The accuracy of the thermodynamic method with taking into account the measurement uncertainties in temperature and pressure can result in a global uncertainty of the B/A estimation within 5%, for tissue and 3% for liquid. The higher uncertainty for tissue samples is due to the inhomogeneous speed of sound [7]. Also Law et al. [7] demonstrates the dependence of B/A on the structural hierarchy of biological chemical composition of biological material solution and the chemical composition of biological solutions.

6.3 The Finite Amplitude Method

The finite amplitude method for B/A measurement was introduced by Beyer [3] based on the Fubini [8] expansion to the nonlinear wave equation for a lossless media. Fubini expansion provided the distortion of acoustic waves measured by the strength of the second harmonics. Cobb [9] improved the results with the incorporation of attenuation and dissipation. Cobb [9]’s method has been widely used in the measurement of the nonlinearity parameter of biomaterials and biological tissues [10, 11]. In this method, the second harmonic pressure p2 was given by [9]:

where p1 = first harmonic, α1,α2 = attenuation coefficients corresponding to p1 and p2 respectively.

In Eq. (6.9), β = B/2A = nonlinearity coefficient. The first term “1” in Eq. (6.9) accounts for the self-convective effect contributing to the nonlinearity of propagation of acoustic waves arising from the continuity equation [11]. Together with B/A which is the inherent material nonlinearity parameter, it describes the distortion of propagating sound waves in the medium.

The experimental measurement of B/A can be performed by the measurement of the values of p2 the second harmonic pressure and p1, the fundamental pressure as well as the corresponding attenuation coefficients α2 and α1. at different locations away from the transducer source by varying x. Then the nonlinearity parameter B/A can be determined by the substitution of the experimental data in Eq. (6.9) [9].

The finite amplitude methods can be classified into three main categories: deriving B/A directly from the wave’s shape, from the second harmonic component and from the fundamental component. A detailed description of each category of methods is given below.

Fig. 6.2 Illustration of the waveform deformation. The image shows the positive half-cycles of an undistorted sinus wave (solid line), and of a distorted wave (dashed line), where the distance w1 is the distance between the maxima of these waves, and w0 is the distance between points C and D, which are the intersections of the tangents at points A and B (After Panfilova [4])

6.3.1 The Wave Shape Method

The earliest works with the observations of the wave shape to measure B/A were performed with optical methods [4] (Fig. 6.2). When an optical wave propagates in a direction perpendicular to the ultrasonic beam, the initially flat wavefront of the optical wave is modulated in phase according to the velocity profile of the ultrasonic wave, and distorted due to the nonlinear propagation. In this way, the measurement of the diffraction of light allows for the reconstruction of the ultrasonic wave’s velocity profile and quantification of distortions by extracting w0 or w1 from its shape (Fig. 6.2). This leads to the derivation of B/A.

6.3.2 Second Harmonic Measuements

The amplitudes of the higher harmonics in the preshock region (σ ≤ 1) of a plane wave in a lossless medium can be given by [13]:

p(0)

p(σ ) = Jn (nσ ) (6.10) nσ

where n = position integer indicating the number of the harmonic: the fundamental p(0) at the source and higher harmonics p, and σ = shock parameter = 2πf p1 (0)Zβ/ρ0c03. Equation (6.10) is the Fubini solution [12] of the lossless Burgers’ equation given by

where τ = retarded time = t − (z/c0).

Expanding the Bessel function as a power series and neglecting higher order terms, one can write the amplitude of the 2nd harmonic as

The above equation shows the quadratic dependence on the transmitted pressure amplitude p1(0) at the source and that the amplitude of the 2nd harmonic increases proportionally to B/A, to the frequency of the signal f and to the distance z from the source.

Dunn et al. [12] considered both attenuation and diffraction effects and obtained the following expression to the 2nd harmonic:

where F(z) = diffraction correction factor, α1, α2 = attenuation coefficients of the fundamental and its harmonic and f = frequency of the transmitted signal.

The above theory is the basis of the finite amplitude method. Numerical works have been performed for the estimation of B/A through 2nd harmonic measurement. In most of these works, the 2nd harmonic is measured in the nearfield of a plane piston source, enabling the plane wave approximation.

6.3.3 Measurement from the Fundamental Component

There is an increase in absorption with signal intensity. At first, this was attributed to cavitation [13]. However, later, it was realized to be the result of energy transfer from the fundamental to higher harmonics [14]. Moreover, nonlinear attenuation increases with nonlinear effects which grow with source amplitude. This limits the sound power that can be delivered to a certain depth.

Due to energy transfer to higher harmonics, in the shock-free or pre-shock region of σ < 1 of a lossless medium, the fundamental component of a plane wave will increase as follows:

For the measurement of B/A in homogeneous media, two modes have been used: the transmission mode and the echo-mode. Hikata et al. [15] used the finite amplitude loss technique and performed the transmission mode measurement. Here the dependence of the received signal pressure p(z) on the intensity of the transmitted signal pressure p(0) was recorded at a fixed distance. They extracted β from the slope