The diffraction of plane waves in a compressible stratified liquid by a strip of finite width

65 6. ARSEN'YEV A.A., Cauchy problems for the linearized Boltrmann equation, Zh. vychisl. Mat. mat. Fiz., 5, 5, 864-882, 1965. 7. ANIKONOV YU.E., The inverse problem for the kinetic equation, in: The Uniqueness, Stability and Methods of Solving Ill-Posed Problems of Mathematical Physics and Analysis, Vychisl. Tsentr SO Akad. Nauk SSSR, 1984. 8. CHERCHIN'YANI K., Theory and Application of the Boltznann Equation, Mir, Moscow, 1978. 9. HILL E. and PHILLIPS R., Functional Analysis and Semigroups, IIL, Moscow, 1962. 10. PRILEPKO A-1. and ORLOVSKII D.G., Inverse problems for evolution semilinear equations, Dokl. Akad. Nauk SSSR, 277, 4, 799-803, 1964. 11. PRILEPKO A.I. and ORLOVSKII D.G., Determination of the parameter of the evolution equation and inverse problems of mathematical physics, I., Diff. Urav. 21, 1, 119-129, 1985. 12. RUNDELL W., Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data, Appl. Analys. 10, 231-242, 1980. 13. EIDEL'MAN YU.S., A two-point boundary value problem for differential equations with a parameter, Dokl. Akad. Nauk Ukr. SSR, Ser. A, 4, 15-19, 1983. 14. PRUSS J., On the spectrum of Co-semigroups,Trans. Amer. Math. Sot. 284, 2, 847-057, 1984.

Translated by R.C.G.

U.S.S.R.

Comput.Maths.Math.Phys

.,Vo1.27,No.6,pp.65-70,1997

Printed in Great Britain

cG41-5553/87 $10.00+0.00 01989 Pergamon Press plc

THE DIFFRACTION OF PLANE WAVES IN A COMPRESSIBLE STRATIFIED LIQUID BY A STRIP OF FINITE WIDTH* R.R. GABDULLIN

The problem of the scattering of plane waves by a horizontal strip immersed in an exponentially stratified compressible liquid is considered. In the treatment the wave propagation process is assumed to be adiabatic. 1. Formulation of the problem. Let us consider the plane motions of a stratified compressible liquid which are described by the equation from /l/:

Eq.(l) describes the small vibrations of a liquid with a specified stratification,that is, with a density in the stationary state po(z)~pp(0)exp(-2?2), $>O in an rOz Cartesian coordinate system. These small vibrations vary with time in accordance withthelaw erp(-ial)* It is assumed that there is a homogeneous gravitational field with an acceleration g-(&-g), that the motions of the liquid are adiabatic and that the adiabatic velocity of sound c is constant. Here, PIP and V-(Vi,Ud, the amplitudes of the dynamic pressure, of the perturbation of the density and of the vector describing the velocities of the liquid particles, respectively, are expressed in terms of Y as follows:

a'-o*/(oo'-o'), k’+-d/c’. p==$-oo’/g, Here and above, where oo'==2pg-g'/c' is the square of the Vaisala-Brunt frequency. When oO*>O, the stratification is assumed to be stable and the inequalities o,O, which corresponds to the caseoftheinternal vibrations of a liquid: Wka, k,==(k,‘-k’a’)“/u by a rigid strip r=(z=O, z,
lZh.vychisl.Mat.mat.Fiz.,27,11,1701-1708,1987 0s~~ 27:6-E

-a’k,),

66

Yyotransports energy from the fourth into the second quadrant of the ZOZ plane. we shall assume that a no-flow condition (the component of the velocity normal to r is zero) is satisfied on the strip r and that the pressure and the vertical component of the velocity are continuous when r=O, ~e[a,~l. Then, if the total field is represented in the we obtain the following problem for '4':it is required to find a function form Y,-YOSY, defined in [R'\r,which satisfies Eq.(l) and the following conditions: Y(z.:), when

z-*0,

r=[z,,zzl,

(24 (2b)

(-g+pY)(z,+O)-(Z+pY)(z,-0)-O

when

z*b,,z,l.

