation f r o m slender bodies
375
where at this stage b~p are arbitrary single variable functions. Outside the body, when r and r ~ 0 it can be shown that pOp _~ oo by using the asymptotic expression of the Bessel functions for small arguments in (2.17). Since pOp must be bounded, we have
b~p(z) = 0
for Iz[ > 1
(2.24)
while from the boundary condition (2.9), we obtain the following expression which determines completely the functions bnp
v.~(z)
b.p(z) = i ~pH~),n[~pF(z)]
for Izl < 1.
(2.25)
From ( 2.11 ), (2.17) and (2.25), we get the leading term of the radiated pressure
p(r, z, qO, ?, ~, E) = ~ ~ e~g"ecos n ~ b n p ( z - r k p / ~ p ) H ~ ' ( ~ p r - ) . n
(2.26)
p
However as (2.26) does not satisfy the radiation condition at infinity, the above expression is only valid in the "near" field at a distance of order O(L) from the body.
2.2. Farfield radiated pressure In order to seek an available expression of the radiated pressure p= that matches (2.26) in the far field, we build up p~ as a superposition of sectorial spherical harmonics which has been previously used [8,9] to represent the sound pressure scattered from a slender-body; these elementary solutions are distributed along the z axis from to oo and their strengths Bnp will be found by matching. Anticipating the matching procedure, let we set p~ in the form k cos nqb r po: = -- ~ V~ ~ J (r/p)~h~)(kp)B~p(t) e ikpr dt "iT n
p
~P
(2.27)
_
where p = ~/(r2 + ( t - z ) 2) ; 7p =Xp/k. In the farfield, we replace h(n 1) in (2.27) by its asymptotic expression
kp>> 1;
h ( ~ l ) ( k p ) = ( - i ) " + l -eikp kp
(2.28)
This yields
-i
e i(kpt + kp)
( -i)"
( r/ p)nB,p( t) - -
poo=--~y'~cosn~ "iT
n
p
"~P
_J
dt .
(2.29)
P
Since the phase of the integrand in the right-hand side of (2.29) is large, the classical saddle-point method can be used to estimate the integral term denoted by l,p. According to Appendix C of reference [ 11 ], l,p can be approximated for I k p l # k b y
I,p( r, z) -- ~/( 27ri/ Xpr) e itkpz+xpr) B,p( z - rkp/ Xp) y p .
(2.30)
Inserting the above expression into (2.29), we have
p= = ~ ~ 1/(2/~rixpr) ( -i)" cos nClgB#p(z-rkp/Xp) e "kpz+xpr) . n
p
(2.31)
M. Tran Van Nhieu /Acoustic radiation from slender bodies
376
Now in order to match p and p~ in the farfield, we reexpress (2.26) in the geometrical variables as
p(r, z, q), e ) = ~ ~ e ik'': COS ndPb.,,(z-rkt,/xp)H~,"(Xj)
(2.32)
and replace the Hankel function by its asymptotic expression for large arguments; this gives
p(r, z, q), e) -- ~ y" (-i)"~/(2/i~rxpr) e "k'':+x~'') cos nq)b,,p(z-rkp/xp) n
(2.33)
p
Then the matching of p and p~ given by (2.31) and (2.33) yields Bnp =
(2.34)
b,,p.
Using (2.24), (2.25), (2.27), (2.34) and after some straightforward calculations, p~ becomes i~ i
P~-
rr ~,, ~
cos nqb (
L,,,p(t) _J, H'"'t~FF(t)],,
y,':+'
(2.35)
(r/P)"h~']'(kp)eikp'dt
which satisfies all the required conditions: (i) the Helmholtz equation; (ii) the radiation condition at infinity and (iii) the matching with p in the farfield. Now we can calculate the far field radiated pressure from (2.35); by multiplying p~ by p, wR ..... Vo and in the limit of Po = ~ + z 2) --* ~, P~ can be expressed in dimensional form as follows L
e ikm
cos nCI)
P ~ - "rrpoL p, wy'~,, ,,
1"i,
( Vo t',,p(Z) ( - i K sin O/Fp)" jLH~t,,[ °
e (K'
K-)Zdz
(2.36)
where 69 is the angle between the Z axis and the direction of observation, l'p = xp/L, K. = K cos 6} and R(Z) is the transverse cross section radius at a distance Z from the origin. Equations (2.26) and (2.36) are the principal results obtained in this paper; they give the expressions of the sound pressure radiated by a slender body of revolution in the near and far fields. We will apply the previous results to compute the acoustic fields in the following applications.
3. Radiation from slender bodies
In this section, we will calculate the farfield pressure radiated from a slowly varying vibration surface and from a long finite cylinder.
3.1. Radiation from a slowly varying vibration surface In the case of a slowly varying surface vibration along the Z axis where [Kt,[ << K and [KI,Rmax[ << 1, ~, = ~/1K~ -K21 can be approximated by K; then from (2.36), we obtain L
P~ -
with
e -iK:z dZ p,,c ~ cos n ~ ( - i sin O)" f H(I),rKR. t,,,(Z) .1! rrpo L ,, l Jx t z ) n L eik'q~
(3.1)
M. Tran Van Nhieu / Acoustic radiation from slender bodies
v.(Z) = Vo E U.p(Z) e i/(pz .
