The jump problem for the Helmholtz equation and singularities at the edges

K e y w o r d s A c o u s t i c scattering, Boundary value problems, Potential theory, Explicit solution, Cuts, Cracks, Screens. 1. I N T R O D U C T I O N In the jump problem for the Helmholtz equation outside cuts in a plane we specie" the jump of the solution and the jump of its normal derivative at the cuts. This problem is closely related to the Dirichlet and Neumann problems outside cuts in a plane, which are used to s tudy wave scattering by cracks in solids or by screens in fluids [1 10]. This problem is also closely related to a transmission problem for the Helmholtz equation, where the jump of unknown flmction and the j ump of its normal derivative are given on the closed curves [11-14]. The a t t empt to fornmlate the jump problem outside an open arc for the 2-D Laplace equatibn with one boundary condition of jump type is contained in [10], but the problent in [10] was not uniquely solvable and so was not well posed. In the present paper, we give a well-posed foi'mulation of boundary value problem outside cuts in a plane for the Helmholtz equation with two boundary conditions of. jump type. Moreover, we construct an explicit solution of our problem in the form of a single layer potential and an angular potential [4]. It should be stressed that our solution is explicit for cuts of an arbi t rary shape. This is the basic advantage of the jump problem over Dirichlet and Neumann problems outside cuts in a plane, since the explicit solution cannot be obtained in Dirichlet and Neumann problems for cuts of an arbi trary shape. In the present paper we also give explicit formulas for singularities of the solution gradient at the ends of cuts. It appears The research was partially, supported by the RFBR Grant 99-01-01063. 0893-9659/00/$ see front mat ter (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by AA/~S-rPE X PI] : S0893-9659(99)00211-6 72 P . A . KRUTITSKII tha t these singularities are weaker than in the Dirichlet and Neumann problems outside cuts in a plane [4,5]. I t is found that singularities in the jump problem are logarithmic, while in the Dirichlet and Neumann problems they are generally of power 1/2. The jump problem for the Helmholtz equation presented in this paper, can be effectively used to model wave propagation in cracked media. Our results also can be helpful in problems on crack determination [15-17]. 2. F O R M U L A T I O N O F T H E P R O B L E M By a simple open curve we mean a nonclosed smooth arc of finite length without self-intersections [5]. In the plane x = (Xl,X2) E R 2 we consider simple open curves F 1 , . . . , FN E C 2'A, N A E (0, 1], so tha t they do not have common points. We put F = Un=lFn. We assume tha t each curve Fn is parametrized by the arc length s Fn = {x: x = x(s) = (xl(s),x2(s)), s e [an, bn]}, n = 1 , . . . , N , so tha t al < bl < . " < aN < bg. Therefore, points x E F and values of the parameter s are in one-to-one correspondence. Below, the set of the intervals on the Os axis N Un=l[an, b~] will be denoted by F also. The tangent vector to F at the point x(s) we denote by T~ = (cosc~(s), since(s)), where cosc~(s) = x~(s), since(s) = x~2(s). Let nx = (sinc~(s) ,-cosc~(s)) be a normal vector to F at x(s). The direction of nx is chosen such that it will coincide with the direction of ~-x if nx is rotated anticlockwise through an angle of 7r/2. We consider F as a set of cuts. The side of F which is on the left when the parameter s increases will be denoted by F + and the opposite side will be denoted by F . We say tha t the function u(x) belongs to the smoothness class K if the following conditions are satisfied: 1) u(x) e C°(R 2 \ F) (3 C2(R 2 \ F) and u(x) is continuous at the ends of F; 2) Vu E C°(R 2 \ F \ X) , where X is a point set, consisting of the endpoints of F : X = N U,,=l(x(a~ ) U x(bn)); 3) in the neighbourhood of any point x(d) E X , for some constants C > 0 and s > 1 , the inequality IW l < ci:~ x (d ) [ ~ (1) holds, where x --* x(d) and d : aN or d : bn for n : 1 , . . . ,N. REMARK. In the definition of the class K we consider F as a set of cuts in a plane. In particular, the notation C°(R 2 \ F) denotes a class of functions, which are continuously extended on F from the left and right, but their values on F from the left and right can be different, so tha t the functions may have a jump across F. Let us formulate the jump problem for the Helmholtz equation in R 2 \ F. PROBLEM V. To find a function u(x) of class K, so that u(x) satisfies the Helmholtz equation in R 2 \ F Au + k2u = O, k = eonst ~ 0, 0 < a rgk < 7r, (2) satisfies the jump boundary conditions u(X)lx(s)~r+ -u(x)Ix(s)er= f l (s), (3a) 0 ~ x(s)er+ 0 ~ x(s)cr= f2(s), (3b) and meets the conditions at infinity. If arg k = 0, tha t is k = Re k > 0, we impose the Sommerfeld radiation conditions at infinity u ( x ) = o 1 O u ( x ) i k u ( ~ ) = o , Ixl = + x~ ~ ~ . (4a) ' 01~l