(2c)

We impose the conditions

$

(%,+z,

lo) -0 (z-“1)

when

Z-+fO,

on the edges O,=(q.O) of the strip I'. These conditions are consequences of the requirement that there should be no additional sources on the edges 0, which is formulated in the form that the total energy flux across the surface encompassing the edges 01,j=i,2 should be equal to zero. We will formulate the conditions at infinity in the form of a requirement that waves which have arisen as a result of diffraction should carry away energy to infinity. For this purpose we shall make use of an analogue of the principle of limiting absorption /2/: when constructing the solution of the problem in question, we shall assume that the quantity a has and, at the same time, k,--k,,+ib(e), where OC6(e)+O a positive imaginary part: a-~,-%~>0 when E-d. We shall seek a solution of problem (l), (2) with complex a and k, in a class of functions which, as r-+w, satisfy the estimate IYI+IVYl
(4)

where r=(~'Cz*)'~, OCA-A(e)= min (6(~),E(k*-CL*)"}. In the final results we pass to the limit as e+O and obtain a solution of the initial problem (l), (2) which describes waves which carry away energy to infinity. 2. Reduction of the problem to a system of integrofunctional equations. Following the Jones' method which has been described in /3, Chapter 5/, we shall reduce the problem in question with complex a and k, to a system of integrofunctionalequations. Let us introduce the following functions of the complex variable a into the treatment: =, Y _ (a, t) = s Y (z, z) e”‘‘-‘1’ dt, .._

Y.(a.z)=

j Y (z,z)e’“‘dt,

*

Y +(a, z) - j Y (t, z) e”(‘+) dz, =, Y(a,

z)==P*Y-(a,

z)+Y~,(a, z)+e’“‘Y+(a,

z).

It follows from the theory of Fourier integrals that, in the case when Y(t,z) belongs to the class of functions with the estimate (41, the analytical properties of the functions Y*. Yyo and, also, (dldz)Y= are such that Y,(c,z) and (d/dz)Y+(a,z)are regular in the Y_(q z) and (d/dz)Y-(a,~) are regular in the half plane half plane lK=(a:Inla>-A}, n_-(a :Ima<.l} while Y,(z,z) is an integral function of a and, at the same time e-'ailYO(cc, z) z) is regular and increases algebraically as (a[+= in the half plane & and e-'z'*Yyo(z, has the same properties in II_. In this case the function Y(a,z) satisfies the equation ($+f)Y(a,z)-0. In order to separate the branches of this function, a where y-y (a)- (a*-kV)“‘/u, 7 (0) ---ik. cut is made which connects the branching points *(a,+ie)k through an infinitely remote point half-plane and vertically downwards in the and which passes vertically upwards in the n+ n- half-plane. Let us seek a solution of Eq.(S) of the form Y(a,z)- (

Y(a,+O)e-"' Y(a,-O)e'l'

when when

z>o, z-=0.

Taking account of conditions (21, it can be shown by simple algebra that

67

* (

ey

_) =

a* -

pfz:

sgn

[e'a*~O,(a)Ce'"~O,(a)1, 2

-s, t

0,(a) -$(a,O)+pY_(a,O)

+s,

ay+

0,(a) =~(a.O)tpY+(a,O)

1

where

p-ipfkr

and, moreover, the relation

(6)

e'"lO,(a)+K(~)cDo(a)+e'""O,(a)=O holds.

Here,

the notation

, . K(a)--(k:b,Sad,J.,,I

[Yy,(a,-O)-Y0(a,+O)l, @0(a)-&x-a(kL-j&y.

has been adopted. The functional Eq.(6) is valid in a strip -L
(7)

where l-zI-r, is the width of the strip I. b-_(l-"/~)esp(ikal+in/4) and &(a)- (x*a)(kaia)-” is the factorization of the function K(z) into a product of the functions &(a)K-(a) which are analytical and differ from zero in the n+ and Ii_ half-planes respectively. Moreover, K-(a)==X+(-a). where -A
3. Asymptotic

behaviour

of

the

solution

of

the problem

when

IloaZ j’i

Then, assuming that solutions F,(a) of Let us use the notation q==min(lka-xl, jk,-kal}. Eqs.(7) exist, one may obtain their asymptotic forms FL(a) when qlwi, which are given by the functional equations

pK+ (a) e-‘L+

pK+ (a) e-**‘* F’(x)

+(a+k,)K_(k,)

‘(a-k,)K,(k,)’

h-*1.

