The Acoustic Wave Equation and Solutions
Introduction
There are actually many different types of pressure fluctuations that can exist in a compressible fluid. Acoustic waves are simply one kind of pressure fluctuation. Others include ultrasonic or infrasonic waves whose frequencies extend beyond the range of human hearing, high-intensity waves that may elicit a pain response as opposed to an audible response, and shock waves generated by explosions or supersonics.
Before we begin, we should define some terminology and variables that will be used.
\vec r is the equilibrium position of a fluid element.
\vec r=x\hat x+y\hat y+z\hat z
Unit vectors are denoted by \hat x, \hat y, and \hat z for the x, y, and z directions respectively.
\vec{\xi} is the particle displacement of a fluid element from its equilibrium position.
\vec{\xi}=\xi_x\hat x+\xi_y\hat y+\xi_z\hat z
\vec u is the particle velocity of the fluid element, defined as the time derivative of the particle displacement.
\vec u=\frac {\partial\vec{\xi}}{\partial t}=u_x\hat x+v_y\hat y+w_z\hat z
\rho is the instantaneous density at (x, y, z)
\rho_0 is the equilibrium density at (x, y, z)
s is the condensation at (x, y, z), defined as
s=\frac {\Delta\rho}{\rho_0}=\frac {\rho-\rho_0}{\rho_0}
Specific Acoustic Impedance
The ratio between the acoustic pressure, \textbf p, and particle speed, \textbf u, in a medium is defined as the specific acoustic impedance, \textbf z. Since both \textbf p and \textbf u are complex, \textbf z\in\mathbb{C}.
\textbf z=\frac {\textbf p}{\textbf u} \tag{1}
For plane waves, the specific acoustic impedance is
\textbf z=\pm\rho_0 c \tag{2}
The sign choice in Eq. 2 is dependent on whether the wave propagates in the +x or -x direction. Units of \textbf z are the rayl, defined as being equivalent to 1 Pa-s/m. Since the term \rho_0c is an important property of the medium, it is called the medium’s characteristic impedance.
Depending on the medium, values for \textbf z can vary across a wide range of numbers. For example, air at atmospheric pressure and 20^{\circ}C has a specific acoustic impedance of
\textbf z=415\text{ Rayl}
whereas distilled water, with its density being much higher than that of air, has a specific acoustic impedance of
\textbf z=1.48\times10^6\text{ Rayl}
Eq. 2 gives a real quantity and is only valid for progressive plane waves (sinusoidal waves propagating in a direction). For standing or diverging waves, \textbf z becomes a complex number.
\textbf z=r+jx
The real part, r, is the specific acoustic resistance and the imaginary part, x, is the specific acoustic reactance of the medium.
Spherical Waves
Some waves are more closely related to spherical waves rather than plane waves. These kinds of waves originate from a point source and propagate uniformly in all directions. In reality, this is not always true due to different mediums offering different levels of resistance. But more often than not, spherical waves are a reasonable approximation to make in most settings.
It is natural to use spherical coordinates when describing spherical waves. The Laplacian operator becomes
\nabla^2=\frac {\partial^2}{\partial r^2}+\frac 2r\frac {\partial}{\partial r}+\frac 1{r^2\sin\theta}\frac {\partial}{\partial\theta}\left(\sin\theta\frac {\partial}{\partial\theta}\right)+\frac 1{r^2\sin^2\theta}\frac {\partial^2}{\partial\phi^2} \tag{3}
where x=r\sin\theta\cos\phi, y=r\sin\theta\sin\phi, and z=r\cos\theta. The angle \theta is the angle between the radial vector and the z-axis, whereas the angle \phi is the angle in the xy-plane between the projected radial vector (on the xy-plane) and the x-axis. Some references will swap the definition of \theta and \phi. If we assume spherical symmetry for spherical waves, then properties like pressure should only depend on time and the radial distance from the source. Therefore, Eq. 3 reduces down to
\nabla^2=\frac {\partial^2}{\partial r^2}+\frac 2r\frac {\partial}{\partial r}
When applied to the pressure field, the wave equation is
\frac {\partial^2p}{\partial r^2}+\frac 2r\frac {\partial p}{\partial r}=\frac 1{c^2}\frac {\partial^2p}{\partial t^2} \tag{4}
The solution to Eq. 4 for a harmonic spherical wave is given by
p=\frac 1rf_1(ct-r)+\frac 1rf_2(ct+r)=\frac {\textbf A}re^{j(\omega t-kr)}+\frac {\textbf B}re^{j(\omega t+kr)}
The first and second terms resemble wave propagations radially in the outwardly and inwardly direction. Waves propagating away from the point source are called diverging waves, whereas those propagating towards the point source are converging waves. The particle velocity and velocity potentials are given by
\begin{aligned} \bm{\Phi} & =-\frac {\textbf p}{j\omega\rho_0} \\ \vec{\textbf u} & =\nabla\bm{\Phi}=\left(1-\frac j{kr}\right)\frac {\textbf p}{\rho_0c}\hat r \end{aligned} \tag{5}
The physically-observed acoustic quantities can be determined by taking the real part of Eq. 5. Due to the presence of the 1-\frac j{kr} term above, the particle velocity is out of phase with the pressure \textbf p. Therefore, the specific acoustic impedance is no longer \rho_0c like that of a plane wave, but rather
\textbf z=\rho_0c\cos\theta e^{j\theta}=\frac {\rho_0krc}{\sqrt{1+k^2r^2}}e^{j\theta}=\rho_0c\left(\frac {k^2r^2}{1+k^2r^2}+\frac {jkr}{1+k^2r^2}\right) \tag{6}
The trigonometric values of \theta can be determined by drawing a right triangle. The side opposite to \theta has length 1 and the hypotenuse is \sqrt{1+k^2r^2}. If we separate the exponential into real and imaginary components, we can derive the specific acoustic resistance and reactance for spherical waves.
\begin{aligned} r & =\frac {\rho_0k^2r^2c}{1+k^2r^2} \\ x & =\frac {\rho_0krc}{1+k^2r^2} \end{aligned}