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SCATTERING FROM PRISMATIC SURFACES
C. Audoly, G. Dumery
To cite this version:
C. Audoly, G. Dumery. SCATTERING FROM PRISMATIC SURFACES. Journal de Physique Col-
loques, 1990, 51 (C2), pp.C2-415-C2-418. �10.1051/jphyscol:1990298�. �jpa-00230727�
COLLOQUE DE PHYSIQUE
Colloque C2, supplement au n02, Tome 51, Fbvrier 1990 ler Congres Frangais d'dcoustique 1990
SCATTERING FROM PRISMATIC SURFACES
C. AUDOLY and G. DUMERY'
Groupe dlEtudes et de Recherches de DBtection S O U S - ~ a r i n e , D.C.A.N.
Toulon, Le Brusc, F-83140 Six-Fours, France
'~niversite de Toulon et du Var, F-83130 La Garde, France
Rgsurng : Les cylindres de forme prismatique peuvent Gtre repr6sentgs par un assemblage de bandes rectilignes. A partir de la connaissance de la matrice de transition d'une bande Blgmentaire, la th6orie de la diffusion multiple permet le calcul de la diffraction par un cylindre prismatique. On traite drabord le cas d'un cylindre dont la frontisre est ferm6e (cylindre 5 section carr6e) et l'on compare les rEsultats 5 une rEsolution directe par Bquation intggrale de Helmholtz. On pr6sente ensuite des rgsultats th6oriques et exp6rimentaux sur des rgflecteurs paraboliques formes de tubes compliants.
Abstract : Prismatic cylinders can be represented by an assembly of strips.
The transition matrix of an individual strip being known, the scattering from a prismatic cylinder can be computed using the multiple scattering theory. Cases where the boundary of the cylinder is closed (square cylinder) are first presented and the results are compared to a direct solution using the Helmholtz integral equation. Then, theoretical and experimental results on parabolic reflectors formed with compliant tubes are given.
1. INTRODUCTION
The multiple scattering method can be applied to a wide varlety of scatte- ring problems. The theory has been presented by many authors, among them Twersky /I/, Varadan et al. /2/ and Dumgry / 3 / . Some new results in the two- dimensional space for elastic obstacles in a fluid, with emphasis on periodic gratings, were recently obtained / 4 , 5 / . The theory applies to f i n ~ t e gratings as well, and this is the case of prismatic cylinders, which can be represented by an assembly of strips.
2. SCATTERING FROM ARBITRARY FINITE GRATINGS
The geometry of an arbitrary grating consisting of obstacles centered at points 0, ,0,.
. .
and the parameters r, ,O, ,rT, ,OT, are defined on fig.1. In the following, the time dependance exp(-iot) is omitted.Fig.1 : Finite grating geometry
Fig.2B : Compliant tube
f Y
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990298
C2-416 COLLOQUE DE PHYSIQUE
A general expression for the scattered pressure field is the sum of outgoing cylindrical waves from each of the N obstacles :
P = Po +
CP,
p E =CCE,
Hn(krE) exp (i n 0 ~ )with k=w/c, c:sound speed in the surrounding fluid,H,: Hankel function of the first kind and order n. Let now consider the obstacle number E. The inci- dent field near 0, is the sum of the direct incident field p, and of the con- tributions scattered by the other obstacles:
P,'(x, Y) = P,(X- Y)+,
5 =,
P,(X, Y)r c e
AS its components are regular at O,, p, takes the form:
+-
p,'(x, y) =
2
(A:+s: )
Jnlk r c l exps n 0.1n-.D
The coefficients A, are related to the direct incident field and S, repre- sent the interactions with the other obstacles. The identification of the two expressions gives: N +-
x
C: ~,,-,[kr,) expjiin -m)e,) S m =7 = 1 n-- r # E
We now introduce the transition matrix T which relates the scattered field to an arbitrary incident field of coefficients B,: +-
Cn-
C Trim
B m",--a
With B,= A,
+
S, , we obtain:c,' - I:
Tnmzx x
C=H,-,(~ ),r exp (i (q-
m) 0,) =C
T,,;A,'m--- T - 1 q - - - "I---
Truncating the series and solving the linear system gives the coefficients C,, then the total scattered field.
3. TRANSITION MATRIX of STRIPS
The geometry of a strip of width 2a is given on fig. 2A. A compliant tube can also be considered as a strip by neglecting the minor axis (see fig.2B).
Numerous methods can be used to compute the transition matrix. We use here an integral equation method. For a given incident field, discontinuities of pres- sure U, and normal derivative p, appear between the sides of the strip.
For a rigid strip : ap,
-
(P)+
Iso4Q)a'
') d S (Q) = 0ax ax, axQ
For a soft strip :
P, (p)
- Is
P~(Q) G (P, Q) dS(Q) = 0For a compliant tube, more computations are necessary /6/. Once u, and p, known, the coefficients of the transition matrix are given by:
i.Q
4. EXAMPLES
The case of a square cylinder, formed with an assembly of 4 strips placed edge to edge (fig.3) is first considered. The computations were done for the reduced frequency ka=2 and for an incident plane wave of direction 0=0. The transition matrices of soft and rigid strips were calculated and the forma- lism of section 2 is used to determine the scattered farfield directivity pattern of the square cylinder. The results are compared to a direct method: the classical Helmholtz integral equation method. For the rigid
cylinder (fig.4), and the soft cylinder (fig.5), the two methods are in good agreement. Note that the multiple scattering method, unlike the Helmholtz integral equation, is insensitive to characteristic frequencies problems.
The multiple scattering method applies also well to open boundary prismatic cylinders. An omnidirectional line source was placed at the focus of a parabolic cylinder formed with glass reinforced composite compliant tubes (Fig.6). Fig.7 gives the calculated and measured directivity patterns for ka=0.645. The width of the parabola can be modified to adjust the main lobe aperture. Fiq.8 shows the patterns for parabolic cylinders formed with 25,15 and 9 tubes at ka=0.43.
- -
- -IT- --- --+- --- -
--., X
Fig.3 : Square cylinder
Fig.4 : Directivity pattern of a rigid square cylinder ka=2.
(- ) Multiple scattering
(---I Integral equation.
Fig.5 : Directivity pattern of a soft square cylinder ka=2.
(- ) Multiple scattering
(---) Integral equation.
COLLOQUE DE PHYSIQUE
1
TRANSDUCERFig.6 :
Parabolic cylinder geometry.
Fig.7 : Directivity pattern of the parabolic cylinder with a line source at the focus, ka=0.645.
( - - - ) Theory
(- ) Experiment
.
Fig.8 : Directivity pattern of parabolic cylinders with a line source at the focus, ka=0 .43.
(- ) 25 tubes
(....-.) 15 tubes
(-
- - -
) 9 tubes.REFERENCES
/l/ TWERSKY V.
,
J.Acoust.Soc.Am. 21(1952)42./2/ VARADAN V.K., VARADAN V.V. a n d 7 ~ 0 Y.-H. , J.Acoust.Soc.Am. %(1978)1310.
/3/ DUMERY G. , Acustica g(19671334.
/ 4 / AUDOLY C. and DUMERY G. , J-Acoustique 1(1988)141.
/5/ AUDOLY C. and DUMERY G. , Etude dr6crans sous-marins constitu6s de tubes Blastiques. To be published in Acustica (1989).
/6/ AUDOLY C. and DUMERY G. , Modeling of compliant tube underwater reflec- tors. To be published in J.Acoust.Soc.Am. (1990).