377 (3.2)
P At low frequency KR << 1, by using the asymptotic expression of the Hankel function for small arguments, we then obtain for the first two terms of the expansion
P~-
eikp°
2rrpoLPaW
L f
[ i ~ r R v o + K s i n 6)cos
dPAvl] e-ig:Z dz
(3.3)
--L where A = "rrR: is the cross section area at a distance Z. Now it is interesting to compare (3.3) with the Chertock results [3] for flexural and accordion vibration modes. For flexural modes when each section vibrates in a direction perpendicular to the Z axis, we have in the Chertock notations
vo(Z) = 0 ,
vl(Z) = - i q o w ~ ( Z )
(3.4)
where qt is the mode function describing the body vibration. Substituting (3.4) into (3.3) yields
P ~ = i ~ eik'q~ qopaKw
L f 2 sin O c o s • A~e-L
K=ZdZ
(3.5)
which is consistent with the results of Refs. [2,3]. For accordion modes where the displacement normal to the body surface is due to a longitudinal displacement of each cross section, the Fourier components of the normal velocity are equal to
v o ( Z ) = i q o w ( ~ - ~ +vRd-d-~),
v,(Z) =0
(3.6)
where uis an effective Poisson's ratio as defined in Ref. [3]. From (3.3) and (3.6) we get the following expression for the farfield radiated pressure
L
P~ - 47rpo~ L qop, Oa2
~--~
+ u2A
e -iK:z
dZ
(3.7)
--L which is equal to the leading term of P~ given by Chertock [ 3 ]. Therefore in these two vibration modes, our general result is in agreement with existing low frequency solutions.
3.2. Radiation from a long cylinder A check on the accuracy of equation (2.36) can be made using the following scheme that is proposed in Ref. [ 13] : for a given object of surface S, we choose a body surface vibration for which an analytical solution exists. Consider one or several point sources of given source strengths located in the region Ri interior to S but that no object is present and assume that all the space is filled with the same fluid. We can easily calculate the normal component of the fluid particle velocity on S and the farfield pressure radiated by these point sources. From the distribution of velocity, we can apply (2.36) and compare the computed results with the acoustic pressure obtained previously. We use the scheme described above for the finite cylinder problem and for point sources symmetrically located on the longitudinal axis of the cylinder with respect to the center of the object. In this case, the expression (2.36) becomes
378
M. Tran Van Nhieu / Acoustic radiation f r o m slender bodies
90
1 5 ~ 0 18~ 0 21o
0
24o 270
Fig. 1. Farfield directivity for a finite cylinder of slenderness ratio l / 10 for Ka = 2.6. The point source solution corresponds to the circle of level 60 dB. 90
1 5 ~ 0 18~ 0
0
21~ 0 3 3 0 SLENDER-BODY
SOLUTION
POINT
SOURCES
SOLUTION
Fig. 2. Farfield directivity for a finite cylinder of slenderness ratio 1/ I 0 for Ka = 2.6. Results on the right h a n d side are obtained with the point sources solution. Results on the left h a n d side are obtained with the slender-body solution.
P~
e ikp°
~po L
2pato
1 Vovop ~p Fp H~o')'( I'pa) K.~
where a is the radius o f the cylinder and
-K:.)L KR - K :
sin(Kp
(3.8)
Kf, = "rrp/L.
It is noticed that in the cylinder case the b o u n d a r y conditions at the two end faces ( Z = ___L) are not specified in the present analysis and therefore (3.8) does not account for their acoustic effects. H o w e v e r it is e x p e c t e d that the contribution o f the end sections are only important near the end-fire direction as we will see in the two f o l l o w i n g examples. W e c o n s i d e r first one point source located at the center o f the cylinder and then two couples o f point sources of
M. Tran Van Nhieu / Acoustic radiation from slender bodies
379
opposite normalized strengths ( _ 1; _ 4) located at Z = _ 0.08L and ___0.14L. The slenderness ratio of the cylinder is equal to a / L = O . 1 and the calculation is made at the reduced frequency K a = 2 . 6 . The computation of both methods are shown in Figs. 1 and 2 where the cylinder is aligned parallel to the 0-180 degrees axis. In the first case (Fig. 1 ) there is a little discrepancy near the end-fire direction which might be explained by the end section effects as previously noted. In the second case, the minima obtained by the slender-body approximation are less important than those of the point sources solution (Fig. 2). Notice that there is a good agreement of both methods near the end-fire direction in this example.
4. Scattering from a slender-body Let a plane wave Pi incident upon a perfectly rigid slender body; the direction of insonification is supposed to be parallel to the plane xOz. The normal surface velocity V is related to P~ by the relationship e i = A e i(K!'x+K~z) ,
(4.1)
0Pi = - ipatoV , On
(4.2)
--
where O/On represents the outward normal derivative on the body surface, K i and K~z are the X and Z components of the incident wavenumber vector. From the classical expansion of plane wave with respect to cylindrical Bessel functions J,, we have
eiKiX=eiKiRc°sq'= E
i~" cos nCrpjn(KixR)
(4.3)
n=O
where En is the Neumann factor (En = 1 for n = 0 and = 2 for n > 0). The expression of the normal velocity V can be deduced from (4.2) and (4.3) as V = i ( A K ~ / p ~ t o ) e ~k~z ~
inE, cos nclgJ'(k~xF)
(4.4)
n=O
with ki= = K~L; k i = K i L a n d / ~ = Eke. From (2.2) and (4.4), we obtain vnp = 0
Vo = A / p a t o L ,
for IPl > 0 ,
V,o(Z) =kxli'"+'E,J~[/dxF(z) l -, '
Kp=o =K'z,
Fp=o =K~x.
