Nonlinear Acoustics -- Perturbation Theory and Webster's Equation

TL;DR

This paper derives and analyzes a nonlinear version of Webster's horn equation using perturbation theory, providing methods to compute frequency corrections and harmonic modes for variable cross-sectional tubes.

Contribution

It introduces a perturbative approach to nonlinear Webster's equation and develops an algorithm for calculating harmonic modes in complex horn geometries.

Findings

First-order frequency corrections for Gabriel's Horn geometry

Perturbative solutions for nonlinear acoustic wave equations

Algorithm for harmonic mode computation in variable cross-section tubes

Abstract

Webster's horn equation (1919) offers a one-dimensional approximation for low-frequency sound waves along a rigid tube with a variable cross-sectional area. It can be thought as a wave equation with a source term that takes into account the nonlinear geometry of the tube. In this document we derive this equation using a simplified fluid model of an ideal gas. By a simple change of variables, we convert it to a Schr\"odinger equation and use the well-known variational and perturbative methods to seek perturbative solutions. As an example, we apply these methods to the Gabriel's Horn geometry, deriving the first order corrections to the linear frequency. An algorithm to the harmonic modes in any order for a general horn geometry is derived.