Amplitude equations for a linear wave equation in a weakly curved pipe

TL;DR

This paper derives amplitude equations for a linear wave in a weakly curved pipe, revealing how boundary curvature causes eigenfrequency shifts through a perturbative analysis with boundary conditions.

Contribution

It introduces a perturbative method that exactly satisfies boundary conditions and derives amplitude equations showing frequency shifts due to boundary curvature.

Findings

Eigenfrequencies are shifted by boundary curvature.

Secular terms in naive perturbation are eliminated with singular perturbation.

Amplitude equations describe wave behavior in curved pipes.

Abstract

We study boundary effects in a linear wave equation with Dirichlet type conditions in a weakly curved pipe. The coordinates in our pipe are prescribed by a given small curvature with finite range, while the pipe's cross section being circular. Based on the straight pipe case a perturbative analysis by which the boundary value conditions are exactly satisfied is employed. As such an analysis we decompose the wave equation into a set of ordinary differential equations perturbatively. We show the conditions when secular terms due to the curbed boundary appear in the naive peturbative analysis. In eliminating such a secularity with a singular perturbation method, we derive amplitude equations and show that the eigenfrequencies in time are shifted due to the curved boundary.