Nonlinear Acoustic Wave Equations for Sound Propagation in Fluids and in Solids

4 2 Nonlinear Acoustic Wave Equations for Sound Propagation in Fluids and in Solids

1 ), ξ = bulk modulus, and ï = shear viscosity.

c p

2.1.1 The Westervelt Equation [1]

By deleting the last term on the right hand side of Eq. (2.4), one will obtain the Westervelt equation [1]:

Westervelt equation can be applied in parametric acoustic array and in medical ultrasound.

2.1.2 The Burgers’ Equation [2]

The Burgers’ equation can be derived from the Navier Stokes equations which can be written as:

and

where ρ = density, p = pressure, v = velocity, and μ = fluid viscosity.

Equations (2.6) and (2.7) are for divergence free incompressible flow given by (2.6) and ρt = 0 and the gravitational effects are neglected. By simplifying this to a one-D problem with propagation only in the x direction, one has.

There is no pressure gradient in one-D problem, then:

which is the inviscid Burgers’ equation.

Burgers’ equation was proposed by Bateman [3] in 1915 and was subsequently analysed by Burgers’ [2] in 1948. This partial differential equation is used in nonlinear acoustics, continuous stochastic processes,shock waves, heat conduction, turbulence, traffic flow, dispersive, water, and traffic flow. It is one of the very few nonlinear partial differential equations that can be solved exactly. It can be considered as a simplified form of Navier Stokes equations of fluid mechanics.

2.1.3 KZK Equation

KZK equation is commonly used in medical ultrasound. The KZK equation describes the propagation of finite amplitude sound waves in thermo-viscous medium. It was derived by Khokhlov and Zabolotskaya [3] and Kuznetsov [4]. KZK equation combines the effects of nonlinearity, diffraction, and absorption. There has been a substantial increase in the use of intense ultrasound in industrial and medical applications in recent years. Nonlinear effects due to intense ultrasound have become very important in many therapeutic and surgical procedures. It is of interest to know that the human body is a nonlinear medium. Ultrasound sources together with finite amplitude effects generate strong diffraction phenomena producing waveform varying from point to point within the sound beam. In addition, the significant absorption of sound due to the biological media also has to be considered.

Ultrasound sources together with finite amplitude effects generate strong diffraction phenomena producing waveform varying from point to point within the sound beam. The KZK equation for sound beam path in the z direction and the (x, y) plane perpendicular to that can be written as

6 2 Nonlinear Acoustic Wave Equations for Sound Propagation in Fluids and in Solids

where p = acoustic pressure, τ = t − z/c0 = retarded time, c0 = sound speed, ρ0 = ambient density, β = nonlinearity coefficient = 1 + 2BA ,

where 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 and

μB = bulk viscosity, κ = the thermal conductivity, cv and c p = specific heat at constant volume and pressure respectively.

2.1.4Nonlinear Acoustic Wave Equations for Sound Propagation in Solids

To describe sound propagation in solids, the elasticity equation has to be used. To describe the elastic behaviour of the material, the elastic energy is expressed as a function of strain by using the Taylor expansion of the elastic strain energy:

where Ci j kl = second order elastic constants, Ci j klmn = third order elastic constants, E = strain tensor, ρ = density, W = elastic strain energy.

For nonlinear acoustics in solids, the Lagrangian description is usually used. The acoustic equation of motion in solids in Lagrangian coordinates is given by

where P = Piola–Kirchhoff strain tensor and U = x − a = Lagrangian displacement or displacement relative to the natural position, a = natural or equilibrium position,

In (2.14), the second term accounts for quadratic nonlinearity and the third term the cubic nonlinearity and

where the first term accounts for material nonlinearity and the second, third, fourth terms the geometric nonlinearity or linear elastic constants.