Governing equations to derive Westervelt equation

Continuity:

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\textbf {u}})=0}$

Conservation of momentum:

${\displaystyle \rho \left({\frac {\partial {\textbf {u}}}{\partial t}}+{\textbf {u}}\cdot \nabla {\textbf {u}}\right)+\nabla p=(\lambda +2\mu )\nabla (\nabla \cdot {\textbf {u}})}$

with Taylor perturbation expansion on density:

${\displaystyle \rho =\sum _{0}^{\infty }\varepsilon ^{i}\rho _{i}}$

where ε is a small parameter, i.e. the perturbation parameter, the equation of state becomes:

${\displaystyle p=\varepsilon \rho _{1}c_{0}^{2}\left(1+\varepsilon {\frac {B}{2!A}}{\frac {\rho _{1}}{\rho _{0}}}+O(\varepsilon ^{2})\right)}$

If the second term in the Taylor expansion of pressure is dropped, the viscous wave equation can be derived. If it is kept, the nonlinear term in pressure appears in the Westervelt equation.

Westervelt equation

The general wave equation that accounts for nonlinearity up to the second-order is given by the Westervelt equation^[2]

${\displaystyle \,\nabla ^{2}p-{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}p}{\partial t^{2}}}+{\frac {\delta }{c_{0}^{4}}}{\frac {\partial ^{3}p}{\partial t^{3}}}=-{\frac {\beta }{\rho _{0}c_{0}^{4}}}{\frac {\partial ^{2}p^{2}}{\partial t^{2}}}}$

where ${\displaystyle p}$ is the sound pressure, ${\displaystyle c_{0}}$ is the small signal sound speed, ${\displaystyle \delta }$ is the sound diffusivity, ${\displaystyle \beta }$ is the nonlinearity coefficient and ${\displaystyle \rho _{0}}$ is the ambient density.

The sound diffusivity is given by

${\displaystyle \,\delta ={\frac {1}{\rho _{0}}}\left({\frac {4}{3}}\mu +\mu _{B}\right)+{\frac {k}{\rho _{0}}}\left({\frac {1}{c_{v}}}-{\frac {1}{c_{p}}}\right)}$

where ${\displaystyle \mu }$ is the shear viscosity, ${\displaystyle \mu _{B}}$ the bulk viscosity, ${\displaystyle k}$ the thermal conductivity, ${\displaystyle c_{v}}$ and ${\displaystyle c_{p}}$ the specific heat at constant volume and pressure respectively.

Burgers' equation

The Westervelt equation can be simplified to take a one-dimensional form with an assumption of strictly forward propagating waves and the use of a coordinate transformation to a retarded time frame:^[3]

${\displaystyle {\frac {\partial p}{\partial z}}-{\frac {\beta }{\rho _{0}c_{0}^{3}}}p{\frac {\partial p}{\partial \tau }}={\frac {\delta }{2c_{0}^{3}}}{\frac {\partial ^{2}p}{\partial \tau ^{2}}}}$

where ${\displaystyle \tau =t-z/c_{0}}$ is retarded time. This corresponds to a viscous Burgers equation:

${\displaystyle {\frac {\partial y}{\partial t'}}+y{\frac {\partial y}{\partial x}}=d{\frac {\partial ^{2}y}{\partial x^{2}}}}$

in the pressure field (y=p), with a mathematical "time variable":

${\displaystyle t'={\frac {z}{c_{0}}}}$

and with a "space variable":

${\displaystyle x=-{\frac {\rho _{0}c_{0}^{2}}{\beta }}\tau }$

and a negative diffusion coefficient:

${\displaystyle d=-{\frac {\rho _{0}c_{0}}{2\beta ^{2}}}\delta }$.

The Burgers' equation is the simplest equation that describes the combined effects of nonlinearity and losses on the propagation of progressive waves.

KZK equation

An augmentation to the Burgers equation that accounts for the combined effects of nonlinearity, diffraction, and absorption in directional sound beams is described by the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, named after Rem Khokhlov, Evgenia Zabolotskaya, and V. P. Kuznetsov.^[4] Solutions to this equation are generally used to model nonlinear acoustics.

If the ${\displaystyle z}$ axis is in the direction of the sound beam path and the ${\displaystyle (x,y)}$ plane is perpendicular to that, the KZK equation can be written^[5]

${\displaystyle \,{\frac {\partial ^{2}p}{\partial z\partial \tau }}={\frac {c_{0}}{2}}\nabla _{\perp }^{2}p+{\frac {\delta }{2c_{0}^{3}}}{\frac {\partial ^{3}p}{\partial \tau ^{3}}}+{\frac {\beta }{2\rho _{0}c_{0}^{3}}}{\frac {\partial ^{2}p^{2}}{\partial \tau ^{2}}}}$

The equation can be solved for a particular system using a finite difference scheme. Such solutions show how the sound beam distorts as it passes through a nonlinear medium.

Sonic boom

The nonlinear behavior of the atmosphere leads to change of the wave shape in a sonic boom. Generally, this makes the boom more 'sharp' or sudden, as the high-amplitude peak moves to the wavefront.

Acoustic levitation

Acoustic levitation would not be possible without nonlinear acoustic phenomena.^[6] The nonlinear effects are particularly evident due to the high-powered acoustic waves involved.

Ultrasonic waves

Because of their relatively high amplitude to wavelength ratio, ultrasonic waves commonly display nonlinear propagation behavior. For example, nonlinear acoustics is a field of interest for medical ultrasonography because it can be exploited to produce better image quality.

Musical acoustics

The physical behavior of musical acoustics is mainly nonlinear. Attempts are made to model their sound generation from physical modeling synthesis, emulating their sound from measurements of their nonlinearity.^[7]

Parametric arrays

A parametric array is a nonlinear transduction mechanism that generates narrow, nearly side lobe-free beams of low frequency sound, through the mixing and interaction of high-frequency sound waves. Applications are e.g. in underwater acoustics and audio.