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# Nonlinear acoustics

Non-linear acoustics is a branch of physics dealing with sound waves being distorted as they travel.

Introduction

A sound wave propagates through a material as a localized pressure change. Increasing the pressure of a gas increases its temperature and the speed of sound in a compressible material increases with temperature; as a result, the wave travels faster during the high pressure phase of the oscillation than during the lower pressure phase. This affects the wave's frequency structure; for example, in a plane sinusoidal wave of a single frequency, the peaks of the wave travel faster than the troughs, and the signal becomes more like a sawtooth wave. In doing so, other frequency components are introduced, which can be described by the Fourier Series. This phenomenon is characteristic of a non-linear system, since a linear acoustic system responds only to the driving frequency.

Additionally, waves of different amplitudes will generate different pressure gradients, contributing to the non-linear effect.

Physical analysis

The pressure changes within a medium cause the wave energy to transfer to higher harmonics. Since attenuation generally increases with frequency, a counter effect exists that changes the nature of the nonlinear effect over distance. To describe their level of nonlinearity, materials can be given a nonlinearity parameter, B/A. The values of A and B are the coefficients of the first and second order terms of the Taylor series expansion of the equation relating the material's pressure to its density. Typical values for the nonlinearity parameter in biological mediums are shown in the following table.[1]

Material B/A

Blood 6.1

Brain 6.6

Fat 10

Liver 6.8

Muscle 7.4

Water 5.2

In a liquid usually a modified coefficient is used known as \beta = 1 + \frac{B}{2A}.

Mathematical model

Westervelt equation

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

\( \, \nabla^{2} p - \frac{1}{c_{0}^{2}} \frac{\partial^{2} p}{\partial t^{2}} + \frac{\delta}{c_{0}^{4}} \frac{\partial^{3} p}{\partial t^{3}} = - \frac{\beta}{\rho_{0} c_{0}^{4}} \frac{\partial^{2} p^{2}}{\partial t^{2}} \)

where p is the sound pressure, c_0 is the small signal sound speed, \( \delta \) is the sound diffusivity, \beta is the non-linearity coefficient and \rho_0 is the ambient density.

The sound diffusivity is given by

\( \, \delta = \frac{1}{\rho_{0}} (\frac{4}{3}\mu+\mu_{B}) + \frac{k}{\rho_{0}} (\frac{1}{c_{v}} - \frac{1}{c_{p}}) \)

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

Burgers equation

The Westervelt equation can be simplified to take a one-dimensional form with the use of a coordinate transformation to a retarded time frame. This is known as the Burgers equation[3]

\( \frac{\partial p}{\partial x} = \frac{\beta p}{\rho_{0} c_{0}^{3}}\frac{\partial p}{\partial \tau} + \frac{\delta}{2 c_{0}^{3}}\frac{\partial^{2} p}{\partial \tau^{2}} \)

where\( \tau = t-z/c_0 \) is retarded time. The Burgers equation is the simplest model that describes the combined effects of nonlinearity and losses on the propagation of plane progressive waves.

KZK equation

An augmentation to the Burgers equation that accounts for the combined effects of non-linearity, diffraction and absorption in directional sound beams is described by the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation.[4] Solutions to this equation are generally used to model non-linear acoustics.

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

\( \, \frac{\partial^2 p}{\partial z \partial \tau} = \frac{c_0}{2}\nabla^2_{\perp}p + \frac{\delta}{2c^3_0}\frac{\partial^3 p}{\partial \tau^3} + \frac{\beta}{2\rho_0 c^3_0}\frac{\partial^2 p^2}{\partial \tau^2} \)

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 non-linear medium.

Common occurrences

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

The practice of acoustic levitation would not be possible without understanding nonlinear acoustic phenomena.[6] The nonlinear effects are particular 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 a better image quality.

References

^ Ultrasonic imaging of the human body, P N T Wells, Rep. Prog. Phys

^ Hamilton, M.F.; Blackstock, D.T. (1998). Nonlinear Acoustics. Academic Press. p. 55. ISBN 0123218608.

^ Hamilton, M.F.; Blackstock, D.T. (1998). Nonlinear Acoustics. Academic Press. p. 57. ISBN 0123218608.

^ Anna Rozanova-Pierrat (PDF). Mathematical analysis of Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie. Retrieved 2008-11-10.

^ V. F. Humphrey (PDF). Non-linear propagation for medical imaging. Department of Physics, University of Bath, Bath, UK. Retrieved 2008-11-10.

^ http://science.howstuffworks.com/acoustic-levitation.htm

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