Nonradiating sources of the biharmonic wave equation

TL;DR

This paper investigates nonradiating sources in biharmonic wave equations, providing characterizations, connections to other wave equations, and examples, highlighting implications for inverse source problem uniqueness.

Contribution

It introduces new characterizations of nonradiating sources for biharmonic wave equations and links them to Helmholtz equations, with explicit examples demonstrating their existence.

Findings

Nonradiating sources exist for biharmonic wave equations.

Connections established between biharmonic and Helmholtz nonradiating sources.

Existence of such sources affects inverse problem uniqueness.

Abstract

This paper offers an extensive exploration of nonradiating sources for the two- and three-dimensional biharmonic wave equations. Various equivalent characterizations are derived to reveal the nature of a nonradiating source. Additionally, we establish the connection between nonradiating sources in the biharmonic wave equation and those in the Helmholtz equation as well as the modified Helmholtz equation. Several illustrative examples are explicitly constructed to showcase the existence of nonradiating sources. One significant implication of the existence of nonradiating sources is that it undermines the uniqueness of the inverse source problem when utilizing boundary data at a fixed frequency.