Sine function with a cosine attitude

TL;DR

This paper constructs two asymptotically sinusoidal solutions to the wave equation with a phase shift, using the J-matrix scattering method, and provides their analytic series representations in one and three dimensions.

Contribution

It introduces a method to construct and analyze sinusoidal solutions with phase shifts using basis functions and the J-matrix scattering approach.

Findings

Analytic series representations of solutions are obtained.

One solution is even and behaves as sin(x), the other is odd and behaves as cos(x).

Eliminating terms in the series makes solutions vanish near the origin.

Abstract

We give a revealing expose that addresses an important issue in scattering theory of how to construct two asymptotically sinusoidal solutions of the wave equation with a phase shift using the same basis having the same boundary conditions at the origin. Analytic series representations of these solutions are obtained. In 1D, one of the solutions is an even function that behaves asymptotically as sin(x), whereas the other is an odd function, which is asymptotically cos(x). The latter vanishes at the origin whereas the derivative of the former becomes zero there. Eliminating the lowest N terms of the series makes these functions vanishingly small in an interval around the origin whose size increases with N. We employ the tools of the J-matrix method of scattering in the construction of these solutions in one and three dimensions.