# Acoustics PDEs

## Contents

# Acoustics PDEs#

Author: Andrew Gibbs

Note

We assume a basic familiarity with the concept of a partial differential equation (PDE). We will not spend time explaining this, as there are plenty of free resources available online. For example, Wikipedia, MathWorld, or this video series by 3blue1brown.

## The Acoustic Wave Equation#

The fundamental PDE we consider, and the starting point for many of technical discussions which follow, is the *Wave Equation*:

where \(\Delta:=\sum_{j=1}^d \frac{\partial}{\partial x_j}\), \(d\) is the number of dimensions, and \(c\) is the speed of sound.

In one dimension, this equation can be derived from fundamental mechanical principles: by treating air molecules as tiny particles connected by vibrating springs, or by considering a vibrating string. Alternatively, the acoustic wave equation can be derived from Euler’s equation, by treating the air as an inviscid fluid, and imposing further physical assumptions [CK19].

Any acoustic wave \(U\) will satisfy (1). To turn this into a problem we can solve (with a unique solution), we must specify initial and/or boundary conditions. For example:

\(U(x,0)=f(x)\) for known \(f\) may describe a wave some initial time, and we may want to model how this evolves for \(t>0\). This is an

*initial condition*.\(U(X,t)=0\) for some \((d-1)\)-dimensional set \(X\) describes a

*sound-soft boundary condition*.More information about boundary conditions is given here.

## The Helmholtz Equation#

In many applications, the source of the sound can be assumed to be consistent. For example, when modelling acoustic noise from a source like a building site or motorway, we may only be interested in times when the noise is emanating consistently from the source. This leads to a simplification of (1) where the unknown does not depend on time.

Mathematically, this can be achieved by assuming a time-harmonic solution to the Wave Equation (1), specifically: \(U(x,t) = u(x) \mathrm{e}^{-\mathrm{i}\omega t}\) with \(\mathrm{i}:=\sqrt{-1}\) and frequency \(\omega>0\). This video by Jon Hargreaves provides a background on time-harmonic signals.

From this we obtain the Helmholtz equation

Here \(k:=\omega/c\) is the *wavenumber*.

As for the Wave Equation, we may impose boundary conditions for the unknown \(u\) to be uniquely determined. It is this boundary value problem (BVP) that we will solve.

Instead of formulating as a BVP, the Helmholtz Equation (2) may represent scattering by *inhomogenous media*, where \(k\) is variable inside of some bounded region. This is sometimes referred to as a *transmission* problem. To the author’s best knowledge, such problems are less common in acoustic modelling; at the time of writing this Knowledgebase does not contain any examples of this type.