Nonlinear acoustics

Nonlinear acoustics
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Nonlinear_acoustics".

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This article is about sound waves being distorted as they travel.

1 Introduction
2 Physical Analysis
3 Mathematical model
4 Common occurrences
- 4.1 Sonic boom
- 4.2 Ultrasonic waves
5 References

Introduction

When a sound wave propagates through a material, it acts as a force that creates localized pressure changes. The speed of sound in a compressible material increases with pressure because the molecules transmitting the energy are closer. 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 plain sinusoidal wave with one frequency, the peaks of the wave travel faster than the dips, and the signal becomes more like a sawtooth wave. In doing so, other frequency components are introduced, as Fourier analysis will show. This phenomenon implies a non-linear system, since a linear system cannot output frequencies that were not a part of the input signal.

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

Mathematical model

The propagation of sound beams in a medium that exhibits non-linearity, diffraction and absorption is described by the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation². 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 written³

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

where $p$ is the sound pressure, $c 0$ is the small signal sound speed, $δ$ is the sound diffusivity, $β$ is the non-linearity coefficient, $ρ 0$ is the ambient density and $τ = t - z / c 0$ is retarded time.

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.

Ultrasonic waves

Because of their relatively high amplitude to wavelength ratio, ultrasonic waves commonly display nonlinear propagation behavior.

References

^ Ultrasonic imaging of the human body, P N T Wells, Rep. Prog. Phys
^ Anna Rozanova-Pierrat. "Mathematical analysis of Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation" (PDF). Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie. Retrieved on 2008-11-10.
^ V. F. Humphrey. "Non-linear propagation for medical imaging" (PDF). Department of Physics, University of Bath, Bath, UK. Retrieved on 2008-11-10.

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