(4.5)
Using these relations, the scattered pressure can be derived from (2.35) to yield 1
k
cos nqb
'rr
sin"~9i
po~ = - - - ~ - n
["
~.i" j
--1
J"
- -
H~nl)'
[[¢ixF( t) ] ( r / p ) " h ~ ) ( k p )
e ik~t dt
(4.6)
where the pressure is normalized by A instead of patoVoRmax and ~9i is the incidence angle defined as cot Oi = i i kz/k~.
Equation (4.6) is identical to our previous results [ 8,9] if terms of order ~ cot Oi which have been assumed to be small are neglected in the calculation. Furthermore it gives a simple and practical expression for the scattered farfield; good agreements have been found with other analytical methods at low frequency (KRm~x << 1 ) [5] and with T matrix computation at medium frequency (KRm~x -- 1 ) [ 14,15 ]. In order to illustrate the accuracy of (4.6), the farfield scattered pressure has been computed and compared with the T matrix calculation [ 15]. The bistatic
380
M. Tran Van Nhieu / Acoustic radiation from slender bodies
90
180~
0
27(? Fig. 3. Farfield scattering from a rigid spheroid of slenderness ratio I / 15 for KR,,,~,~= 4 ( 30 degrees relative to the axis of symmetry).
90
180/
f
I °
270 Fig. 4. Fartield scattering from a rigid spheroid of slenderness ratio 1/ 15 for KR.I~,x= 4 (60 degrees relative to the axis of symmetry ). angular distributions for a rigid spheroid of slenderness ratio 1 / 15 for K R ..... = 4 are shown in Figs. 3 - 5 at several aspects o f insonification. The spheroid is aligned along the 0 - 1 8 0 deg. axis and the arrow indicates the direction o f the incident plane wave. As can be seen, a c o m p a r i s o n with Figs. 2 - 4 o f Ref. [ 15] lends further support to the present method. In the same fashion, analytical expressions of the pressure scattered from a long finite cylindrical shell can be d e d u c e d f r o m equations (2.26) and ( 2 . 3 6 ) ; again the results are identical to those g i v e n in Ref. [ 11 ].
Acknowledgments
T h e present study was supported by the " D i r e c t i o n des R e c h e r c h e s et Etudes T e c h n i q u e s " , France, Contract
M. Tran Van Nhieu / Acoustic radiation from slender bodies
381
90
Fig. 5. Farfield scattering from a rigid spheroid of slenderness ratio 1/ 15 for KRmax= 4 (90 degrees relative to the axis of symmetry). D R E T N o . 9 1 3 4 8 5 . I w o u l d like to t h a n k M i s s L a u r e n c e G u e r r e f o r h e r c o m p u t a t i o n a l a s s i s t a n c e c o n c e r n i n g t h e radiation problems.
References [ 1] H.L. Pond, "Low frequency sound radiation from slender bodies of revolution", J. Acoust. Soc. Am. 40, 711-720 (1966). [ 2 ] M.C. Junger, Sound radiation by resonances of free-free beams, J. Acoust. Soc. Am. 52, 332-334 (1972). [3] G. Chertock, "Sound radiation by low-frequency vibrations of slender bodies", J. Acoust. Soc. Am. 57, 1007-1016 (1975). [ 4] J.G. Geers, "The scattering of scalar wave by a slender body of revolution", SIAM, J. Appl. Math. 34, 348-370 (1978). [ 5 ] M.V. Fedoryuk, "Scattering of sound waves by a thin acoustically rigid body of revolution", Sot,. Phys. Acoust. 27, 336-338 ( 1981 ). [6] D.S. Ahluwalia and J.B. Keller, "Scattering by a slender body", J. Acoust. Soc. Am. 80, 1782-1797 (1986). [ 7 ] M.D. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York (1964). [8] M. Tran Van Nhieu, " A singular perturbation problem: scattering by a slender-body", J. Acoust. Soc. Am. 83, 68-73 (1988). [ 9] M. Tran Van Nhieu, " A slender body approximation in scattering theory", J. Acoust. Soc. Am. 85, 1834-1840 (1989). [ 10] M. Tran Van Nhieu, "On the asymptotic solution of scattering from slender bodies by the two-variable technique", J. Acoust. Soc. Am. 91,495-497 (1992). [ 11] M. Tran Van Nhieu, "Scattering from a finite cylindrical shell", J. Acoust. Soc. Am. 91,670--679 (1992). [ 12 ] J. Kevorkian and J. Cole, Perturbations methods in Applied Mathematics, Springer, New York ( 1981 ). [ 13] H.A. Schenck, "Improved integral formulation for acoustic radiation problems", J. Acoust. Soc. Am. 44, 41-58 (1968). [ 14] M.F. Werby and R.B. Evans, "Scattering from objects submerged in unbounded and bounded oceans", IEEE Trans. Oceanic Eng. 0E-12, 380-394 (1987). [ 15] M.F. Werby, G.J. Tango and L.H. Green, "Computational acoustics: algorithms and applications", 1st IMACS Symposium on Computational Acoustics, New Haven, CT, USA (1986).
