A soft spherical “cap” in the field of a plane sound wave

244

s. s. vinogradov REFERENCES

1.

FELLER, W. Intduction to probabiIi@ theory and its applications (Vvedenie v teoryu veroyatnostei i ee prilozheniya), Vol. 1, “MY, Moscow, 1967.

2.

KNUTH, D. E. The art of computer programming (Iskusstvo programmirovaniya FundamentaJ algorithms (Osnovnye algoritmy), “MY’, Moscow, 1976.

3.

AHRENS, J. H., DIETER, U. and GRUBE, A. Pseudo-random numbers: a new proposal for the of multiplicators.Computing, 6, No. l/2,121-128,1970.

4.

MIKHAILOV, G. A. Some topics in the theory of Monte Carlo methods (Nekotorie

dlya EVM), Vol. 1,

choice

voprosy teorii

metodov Monte-Karlo),“Nauka”,Novosibirsk,1976. 5. SHELL,D. L. A highspeedsortingprocedure.Communr ACM, 2,7,30-32,1959. US.S.R. Comput. MathsMath. whys. Vol. 18, pp. 244-249 8 Pergamon Press Ltd. 1979. Printed in Great Britain.

0041-5553/78/1001-0244$07.50/O

A SOFT SPHERICAL “CAP” IN THE FIELD OF A PLANE SOUND WAVE* s. s. VINocRADov Khar’kov (Received 8 December 1977)

THB PROBLEM of the scattering of a plane wave by a soft “cap” is reduced to the solution of an inftite system of linear algebraic equations of Fredhohn type. In the case of Rayleigh scattering an analytic expression is obtained for the complete scattering cross-section, uniform over the aperture angle of the sphere. The perturbing action of a small circular aperture on the spectrum of the oscillations of a spherical cavity with a soft boundary is investigated. Previously we obtained a rigorous solution of the problem of the scattering of a plane sound wave by a rigid sphere with an aperture [ 1] . Its solution was based on the method, developed in [2] , for solving paired summation equations, containing in the general case the associated Legendre polynomials Pnm(cos 13). In this paper the method of [2] is used to obtain a rigorous solution of the problem of the scattering of a plane wave propagating along the axis of symmetry of a spherical “cap”. On the surface of the spherical cap the Dirichlet boundary conditions are satisfied:

1. Statment of the problem The disposition of the spherical cap in space is described in the spherical coordinate system r, 0, cpby r=a, 8=8,,, O=qc%z. The aperture angle of the sphere is denoted by &=n-fjo. The velocity potential of the plane wave propagating in the direction 8 = 0 is represented by the scalar function OD U(O)(r, 0) - exp (Ikr cos 0) -

P(2n+l)j,(kr)P,(cos f:

?I.-0

*Zh. vj&hisl.Mat. mat. Fiz., 18,5,1320-1324,1978.

f3).

245

Short communications

A solution satisfying the Hehnholtz equation, the condition of continuity of the velocity potential on the surface of the sphere (r = a), and also the condition of radiation at infhrity, will be sought in the following form:

hf)

00

(ka) -in(kr)Pn(cos i,(ka)

Ic

i”(2n+l)b,

e),

r
n-0

U(r, (3)= V(O)(ra 6) +

(l-1) 00

in(2n+1)b,h~‘(kr)P,(cos Is 7L=O

r>a,

CO,

are Bessel spherical functions, P,, (cos 0) are Legendre polynomials, and where j,,(kr), h,(Q(kr) UJ$ow is an infinite sequence of amplitude coefficients required in the definition. We impose on (1 .l) the exact boundary conditions on the surface of the sphere

VIrna-o=vIrlo+O=O,

oG3ce0,

au

au

a, I rl(I-0 =,

eo-=eGn,

I rlo+Op

and having satisfied these we establish the system of summatory functional equations 0

z c

in(2n+l)h~i’(ka)b,P,(~~s

e)=

-V(O)(a, Cl),

06ece0,

?L=O

(1.2)

00

b, P(2n+l) -Pp,(cos

in@4

T%=O

9) = 0,

eoce==R.

