# Analysis of Elastic Nonlinearity Using Continuous Waves: Validation and Applications
## 3 Validation
### Quantitative Behaviour  
As discussed in Section 3.1, the parameters derived by fitting the curves of the relative variations of velocity and attenuation vs. amplitude do not correspond to those theoretically expected. This is not surprising, since \(S\) and \(\lambda\) (and hence \(c\) and \(\alpha\)) are defined as a function of strain. In addition, detected signals (both numerically and experimentally obtained) are resulting from waves travelling in space and time and are analysed using a long time window (a few cycles of the signals). Hence,  
Figure 6: Validation of the method: behaviour of \(\Delta c/c_{0}\) and \(\Delta\alpha/\alpha_{0}\) vs. displacement amplitude for hysteretic nonlinearity. Arrows indicate branches for increasing and decreasing amplitudes of excitation. the nonlinear signature contained in the signal is the result of averaging nonlinearities over space and time. Equivalently, it is feasible asserting that for each amplitude A, the signals are influenced by an averaged strain. Thus, a comparison of the theoretical values of the nonlinear parameters (\(\delta_{c}^{th}\), \(\gamma_{c}^{th}\) and \(\delta_{a}^{th}\), i.e., the input parameters of the simulations derived in the table of Figure 1) and the coefficients of the polynomial fitting of the curves obtained from the analysis of numerical data of curves reported in Figure 3 (\(\delta_{c}^{fit}\), \(\gamma_{c}^{fit}\) and \(\delta_{a}^{fit}\)), must be carefully performed. Indeed, we could expect that:  
\[c_{th}=c_{0}\left(1+\delta_{c}^{th}\epsilon^{2}+\gamma_{c}^{th} \epsilon^{4}\right)\] \[c_{measured}=c_{0}\left(1+\delta_{c}^{fit}<\epsilon^{2}>+\gamma_{ c}^{fit}<\epsilon^{4}>\right) \tag{18}\]  
and similarly for \(\alpha\). In Equation (18), the symbols \" \(<>\)\" represent the average over space (between 0 and the sample length \(L\)) and time (between 0 and one period \(T\)). Furthermore, as shown in Appendix C, averaged quadratic and quartic strains are proportional to the amplitude \(A\) of the displacement at \(x=L\) (hence, the preservation of proportionality discussed before). Therefore,  
\[<\epsilon^{2}(x,t)>=K_{1}^{2}A^{2}\] \[<\epsilon^{4}(x,t)>=K_{2}^{4}A^{4} \tag{19}\]  
where \(K_{1}\
eq K_{2}\) and both could be analytically calculated and depend on \(S_{0}\), \(\lambda_{0}\) (connected to \(c_{0}\) and \(\alpha_{0}\) as discussed previously) and \(L\), as shown in Appendix C. It follows that  
\[c_{measured}=c_{0}\left(1+\delta_{c}^{fit}K_{1}^{2}A^{2}+\gamma_ {c}^{fit}K_{2}^{4}A^{4}\right)\] \[\alpha_{measured}=\alpha_{0}\left(1+\delta_{a}^{fit}K_{1}^{2}A^{2 }\right) \tag{20}\]  
Thus, when fitting the measured curves (e.g., those in Figure 3) with a polynomial function,  
\[\Delta c_{measured}/c_{0}=q_{1}A^{2}+q_{2}A^{4}\] \[\Delta\alpha_{measured}/\alpha_{0}=r_{1}A^{2}\]  
we obtain the estimated nonlinear parameters as:  
\[\delta_{c}^{fit}=q_{1}/K_{1}^{2}\] \[c_{th}=c_{0}\left(1+\delta_{c}^{th}\epsilon^{2}+\gamma_{c}^{th} \epsilon^{4}\right)\]0 \[c_{th}=c_{0}\left(1+\delta_{c}^{th}\epsilon^{2}+\gamma_{c}^{th} \epsilon^{4}\right)\]1  
Calculations were performed and results are shown in Table 1 for the data reported in \[c_{th}=c_{0}\left(1+\delta_{c}^{th}\epsilon^{2}+\gamma_{c}^{th} \epsilon^{4}\right)\]2 We obtain an excellent agreement between measured nonlinear parameters with those theoretically expected in Cases 1, 2 and 3. The efficiency of the approach in detecting accurately effects of nonlinearities in modulus/damping on nonlinearities in velocity/attenuation (Case 3) is remarkable. The estimation of nonlinear parameters of velocity is poorer when the dependence of the nonlinear parameter is a combination of two power laws (quadratic and quartic). We have a reasonable (at least as far as the order of magnitude is concerned) estimation of both quadratic and quartic nonlinear parameters, but the error increases to about 15-20%. This is probably due to the fact that the hypothesis of Equation (18) is no longer valid. The estimated parameters for attenuation are still well predicted.