Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. 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). These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.
Introduction A sound wave propagates through a material as a localized pressure change. Increasing the pressure of a gas or fluid increases its local temperature. 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 affects the wave's frequency structure; for example, in an initially plain sinusoidal wave of a single frequency, the peaks of the wave travel faster than the troughs, and the pulse becomes cumulatively more like a sawtooth wave. In other words, the wave distorts itself. In doing so, other frequency components are introduced, which can be described by the Fourier series. This phenomenon is characteristic of a nonlinear system, since a linear acoustic system responds only to the driving frequency. 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. Additionally, waves of different amplitudes will generate different pressure gradients, contributing to the nonlinear 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 countereffect 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 {\displaystyle B/A} . The values of A {\displaystyle A} and B {\displaystyle 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. The Taylor series has more terms, and hence more coefficients (C, D, ...) but they are seldom used. Typical values for the nonlinearity parameter in biological mediums are shown in the following table.
In a liquid usually a modified coefficient is used known as β = 1 + B 2 A {\displaystyle \beta =1+{\frac {B}{2A}}} .
Mathematical model
Governing equations to derive Westervelt equation Continuity:
∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\textbf {u}})=0}
Conservation of momentum:
ρ ( ∂ u ∂ t + u ⋅ ∇ u ) + ∇ p = ( λ + 2 μ ) ∇ ( ∇ ⋅ u ) {\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:
ρ = ∑ 0 ∞ ε i ρ i {\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:
p = ε ρ 1 c 0 2 ( 1 + ε B 2 ! A ρ 1 ρ 0 + O ( ε 2 ) ) {\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
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