Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

TL;DR

This paper investigates the separation of variables for the bi-Helmholtz equation, revealing it does not admit regular separation but allows non-regular separation in certain coordinate systems, with applications to vibrations of circular plates.

Contribution

It applies geometric theory to analyze separation of variables for the bi-Helmholtz equation, showing the absence of regular separation and identifying non-regular separability in specific coordinates.

Findings

No regular separation in any coordinate system in pseudo-Riemannian spaces.

Non-trivial non-regular separation exists in Cartesian and polar coordinates.

Only trivial separability in parabolic and elliptic-hyperbolic coordinates.

Abstract

The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic-hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation.