Approximation of Radiation Boundary Conditions

Radiation boundary conditions appear in a wide variety of physical problems involving wave propagation [ 1,2]. For these problems, boundary conditions should be specified for dynamical quantities propagating on characteristics entering the domain of integration, while no boundary conditions are necessary for quantities propagating on characteristics leaving the domain. If the domain of integration is finite, numerical solutions of such a problem may be effected by standard techniques, such as the finite difference method. On the other hand, if the desired region of integration is infinite, numerical solution encounters difficulties because of the necessary limitation of the computational domain to a finite region. With initial-value or radiation problems, the appropriate boundary conditions at infinity are radiation boundary conditions; that is, the amplitude of waves entering from infinity is required to be zero, while no conditions are placed on outgoing waves propagating to infinity. With scattering problems, the amplitude of an incoming wave from infinity is specified and the amplitudes of the scattered outgoing waves emanating from the physical domain are sought; scattering problems can be easily reformulated as radiation problems. In this paper we discuss the approximation of wave propagation problems in infinite regions by finite discrete problems. A prototype radiation problem is given by the n-dimensional wave equation