At the same time the following estimate, which is uniform with respect to a in holds: [F,(a)-F,(a)

Im aad,

1 Ika+al'b~c~.

Eqs.(S) are simply solved: in order to do this it is necessary to consider them when and to solve the resulting system of linear equations for FL(kn) and FL(~). The determinant which arises when this is done is equal to a-ka,x

be”‘(z)

%

and differs from zero, for exmaple, when

Here, w(6)-wd-f(kaff)l),

where

W(X)--a (2ka)

%

68

~ -

(2)-

%-lb

r(nii)e”‘z’“W-,,+,,,,.,,,(z),

is Whittaker's function /4/. w,, The above-mentionedproperty of the determinant becomes obvious if account is taken of the inequality and

when

Re(ka+~)>O

+=

and

(k'-p')"'>O.

Then, by carrying out an inverse Fourier transformation from Y'(a,z), we obtain that the solution of problem (11, (2) with complex a and k, can be represented in the form Y (2, Z)==Ti(I, z)+F*(z,

z)-iR(z,

z),

(94

_

exp[-iy 1z 1--ia(z--z,) I

where @2(a)+k~,(a)==Fr(a), h=*L

@,(a)

i-1,2,

that is,

pK+(a)e-‘**‘* Q’(a) -(a-k,)K+(k,)

@I(a) da.

pfiy sgn z

-=

(_%)+

b w(a)-d--x) (ka+a)%

ka-x ‘b a,(xj+ ( ka-a 1

,w(-a)-w(-4 (ka-a)”

- “‘”

pK_ (a) e-J~x~ -----e’~x (a-kdK-(kd

G,(-W,

G(k4.

(lob)

We note that the quantities a,(-x),@z(x),a,(-ka),di,(ka)in (10) are determined from which have been found, but explicit expressions for them are not F&(x) and F%(h), presented here on account of their complexity. The estimate IR(z,z)~~const(qZ)-'. (11) which is uniform with respect to (I,z)~C?\~,holds in the case of the function R(z,z) in (9a). Hence, ~,(z,z)+~,(z,L) describes the solution well when the width of the strip Blkaj-‘, the characteristic wavelength for the liquid and, at the same time, the quantity p is not too small: ka-x-ka-a(k*-p*)“-ka and, also, the angle of incidence of the scattered wave, measured from the Ox-axis, is not too small: k,-ka-ka. Let us now pass to the limit as e-0. BY making use of the fact that the quantity (ka)-' has the dimensions of length, let us transform to the dimensionless variables alka, kax, kaz, kal, p/ka, k,/ka , etc. while retaining the previously used notation for them. The large parameter oftheproblem is now 1 while i-x and k,-1 are quantities of the order of magnitude of 1. After some reduction, we obtain that the solution of problem (11, (2) can be represented by formula (gal, where

(12)

_“,(-fl(i~a))--w,(--L1(l--x)) DAa)

--

A,=-

p (IFk,)“’ 2x

xFk,

c,=- 2”, &C-W,

’

E

(a)da

I

t

B* -

E,(a)-exp[-iylzl+fa(I--Z,)l,

Dj(a)=(lTa)'A(piiy sgnz),

T=-y(a)-(a'-l)"/a.

i-L2

and, in all of the formulae, the upper signs are to be taken when j-i and the lower signs when j-2. The integration in (12) is carried out along the real axis of the a plane, circumventing the singular points of the integrands. Positive singular points are passed around from below and negative singular points from above.

4. Analysis of the resulting asymptotic forms. We will obtain formulae from the integral representationsof the leading terms of the asymptotic forms which enable us to present a fairly explicit picture of the process under consideration. In order to do this we introduce the following polar coordinate systems:(r,, 8,) with centre at the point 0, and (rl, 8,) with centre at the point '0,.At the same timewe shall

69

measure the angles e,_j-i,2 from the rays (~=O,Z>Z~} in an anticlockwise sense and -n<3,< The integrals in (12) will be treated in the coordinate system (r,,&) when j-l and the coordinate system (~~~32)when j==Z. Let us use the notation 3.-arctga,BP= arctg(a'k,/k,) where 8. is equal to the magnitude of the angle which the straight lines forming the characteristic cone of Eq.(l) subtend with the Or-axis, and n-80 is the magnitude of the angle which the group velocity vector of the incident wave YyJ subtends with the&-axis. On the basis of an asymptotic analysis of the integrals in (12) using an analogue of the method of steepest descent, it is possible to obtain the following representation for the total wave field: Yu,-Y,~+~,+~,d+~~+R(I,I),

