## 2 Theoretical Background  
In general, an ultrasonic pulse propagating through a nonlinear medium exhibits dispersion, i.e., each frequency of the pulse wave propagates at a different velocity value, and the pulse is distorted. This could be presented at the A-scan acoustic pressure as a function of time. In the frequency domain (the signal magnitude as a function of frequency), there is a graph with a maximum value at the central frequency of the pulse. In the present study, monochromatic ultrasonic waves, i.e., waves with a single frequency that are induced in the material, are used. In the frequency domain, the magnitude over frequency is, therefore, a single vertical line, which is described by the amplitude of the characteristic wave propagating through the material. According to nonlinear acoustics, A\({}_{0}\) is the amplitude (magnitude) of the fundamental frequency that is induced in the material. A\({}_{1}\), A\({}_{2}\), and A\({}_{3}\) are respectively the amplitudes (magnitudes) of the waves that are received after they propagated through the material; A\({}_{1}\) is the amplitude of the fundamental frequency, A\({}_{2}\) of the 2 nd harmonic and A\({}_{3}\) of the 2 nd harmonic. While a pure fundamental frequency f is induced in the material, the material itself being a nonlinear medium generates the harmonics 2\(t\), 3\(f\) and higher. In brief, the combination of the fundamental and harmonics amplitudes, as shown in the theoretical background, represents the way that these harmonics interact in the medium during the hardening process. Inherently, the materials that are applied in technological needs are nonlinear media and this has as a result, the deformation of a fundamental elastic wave (sin \(\omega\)t) when the wave is propagating in such material [42; 43; 44]. The expansion of acoustic wave propagation equation by introducing nonlinear terms into the stress-strain relationship, leads to the estimation of higher-order elastic constants of a structure. The parameter that is symbolized with \(\beta\) is the expression of the material's intrinsic nonlinearity. The generation of second- and higher-order harmonics is derived from a monochromatic longitudinal wave propagating through a solid medium. Equation (1) refers to the differential equation of motion which presents a coefficient for each term that distinguishes the linear and nonlinear terms,  
\[\rho\ddot{u}=\alpha\frac{\partial^{2}u}{\partial a_{i}^{2}}+\delta\frac{ \partial u}{\partial a_{i}}\left(\frac{\partial^{2}u}{\partial a_{i}^{2}}\right) \tag{1}\]  
the values \(\alpha\) and \(\delta\) express the elastic constants. In addition, the coefficient \(\alpha\) is expressed only in terms of the second-order elastic constants; however, the coefficient \(\delta\) is expressed in terms of elastic constants of both the second and third orders [44; 45]. An approximate solution involving the second harmonic can be obtained by considering a solution to Equation (1) in the form \(u=uo+u^{\prime}\) and by performing suitable substitutions:  
\[u(a,t)=Asin(ka-\omega t)-\frac{a\,\delta k^{2}A^{2}}{8\,\rho_{0}C_{0}^{2}}cos [2(ka-\omega t)] \tag{2}\]  
where \(C_{0}=(\alpha/\rho_{0})^{1/2}\) is the ultrasonic phase velocity. In the wave equation, the nonlinear acoustics parameter, \(\beta\), is defined as the ratio of the nonlinear to the linear term. It is then possible to rewrite Equation (2) as:  
\[u(a,t)=Asin(ka-\omega t)+\frac{a\beta_{2}(kA)^{2}}{8}cos[2(ka-\omega t)] \tag{3}\]  
The amplitude A in the first term and the expression \(\frac{a\beta_{2}(kA)^{2}}{8}\) in the second term of Equation (3), are respectively, the fundamental amplitude \(A_{1}\) and the second harmonic amplitude \(A_{2}\). The nonlinearity parameter of the second order, \(\beta_{2}\), can therefore be expressed in terms of amplitudes \(A_{1}\) and \(A_{2}\):  
\[\beta_{2}=\frac{8}{ak^{2}}\left(\frac{A_{2}}{A_{1}^{2}}\right) \tag{4}\]where \(a\) is the propagation distance and \(k=2\pi/\lambda\) is the wavenumber.