371
Acoustic radiation from slender bodies M. Tran Van Nhieu Thomson Sintra ASM, 94117 Arcueil, France
Received 17 August 1992, Revised 15 June 1993
A general approach is proposed to compute the acoustic field radiated by axisymmetric slender bodies at medium (Ka -- 1) and low (Ka << 1) frequencies. A multiple-scales technique is developedto determine the sound pressure in the near field. The far field solution is built up as a continuous distribution of sectorial spherical harmonics and their densities are deduced by matching with the near field solution. Some applications are presented to support the analytical results.
1. Introduction Because of their particular geometry, slender bodies have been the subject of numerous studies and several approximate methods have been proposed to calculate the sound pressure field; among these papers we mention in particular Pond [ 1 ], Junger [2], Chertock [3] for radiation problems and Geer [4], Fedoryuk [5], Ahluwalia and Keller [6] for scattering problems. All methods are based upon an asymptotic expansion of the solution with respect to the slenderness ratio e which is the ratio of the maximum body cross-section radius Rma x tO the longitudinal body half-length L. However most of them are restricted to low frequency as they are similar to the classical hydrodynamic slender-body theory which can be rigorously derived from the matched asymptotic expansion method [ 7]. Two key points limit their extension to high frequency: i) in the "inner region" scaled by Rma x the boundary condition is assumed to vary slowly along the longitudinal axis and ii) the "outer solution" that is valid on the lengthscale of order L has been built up as a continuous distribution of monopoles and dipoles along the body axis. In a series of papers, we proposed a formalism based on the matched asymptotic method to deal with scattering by perfectly rigid slender bodies at medium frequencies [ 8-9 ]. The extension to higher frequency has been made possible firstly by considering the inner region as a local region scaled in all directions by Rma x around a current point on the body axis; as a consequence of the change in the spatial structure, the inner solution is then function of two stretched coordinates instead of only the radial boundary-layer variable as in previous methods and secondly by describing the outer solution as a continuous distribution of sectorial spherical harmonics instead of sources and doublets; the choice of such elementary singularities was suggested by the form of the inner solution and by the matching condition. In many applications in acoustics, it is interesting to compute the sound pressure radiated from a vibrating slender shell excited by mechanic forces or incident acoustic waves. In these cases strong fluid-solid interactions can occur and resonance effects could be used to determine the elastical parameters of the shell. Very often the frequency band of interest which leads to acoustic wavelengths of order Rmax is in the medium frequencies ( K R m a x = 1 ) and the vibrations of the shell are described in terms of the in vacuo eigenfunctions with structural wavelengths of different orders of magnitude. In order to address this problem, we have shown that the multi-variable expansions technique is available for treating this kind of singular perturbation problem; the method leads to a solution which 0165-2125/93/$06.00 © 1993- Elsevier SciencePublishers B.V. All rights reserved
372
M. Tran Van Nhieu / Acoustic radiation,[}'ool slender bodie,s
is asymptotically equivalent with our previous results in the "near and intermediate" fields for rigid bodies [ 10] and is able to deal with the fluid-solid coupling of a finite cylindrical shell insonified by an incident plane wave Ill]. The purpose of this paper is to extend the work presented in Ref. [ 11] and to propose an unified approach for slender-body acoustics. The development of the theory is presented in Section 2 where the derivation of the governing equations is restricted to axisymmetric slender bodies. In order to illustrate the application of the formulas derived here, several examples concerning radiation and scattering from slender bodies are presented in Sections 3 and 4. It is found that the analytical results of previous works are reproduced in a systematic fashion while numerical computations show good agreement with benchmark cases and T matrix calculations.
2. T h e o r y
Consider a slender body of revolution immersed in an unbounded fluid. The body is placed with its longitudinal axis on the Z axis; let the equation of the body surface in cylindrical coordinates be
R=Rma,,F(Z/L)
(2.1)
where R is the radial coordinate. We denote the body half-length by L and the maximum transverse section radius A sound pressure field is induced by the body surface vibration; we assume that the normal component V of the velocity can be represented by a discrete double Fourier series with respect to the longitudinal variable Z and the azimuth angle q~; for simplicity it has been taken to be a cosine series of q~.
b y R m a x.
V = V o e i~,, ~ n~O
U,,p(Z/L) cosnq~e ix'z
~ p=
(2.2)
--~
where the axial wavenumbers Kp and the complex coefficients v,,i, are given by the Fourier analysis of the body surface vibration. In the above, capital letters denote dimensional variables. It is convenient to introduce dimensionless variables defined as follows
r=R/L;
z=Z/L;
e.=Rmax/L;
k~, =KpL;
k=KL;
t,= V/Vo;
p = P / p , wR . . . .