2. Regdarization of the functional equations The regularization of (1.2) is carried out in several stages. We first introduce the coefficients The functional equations (1.2) are transformed to the form B,==PbJj,(ka). B,(2n+I)jn(ka)h~)(ka)P,(cos

fI)-

-U(O)(a, e),

e)= 0,

eoce
oteceO,

(2.1)

c (D

c

B,(2n+l)P,(cos

n-o

(2.2)

Since the expansion OD

exp(ikR)Il? -

i

c n-o

(2n+l)jn(ka)h?

(ka)P,(cos

e),

(2.3)

S. S. Vinogadov

246

holds, it is obvious that the left side of the functional equation (2.1) is the expansion of the quantity -v(O)@, 6) in a series of spherical wave function of a surface source situated at the point with coord~ates r = a, @= 0, the further regularkation of (2.1), (2.2) is based on the separation in (2.1) of the static and dynamic parts in each harmonic of the elementary source (2.3):

where

Therefore, the functional equation (2.1), after separating it into static and dynamic parts, is transformed to the form

~,(f-p~)~,(eos

e)=

-ikuU(O)(u, 01,

OG940.

(2.4)

We will search for a solution of (2.4), (2.2) in the class of complex numerical sequences {Z3,,}o~~Z~, defined by the condition OD P*l “(06. z a-0 Using the results of [ 1,2 ] and the method of semi-mversion [3] , we can establish the system of linear algebraic equations of the second kind

where Qnm(00)= +

[

sin(n-m)90 n-m

sin(n+m*l)Bo +

n+m+l

-1’

The operator A defined by the matrix llu,Q,ll, is completely cont~uous in the Hilbert space of complex sequences {B$ooD=12, since, as follows from the es~ation of the absolute norm of the operator N(A), the latter is bounded for all parameters of the problem:

where C is Euler’s constant.

241

Short communications

Moreover, it can be shown that the absolute norm N(B), where B is the operator determined by the column of right sides of the infinite system (2.9, is also bounded for all values of ka and

xc OD

W(B)=(ka)*

(D

m-o <$(kcr)’

2

l”(Zn+I)in(ka)Q.,(eO)

nlo

I

(2.7)

I+$(k@

.

C

I

The estimates (2.6), (2.7) imply that the system (2.5) is of Fredholm type, and since the Hilbert alternative holds for it [4], it can be solved by the method of reduction.

3. Perturbation of the spectrum of the self-oscillations of the spherical cavity for 01 < 1 In the case of Rayleigh scattering (ka g 1) or of small angles 00 Q 1, where @(A) < 1, the solution of (2.5) can be obtained in analytic form by the method of successive approximations. As 80 + n it is useful to introduce a new system of algebraic equations, which follows from (2.9, if we use the relation

Qm(Bo) =6nm- (-1)-%m(ei),

ft,,(ed =

1Jr [ sin(n-m)& n-m

sin(n+m+l) 8i n+m+l

1.

(3.1)

Therefore, using (3.1) and (2.5), we arrive at the system 0 (1-p&L

+

c

x+R,

(e d

n=o

ca =

-ika

x,=

Lx n-”

i~(21a+l){(-l)mSnm-(-l)nR,m(e1)),

(3.2)

(-i)rnB,.

Equating to zero the determinant of the system (3.2), we’obtain the dispersion equation for determining the eigenvaiues of the part of ,the spectrum of the self-oscillations of a sphere with an aperture, which can be excited by a plane wave propagating along the axis of symmetry of the screen. Provided the aperture is small (01 Q 1) we expand the determinant by the zeroth row (m = 0), and retaining the quadratic form of the angular functions R,, (0 I), we arrive at the dispersion equation determining the eigenvalues of the relative wave number:

We denote the roots of the dispersion equation by yip; then its solution can be represented in the form

248

S. S. Vinogmdov

Y ip=%p

+i&p,

j=(_l)'h,

Btp = - (2i+1) --i

c m#i

ar*=sjV{~-(2t+l)-1Rii},

(3.3)

&A1 (2s+U W(Zig)

+ n,a(z(v)]

