Nonlinear acoustics
Large amplitudes require using full systems of governing equations of fluid dynamics (for sound waves in liquids and gases) and elasticity (for sound waves in solids).
The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.
A sound wave propagates through a material as a localized pressure change.
The local 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 always occurs but the effects of geometric spreading and of absorption usually overcome the self-distortion, so linear behavior usually prevails and nonlinear acoustic propagation occurs only for very large amplitudes and only near the source.
The pressure changes within a medium cause the wave energy to transfer to higher harmonics.
Since attenuation generally increases with frequency, a countereffect exists that changes the nature of the nonlinear effect over distance.
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.
Continuity: Conservation of momentum: with Taylor perturbation expansion on density: where ε is a small parameter, i.e. the perturbation parameter, the equation of state becomes: If the second term in the Taylor expansion of pressure is dropped, the viscous wave equation can be derived.
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] where
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.
Such solutions show how the sound beam distorts as it passes through a nonlinear medium.
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.
[6] The nonlinear effects are particularly evident due to the high-powered acoustic waves involved.
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.
[7] 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.