Vo,
(2.3) where e is the slenderness of the body, Pa is the fluid density, K = w/c is the acoustic wavenumber, Vo is a measure of the velocity amplitude and P is the radiated sound pressure. Then the Helmholtz equation becomes 1 Op 1 OZp O2p 82P + - - - + - - + -Or 2 r Or r 2 0(~ 2 OZ 2
__
(2.4)
+kZp=0
and the boundary condition on the surface reduces to
r=EF(z);
Izl <1:
OP - ¢ F ' ( z ) Oz Op = i , - I E E v,,,,(z) cos nq~e 'k'~ 0--7 n
(2.5)
p
where the symbol ' denotes the derivative with respect to the argument; the e - i ' ° t time dependence and the double summation limits are suppressed in the above equation as in the remainder of this paper. The present analysis is made under the following assumptions: (i) the body is slender: E << 1; F and its derivatives are regular functions of z and are of order unity. (ii) the n, p component of the normal velocity is the product of a regular slowly varying function t'np with a rapidly
M. Tran Van Nhieu / Acoustic radiation from slender bodies
373
oscillating phase function along the z axis: vnp are supposed to be of order unity and kp of order 6 - ~ with 8<<1. (iii) the normalized acoustic wavenumber k is large k >> 1.
2.1. Multi-variable expansion procedures In the following, we propose a general formalism based on the multiple scale method to deal with the acoustic radiation from slender bodies. In addition to the geometrical variables (r, z, ~), we introduce the boundary-layer variable r and the short longitudinal scale g defined as
F=rlE,
£=Z/6.
(2.6)
According to the multiple scale method [ 12 ], the radiated pressure p is considered as a function of the independent variables previously introduced. We replace the r and z derivatives by 0 Or
--
--*
8 Or
--
+
l0 . . e OF'
.
0 3z
.
*--
0 18 + - - Oz ~ O~
(2.7)
and using (2.7), we find the following equation from eq. (2.4) __
_ _
LOOp
OZp + 1F Op + F2 O ~ OF 2 OF
2
+(e/6)z
OzOf]
+2EL0r0F
0£ 2
+(kE)Z p
L Orz + r O---r+ 0z21 = 0
(2.8)
while the boundary condition (2.5) becomes
F=F(z);
Izl < 1 :
0p + E l 0~rp - ( e / 6 ) F ' ( z ) . ~ ] O-r
Op = i E E Unp(Z) COS n t l ~ e i(/p')z-. -- E 2 F ' ( z ) -~Z n p
(2.9)
To proceed further, we regard 8, k and kp as unknown functions of e and their relative orders of magnitude are chosen by applying the principle of least degeneracy (in French "principe de moindre drgrnrrescence" ). It can be easily shown that the richest equation which retains the largest possible number of terms occurs if 6 = O(E) and k - 1= O(E). So we take 3-- E and introduce the e-independent parameters/~ and/~p defined by (2.10)
=,kp = 6i..
Now we will construct an approximate solution of (2.8)-(2.9) in the form oo
p(r, z, q~, F, ~, e) = y ' ~ ~ e" elg"Z-cosn*P'~p(r, Y, z) m=O n
(2.11)
p
where Pn'p and its derivatives are bounded functions of (r, f, z). Substitution of the expansion (2.11 ) into (2.8) gives Y'. E cos nCI)ei~pe[L~oP~,p+ e(L~oP~,p + L~IP~p) +" "'] = 0 n
p
where the operators LP~oand L~l are defined by (2.13)
(2.12)
M. Tran Van Nhieu /Acoustic radiation from slender bodies
374
a2
13
n2
(0 2
U"°= --OF 2 + ---F OF +)~2_ ?~'
O)
L~, =2 ~ - ~ +i/~, ~
(2.13)
and
= f ( k 2 - k ~ ) ~/2 Xp
/.i(k 2 _ k 2),/2
for Ik, I < k for Ik~,l > k
(2.14)
with .~p = eXp. Since the terms in brackets in the left member of (2.12) are independent of • and ~, we have at the first two orders L,,nOuo -- nO = 0 ,
(2.15)
p 1 p 0 L,oP,r = --L,lP,,p •
(2.16)
We seek a solution of (2.15 ) in the form
P~',, =anp( r, z)H~"t)~p?)
(2.17)
and substituting this expression into the right member of (2.16) yields p 1 L.oP~p =g~p(t7)
(2.18)
where gnp(r-) = --2
Oanp
Oanp
~,--~r H~'~'+ifcp ~
H'",
)
.
(2.19)
The determination of anp is obtained by requiring that the right-hand side of (2.18) contains no secular terms in the limit of f ~ ~; in order to analyze the asymptotic behaviour of the solution of (2.18) for large ~, let we set
F--+~: P],p= ~,p/X/~.
(2.20)
Inserting (2.20) into (2.18) and expanding the different terms in 1 / ~ in the two members, we find the following differential equation for ~np at the first approximation
dzaglnp ( XP--~r Oa,p +fcP 0a"p]eiX/ dr 2 +X2aIe"p=-2(-i)"I/(2i/TrXP) OZ ]
(2.21)
where )~2 are real positive or negative numbers. It is readily seen that the right member of (2.21 ) gives rise to a term in the solution that is proportional to g. Now according to the classical multi-variable expansion procedures, the expansion (2. l 1 ) is supposed to be uniformly valid in the variable ~ and therefore this term must vanish to keep (2.11 ) well-ordered with respect to e when r-~ ~; it follows that ~)anp + f~p ~)anp
-ff-r
=0.