9

where Xip is the p-th root of the Bessel function ji(xip) = 0. The last equation is the characteristic equation for determining the self-oscillations of a spherical cavity with the Dir-i&let boundary conditions on its surface (soft boundary). It follows from (3.3) that for 81 < 1 the perturbing action of the aperture on the spectrum of the self-oscillations of the spherical is manifested in the appearance of a small imaginary addition, leading to the finite figure of merit of the oscillations and to some decrease in the eigenvalue of the oscillations of the spherical cavity xip. This decrease is proportional to (21+1)-%#I~) Q (21+1)8,~/&~. We return to the case of Rayleigh scattering. We will consider the behaviour of the scattered Velocity potential PC in the far zone krBka, krwi : w f(B) = (ik) --i

exp(tkr),

USC= f(O)+

x

(~~+~)~,P,(cos

e).

n-0

The complete scattering cross-section, normalized to the area of the great circle of the sphere can be represented in the form u

4

naa

(k4’

-e-

c -

(2n+l)in2(ka)

[&I’.

n-o

In the limiting case of long waves it follows from the estimate (2.6) that @(A) -%1 and the solution of the system (2.5) can be found in analytic form by the method of successive approximations. To a first approximation we find an analytic expression for the complete scattering cross-section, uniform with respect to the angle 190,which is identical with the result obtained in [5] : 3

= 4

QooVeo)- f

+aQoo (eo) Qoz (eo) -4QoiYeo)

W2[Qoo2(eo) I *WW

‘1

.

Short communications

249

In the case 8o = ?Tthere follows from this the well-known expression for the cross-section of scattering of a soft sphere o=4naa ’

1 3 Pala +O((kaN~

i--

I 1

{

and for f30% 1 that for a circular disk of radius a&-~.For arbitrary parameters ku and 80 system (2.5) was solved by the method of reduction on the M-222 computer. The figure shows the complete scattering cross-section u as a function of the relative wave number ka for a fared aperture angle of the sphere 0 1. l’kanslated by J. Berry. REFERENCES 1.

VINOGRADOV, S. S. and SHESTOPALOV, V. P. The rigorous theory of the scattering of waves by a sphere with an aperture (the Helmholtz resonator). Ninth All-Union acoustics conference, Moscow, 1977 (IX VW. akustich. konf., Moskva, 1977), Subjects ofreports ofsection A (Tezisy dokl. Sekts. A), 79-81, Akistich. in-t, Moscow.

2.

RADIN, A. M. and SHESTOPALOV, V. P. Diffraction of waves by a sphere with an aperture. Dokl. Akad. Nauk SSSR, 212,4,838-841,1973.

3.

SHESTOPALOV, V. P. et aL Diffraction of wavesby lattices (Difraktsiya voln na reshetkakh), Izd-vo Khar’kovsk. un-ta, Khar’kov, 1973.

4.

KANTOROVICH, L. V. and AKILOV, G. P. Functional analysisin normalized spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh), Flzmatgiz, Moscow, 1962.

5.

THOMAS, D. P. Diffraction by a spherical cap. Proc. timbridge Philos. Sot., 59,197-209,1963.

U.S.S.R. Comput. Maths Math. Phys Vol. 18, pp. 249- 254 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

0041-5553/78/1001-0249$07.58/o

SOLUTION OF THE PROBLEM OF THE DIFFRACTION OF AN ACOUSTIC PULSE BY A WEDGE USING WAVE POTENTIALS* A. A. GLADKOV

Moscow (Received 19 October 1977; revised 19 December 1977)

THE FORMULATION and solution of the problem of the diffraction of an acoustic pulse by a wedge using integral equations for the wave surface potentials is considered. The numerical and analytic solutions are compared. 1. The solution of the problem of the diffraction of an acoustic pulse by a wedge has attracted many investigators from the beginning of the century [ 1,2] up to the present time. Especially valuable is [3], which determined the direction of investigations for many years. Below, using the example of the symmetric flow past a wedge we consider the numerical solution of the pulse diffraction problem obtained by using integral equations for the surface wave potentials [4].

*Zh. vjihisl. Mat. mat. Fiz., 18,5, 1324-1329.1978.