(13)

where the approximation of geometrical optics Y pa-

YQ Y’-iY” 0

-zfe,ce,<7t-eo when when 8,-C-n+%, e,h-e. when

-7t+e,ce,cn-e,, eI>-n+e,,

or

and and e,o--eo

(here Y"--P(kL-iCc)-'exP(-ik,z+ik,l) is the wave which has been rsflected from the strip I?) and when when

e.< n-e., I BjI

j=i,2 and, moreover, the following asymptotic estimates when kr,-)w,j,m=l,2 hold for the terms Yin: w~(-il(lsaim))-W,(-il(i-X)) x x-a;" BJ Y,"- =fA, (14) 3 D,(ajm) (aim-X.1) + D,(a,“‘) ’ ” &(a8

I

where EJk3p aJ’aa--

[ ir,q(e,)+i+ CO9 eJ 903,)'

1 ,

1,

[

E?=exp -r,m(e,)+i$

:_hseJ aJ

a

9@J) '

In (14) the upper signs are to be taken when j-1 and the lower signs when j-2 Formulae (14) are obtained by applying the method of steepest descent to the Integrals in (12) and, although there is a large parameter in the functions wo(-iL(l*a)) in the third integral of (12), the estimate (14) holds since, when a+*i, the functions wo(-il(l*a)) are expanded in a Taylor's series in negative powers of the parameter 1. In (13),Yk represents a Kelvin wave near the strip which only exists in the domain (t>O, z=[z,,z~]} when u>O and in {z
(l-~,~t)c-““-4’+(8,-g,)e-‘“‘=-*’ e-l”4

61”(i-8,‘) where the notation

has been adopted and to-(1,+zr)/2 is the middle of the strip. It can be seen from formula (15) that Yr is a wave which decays exponentially with respect to s as it becomes more remote from the strip r. With respect to the z-axis it represents the superposition of two waves travelling in the positive and negative directions of the Ox-axis with a velocity c. The cases when one of these waves is not present or when they have amplitudes with the same modulus and they produce a standing wave (with respect to r) are of interest. If 61=& or

~=++~),,

n-0,*1,*2,....

then just the wave travelling in the positive z-direction may not be present (if k,>O). For each fixed value of 1 this condition is realized for a denumerable set of angles of incidence of the scattered wave Y0 and at each angle of incidence for a denumerable set of 1. The condition for a standing Kelvin wave to occur is J&-6,]-ll-&&l or _cos(k,Z)sin(xl)- k, alpI(k,'-1)'" IultgC 1

70

and cannot be satisfied in view of the inequalities tg8~
5. Discussion of the results. In describing the total wave field represented by (13) it may be said that, apart from it is composed of two types of components: of the approximation of a small term R(z,z), geometricalopticswhichdescribesthe incidentandreflected waves in the region of the shadow behind the strip and of diffraction terms. The latter constitute the internal Kelvin wave q& and the terms 'Yp,j=l,2 which are specific for the internal vibrations which are not qualitatively different from their analogues in /l/. At the same time, if one expands the square brackets in formulae (14), describing 'f':,it can be seen that the first two terms can be interpreted as the diffraction of the strictly incident wave and the Kelvin wave excited by it on the edge of the strip r while the third term can be interpreted as the result of the diffraction of a wave, which has already been diffracted at one edge, on the other edge. The Kelvin wave near the strip (above or below r, depending on the sign of p) is a wave which is exponentially dependent on z and a travelling wave with respect to r which is reflected onpropagation fromthe boundaries of the strip, and is therefore representable in the form of the superpositionof two waves travelling in the positive and negative directions of the Or-axis. Also, the amplitude of the first wave is always smaller (in modulus) than the amplitude of the second wave. When conditions (16) are satisfied, there is no reflection from the boundaries of the strip. The term R(r, 2) in (13) has the estimate (11) and does not make any appreciable contribution to the total wave field when p1>1. We note that, by letting 2,---m and passing to the limit as e-0. it is possible to obtain the solution of the problem of diffraction by the half plane (z=O,t
Translated by E.L.S.