(2.22)
Thus the functions anp verify a first partial derivative equation in the geometrical variables (r, z) ; the integration of (2.22) is straightforward and yields
a,p( r, Z)=bnp(Z-rkn/X p)
(2.23)
M. Tran Van Nhieu / Acoustic radiation f r o m slender bodies
375
where at this stage b~p are arbitrary single variable functions. Outside the body, when r and r ~ 0 it can be shown that pOp _~ oo by using the asymptotic expression of the Bessel functions for small arguments in (2.17). Since pOp must be bounded, we have
b~p(z) = 0
for Iz[ > 1
(2.24)
while from the boundary condition (2.9), we obtain the following expression which determines completely the functions bnp
v.~(z)
b.p(z) = i ~pH~),n[~pF(z)]
for Izl < 1.
(2.25)
From ( 2.11 ), (2.17) and (2.25), we get the leading term of the radiated pressure
p(r, z, qO, ?, ~, E) = ~ ~ e~g"ecos n ~ b n p ( z - r k p / ~ p ) H ~ ' ( ~ p r - ) . n
(2.26)
p
However as (2.26) does not satisfy the radiation condition at infinity, the above expression is only valid in the "near" field at a distance of order O(L) from the body.
2.2. Farfield radiated pressure In order to seek an available expression of the radiated pressure p= that matches (2.26) in the far field, we build up p~ as a superposition of sectorial spherical harmonics which has been previously used [8,9] to represent the sound pressure scattered from a slender-body; these elementary solutions are distributed along the z axis from to oo and their strengths Bnp will be found by matching. Anticipating the matching procedure, let we set p~ in the form k cos nqb r po: = -- ~ V~ ~ J (r/p)~h~)(kp)B~p(t) e ikpr dt "iT n
p
~P
(2.27)
_
where p = ~/(r2 + ( t - z ) 2) ; 7p =Xp/k. In the farfield, we replace h(n 1) in (2.27) by its asymptotic expression
kp>> 1;
h ( ~ l ) ( k p ) = ( - i ) " + l -eikp kp
(2.28)
This yields
-i
e i(kpt + kp)
( -i)"
( r/ p)nB,p( t) - -
poo=--~y'~cosn~ "iT
n
p
"~P
_J
dt .
(2.29)
P
Since the phase of the integrand in the right-hand side of (2.29) is large, the classical saddle-point method can be used to estimate the integral term denoted by l,p. According to Appendix C of reference [ 11 ], l,p can be approximated for I k p l # k b y
I,p( r, z) -- ~/( 27ri/ Xpr) e itkpz+xpr) B,p( z - rkp/ Xp) y p .
(2.30)
Inserting the above expression into (2.29), we have
p= = ~ ~ 1/(2/~rixpr) ( -i)" cos nClgB#p(z-rkp/Xp) e "kpz+xpr) . n
p
(2.31)
M. Tran Van Nhieu /Acoustic radiation from slender bodies
376
Now in order to match p and p~ in the farfield, we reexpress (2.26) in the geometrical variables as
p(r, z, q), e ) = ~ ~ e ik'': COS ndPb.,,(z-rkt,/xp)H~,"(Xj)
(2.32)
and replace the Hankel function by its asymptotic expression for large arguments; this gives
p(r, z, q), e) -- ~ y" (-i)"~/(2/i~rxpr) e "k'':+x~'') cos nq)b,,p(z-rkp/xp) n
(2.33)
p
Then the matching of p and p~ given by (2.31) and (2.33) yields Bnp =
(2.34)
b,,p.
Using (2.24), (2.25), (2.27), (2.34) and after some straightforward calculations, p~ becomes i~ i
P~-
rr ~,, ~
cos nqb (
L,,,p(t) _J, H'"'t~FF(t)],,
y,':+'
(2.35)
(r/P)"h~']'(kp)eikp'dt
which satisfies all the required conditions: (i) the Helmholtz equation; (ii) the radiation condition at infinity and (iii) the matching with p in the farfield. Now we can calculate the far field radiated pressure from (2.35); by multiplying p~ by p, wR ..... Vo and in the limit of Po = ~ + z 2) --* ~, P~ can be expressed in dimensional form as follows L
e ikm
cos nCI)
P ~ - "rrpoL p, wy'~,, ,,
1"i,
( Vo t',,p(Z) ( - i K sin O/Fp)" jLH~t,,[ °
e (K'
K-)Zdz
(2.36)
where 69 is the angle between the Z axis and the direction of observation, l'p = xp/L, K. = K cos 6} and R(Z) is the transverse cross section radius at a distance Z from the origin. Equations (2.26) and (2.36) are the principal results obtained in this paper; they give the expressions of the sound pressure radiated by a slender body of revolution in the near and far fields. We will apply the previous results to compute the acoustic fields in the following applications.
3. Radiation from slender bodies
In this section, we will calculate the farfield pressure radiated from a slowly varying vibration surface and from a long finite cylinder.
3.1. Radiation from a slowly varying vibration surface In the case of a slowly varying surface vibration along the Z axis where [Kt,[ << K and [KI,Rmax[ << 1, ~, = ~/1K~ -K21 can be approximated by K; then from (2.36), we obtain L
P~ -
with
e -iK:z dZ p,,c ~ cos n ~ ( - i sin O)" f H(I),rKR. t,,,(Z) .1! rrpo L ,, l Jx t z ) n L eik'q~
(3.1)
M. Tran Van Nhieu / Acoustic radiation from slender bodies
v.(Z) = Vo E U.p(Z) e i/(pz .
