Semi-analytic time-domain solutions to wave-scattering problems

Abstract

Scattered waves are constructed in the time domain by explicitly evaluating the inverse Laplace transform integral as a summation of complex residues. The method is applied to two problems: a bead on a semi-infinite string and acoustic scattering by a sphere. In the first problem, the end of an infinitely long string is oscillated in a specified way. The incident wave is scattered by a bead of negligible width and specified mass, some given distance from the end of the string. The initial boundary value problem for the total wave is solved in the frequency domain. The corresponding time-domain solution is obtained with complex calculus; the Laplace transform is inverted by closing the Bromwich contour and applying a rotated Jordan’s lemma, allowing the time-domain solution to be written as a sum of complex frequency-domain residues. In the second problem, a rotationally-symmetric acoustic wave is scattered by a soft sphere. The same method of closing the Bromwich contour and applying Jordan’s lemma is used to write the time-domain solution as a sum of residues. Mathematically equivalent to the spherical scattering problem, the residue solution is also constructed for a point source. The closed-form solution for the point source problem is known, and evaluation on the surface of a sphere gives the appropriate boundary conditions for the residue solution. Application of Jordan’s lemma is considered in three space-time regions, corresponding to times where the incident wave is contained in the sphere, times where the incident wave has begun to exit the sphere, and times where the incident wavefront has completely exited the sphere. Solving in the frequency domain and writing the inverse Laplace transform as a summation of residues avoids the accumulation of error associated with time-stepping methods, making the method especially useful in evaluating late-time solutions. The method is entirely mesh-free, allowing evaluation of the solution anywhere in the space-time domain.