Wave Propagation in 1-D Spiral geometry

TL;DR

This paper explores wave propagation in a 1-D spiral-shaped string, demonstrating that the problem reduces to a Bessel differential equation, linking it to radial waves in 2-D circular membranes.

Contribution

It introduces a novel analysis of wave behavior in spiral geometries, connecting 1-D spiral wave modes to classical Bessel equations and 2-D circular wave problems.

Findings

Wave equation in spiral geometry reduces to Bessel differential equation.

Spiral wave modes are analogous to radial waves in circular membranes.

Provides a mathematical framework for analyzing spiral wave propagation.

Abstract

In this article, we investigate the wave equation in spiral geometry and study the modes of vibrations of a one-dimensional (1-D) string in spiral shape. Here we show that the problem of wave propagation along a spiral can be reduced to Bessel differential equation and hence, very closely related to the problem of radial waves of two-dimensional (2-D) vibrating membrane in circular geometry.