377 (3.2)
P At low frequency KR << 1, by using the asymptotic expression of the Hankel function for small arguments, we then obtain for the first two terms of the expansion
P~-
eikp°
2rrpoLPaW
L f
[ i ~ r R v o + K s i n 6)cos
dPAvl] e-ig:Z dz
(3.3)
--L where A = "rrR: is the cross section area at a distance Z. Now it is interesting to compare (3.3) with the Chertock results [3] for flexural and accordion vibration modes. For flexural modes when each section vibrates in a direction perpendicular to the Z axis, we have in the Chertock notations
vo(Z) = 0 ,
vl(Z) = - i q o w ~ ( Z )
(3.4)
where qt is the mode function describing the body vibration. Substituting (3.4) into (3.3) yields
P ~ = i ~ eik'q~ qopaKw
L f 2 sin O c o s • A~e-L
K=ZdZ
(3.5)
which is consistent with the results of Refs. [2,3]. For accordion modes where the displacement normal to the body surface is due to a longitudinal displacement of each cross section, the Fourier components of the normal velocity are equal to
v o ( Z ) = i q o w ( ~ - ~ +vRd-d-~),
v,(Z) =0
(3.6)
where uis an effective Poisson's ratio as defined in Ref. [3]. From (3.3) and (3.6) we get the following expression for the farfield radiated pressure
L
P~ - 47rpo~ L qop, Oa2
~--~
+ u2A
e -iK:z
dZ
(3.7)
--L which is equal to the leading term of P~ given by Chertock [ 3 ]. Therefore in these two vibration modes, our general result is in agreement with existing low frequency solutions.
3.2. Radiation from a long cylinder A check on the accuracy of equation (2.36) can be made using the following scheme that is proposed in Ref. [ 13] : for a given object of surface S, we choose a body surface vibration for which an analytical solution exists. Consider one or several point sources of given source strengths located in the region Ri interior to S but that no object is present and assume that all the space is filled with the same fluid. We can easily calculate the normal component of the fluid particle velocity on S and the farfield pressure radiated by these point sources. From the distribution of velocity, we can apply (2.36) and compare the computed results with the acoustic pressure obtained previously. We use the scheme described above for the finite cylinder problem and for point sources symmetrically located on the longitudinal axis of the cylinder with respect to the center of the object. In this case, the expression (2.36) becomes
378
M. Tran Van Nhieu / Acoustic radiation f r o m slender bodies
90
1 5 ~ 0 18~ 0 21o
0
24o 270
Fig. 1. Farfield directivity for a finite cylinder of slenderness ratio l / 10 for Ka = 2.6. The point source solution corresponds to the circle of level 60 dB. 90
1 5 ~ 0 18~ 0
0
21~ 0 3 3 0 SLENDER-BODY
SOLUTION
POINT
SOURCES
SOLUTION
Fig. 2. Farfield directivity for a finite cylinder of slenderness ratio 1/ I 0 for Ka = 2.6. Results on the right h a n d side are obtained with the point sources solution. Results on the left h a n d side are obtained with the slender-body solution.
P~
e ikp°
~po L
2pato
1 Vovop ~p Fp H~o')'( I'pa) K.~
where a is the radius o f the cylinder and
-K:.)L KR - K :
sin(Kp
(3.8)
Kf, = "rrp/L.
It is noticed that in the cylinder case the b o u n d a r y conditions at the two end faces ( Z = ___L) are not specified in the present analysis and therefore (3.8) does not account for their acoustic effects. H o w e v e r it is e x p e c t e d that the contribution o f the end sections are only important near the end-fire direction as we will see in the two f o l l o w i n g examples. W e c o n s i d e r first one point source located at the center o f the cylinder and then two couples o f point sources of
M. Tran Van Nhieu / Acoustic radiation from slender bodies
379
opposite normalized strengths ( _ 1; _ 4) located at Z = _ 0.08L and ___0.14L. The slenderness ratio of the cylinder is equal to a / L = O . 1 and the calculation is made at the reduced frequency K a = 2 . 6 . The computation of both methods are shown in Figs. 1 and 2 where the cylinder is aligned parallel to the 0-180 degrees axis. In the first case (Fig. 1 ) there is a little discrepancy near the end-fire direction which might be explained by the end section effects as previously noted. In the second case, the minima obtained by the slender-body approximation are less important than those of the point sources solution (Fig. 2). Notice that there is a good agreement of both methods near the end-fire direction in this example.
4. Scattering from a slender-body Let a plane wave Pi incident upon a perfectly rigid slender body; the direction of insonification is supposed to be parallel to the plane xOz. The normal surface velocity V is related to P~ by the relationship e i = A e i(K!'x+K~z) ,
(4.1)
0Pi = - ipatoV , On
(4.2)
--
where O/On represents the outward normal derivative on the body surface, K i and K~z are the X and Z components of the incident wavenumber vector. From the classical expansion of plane wave with respect to cylindrical Bessel functions J,, we have
eiKiX=eiKiRc°sq'= E
i~" cos nCrpjn(KixR)
(4.3)
n=O
where En is the Neumann factor (En = 1 for n = 0 and = 2 for n > 0). The expression of the normal velocity V can be deduced from (4.2) and (4.3) as V = i ( A K ~ / p ~ t o ) e ~k~z ~
inE, cos nclgJ'(k~xF)
(4.4)
n=O
with ki= = K~L; k i = K i L a n d / ~ = Eke. From (2.2) and (4.4), we obtain vnp = 0
Vo = A / p a t o L ,
for IPl > 0 ,
V,o(Z) =kxli'"+'E,J~[/dxF(z) l -, '
Kp=o =K'z,
Fp=o =K~x.
(4.5)
Using these relations, the scattered pressure can be derived from (2.35) to yield 1
k
cos nqb
'rr
sin"~9i
po~ = - - - ~ - n
["
~.i" j
--1
J"
- -
H~nl)'
[[¢ixF( t) ] ( r / p ) " h ~ ) ( k p )
e ik~t dt
(4.6)
where the pressure is normalized by A instead of patoVoRmax and ~9i is the incidence angle defined as cot Oi = i i kz/k~.
Equation (4.6) is identical to our previous results [ 8,9] if terms of order ~ cot Oi which have been assumed to be small are neglected in the calculation. Furthermore it gives a simple and practical expression for the scattered farfield; good agreements have been found with other analytical methods at low frequency (KRm~x << 1 ) [5] and with T matrix computation at medium frequency (KRm~x -- 1 ) [ 14,15 ]. In order to illustrate the accuracy of (4.6), the farfield scattered pressure has been computed and compared with the T matrix calculation [ 15]. The bistatic
380
M. Tran Van Nhieu / Acoustic radiation from slender bodies
90
180~
0
27(? Fig. 3. Farfield scattering from a rigid spheroid of slenderness ratio I / 15 for KR,,,~,~= 4 ( 30 degrees relative to the axis of symmetry).
90
180/
f
I °
270 Fig. 4. Fartield scattering from a rigid spheroid of slenderness ratio 1/ 15 for KR.I~,x= 4 (60 degrees relative to the axis of symmetry ). angular distributions for a rigid spheroid of slenderness ratio 1 / 15 for K R ..... = 4 are shown in Figs. 3 - 5 at several aspects o f insonification. The spheroid is aligned along the 0 - 1 8 0 deg. axis and the arrow indicates the direction o f the incident plane wave. As can be seen, a c o m p a r i s o n with Figs. 2 - 4 o f Ref. [ 15] lends further support to the present method. In the same fashion, analytical expressions of the pressure scattered from a long finite cylindrical shell can be d e d u c e d f r o m equations (2.26) and ( 2 . 3 6 ) ; again the results are identical to those g i v e n in Ref. [ 11 ].
Acknowledgments
T h e present study was supported by the " D i r e c t i o n des R e c h e r c h e s et Etudes T e c h n i q u e s " , France, Contract
M. Tran Van Nhieu / Acoustic radiation from slender bodies
381
90
Fig. 5. Farfield scattering from a rigid spheroid of slenderness ratio 1/ 15 for KRmax= 4 (90 degrees relative to the axis of symmetry). D R E T N o . 9 1 3 4 8 5 . I w o u l d like to t h a n k M i s s L a u r e n c e G u e r r e f o r h e r c o m p u t a t i o n a l a s s i s t a n c e c o n c e r n i n g t h e radiation problems.
References [ 1] H.L. Pond, "Low frequency sound radiation from slender bodies of revolution", J. Acoust. Soc. Am. 40, 711-720 (1966). [ 2 ] M.C. Junger, Sound radiation by resonances of free-free beams, J. Acoust. Soc. Am. 52, 332-334 (1972). [3] G. Chertock, "Sound radiation by low-frequency vibrations of slender bodies", J. Acoust. Soc. Am. 57, 1007-1016 (1975). [ 4] J.G. Geers, "The scattering of scalar wave by a slender body of revolution", SIAM, J. Appl. Math. 34, 348-370 (1978). [ 5 ] M.V. Fedoryuk, "Scattering of sound waves by a thin acoustically rigid body of revolution", Sot,. Phys. Acoust. 27, 336-338 ( 1981 ). [6] D.S. Ahluwalia and J.B. Keller, "Scattering by a slender body", J. Acoust. Soc. Am. 80, 1782-1797 (1986). [ 7 ] M.D. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York (1964). [8] M. Tran Van Nhieu, " A singular perturbation problem: scattering by a slender-body", J. Acoust. Soc. Am. 83, 68-73 (1988). [ 9] M. Tran Van Nhieu, " A slender body approximation in scattering theory", J. Acoust. Soc. Am. 85, 1834-1840 (1989). [ 10] M. Tran Van Nhieu, "On the asymptotic solution of scattering from slender bodies by the two-variable technique", J. Acoust. Soc. Am. 91,495-497 (1992). [ 11] M. Tran Van Nhieu, "Scattering from a finite cylindrical shell", J. Acoust. Soc. Am. 91,670--679 (1992). [ 12 ] J. Kevorkian and J. Cole, Perturbations methods in Applied Mathematics, Springer, New York ( 1981 ). [ 13] H.A. Schenck, "Improved integral formulation for acoustic radiation problems", J. Acoust. Soc. Am. 44, 41-58 (1968). [ 14] M.F. Werby and R.B. Evans, "Scattering from objects submerged in unbounded and bounded oceans", IEEE Trans. Oceanic Eng. 0E-12, 380-394 (1987). [ 15] M.F. Werby, G.J. Tango and L.H. Green, "Computational acoustics: algorithms and applications", 1st IMACS Symposium on Computational Acoustics, New Haven, CT, USA (1986).