Publication Date

Spring 5-16-2024

Abstract

The 3-space, 1-time dimensional scalar wave equation, or 3+1 wave equation, describes the propagation of scalar or acoustic waves. The unforced homogeneous equation admits a class of outgoing solutions relative to a chosen fixed center, so called “multipole” solutions. This thesis examines near-to-far signal propagation in the context of these multipole solutions. Given a time-series (history of values) for a multipole solution recorded at a radius r1, near- to-far signal propagation recovers the corresponding time-series at larger radius r2 ≫ r1. This propagation takes into account both the appropriate time delay r2 − r1 and corrections to the wave shape. Mathematically, the propagation is described by a Laplace convolution

involving the time-series at r1 and a “kernel” which is a weighted sum of time-dependent exponential functions. This thesis studies the Alpert-Greengard-Hagstrom [5] algorithm for rational approximation. It applies the algorithm, with some improvements not considered in [4], to approximation of a near-to-far kernel for l = 64 (a moderately large l). As an exercise, it also considers approximation of the scaled Bessel function J1(t)/t. Precisely, the thesis considers approximation of its Laplace transform as a rational function.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Stephen Lau

Second Committee Member

Jens Lorenz

Third Committee Member

James Degnan

Project Sponsors

Stephen Lau

Language

English

Keywords

Radial wave equation; 1 + 1 wave equation; 3 + 1 wave equation; spherical coordinates; multipole expansion; Nabla operators; Laplace transform; Near-to-far kernel; Bessel functions; Rational and polynomial approximation; propagation; FORTRAN implementation; Gram-Schmidt orthogonalization and re-orthogonalization.

Document Type

Thesis

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