Reflectionless boundary propagation formulas for partial wave solutions to the wave equation

Abstract

dimensions whose data is compactly supported at some initial time. For points outside a ball containing the initial support, we develop an outgoing wave condition, and associated one-way propagation formula, for the partial waves in the spherical-harmonic decomposition of the solution. The propagation formula expresses the $l$-th partial wave at time $t$ and radius $a$ in terms of order-$l$ radial derivatives of the partial wave at time $t-Delta t$ and radius $a-Delta t$. The boundary propagation formula can be applied to any differential equation that is well-approximated by the wave equation outside a fixed ball.

Keywords

• One-sided wave propagation
• Wave equation
• Reflectionless boundary conditions
• Partial waves
• Spherical-harmonic decomposition
• Open-space boundary conditions.