# Analysis of Elastic Nonlinearity Using Continuous Waves: Validation and Applications
## 2 Semianalytical Approach for the Determination of Modulus and Damping
### Numerical Solutions
#### 2.2.2 Estimation of Nonlinear Parameters on Velocity and Attenuation  
From Equations (3) and (4), it is possible to derive the expressions which evidence how nonlinearities on modulus and damping may determine nonlinearities on velocity and attenuation:  
\[c=\sqrt{\frac{S}{\rho}\frac{2}{(1+\lambda^{2}/(\rho^{2}\omega^{2}))}}\] \[\alpha=\frac{c\lambda}{2S} \tag{13}\]  
It follows that nonlinearity on velocity \(c\) is due to both nonlinearities on modulus \(S\) and damping \(\lambda\), and similarly for what concerns nonlinearity on attenuation \(\alpha\).  
Normally, in the literature, it is assumed that nonlinearities on \(c\) and \(\alpha\) are decoupled, in the sense that, assuming small \(\lambda\) and small nonlinearity on \(S\), we have:  
\[\frac{\Delta c}{c_{0}}=\frac{1}{2}\frac{\Delta S}{S_{0}}\] \[\frac{\Delta\alpha}{\alpha_{0}}=\frac{\Delta\lambda}{\lambda_{0}} \tag{14}\]  
However, as we shown here, this is not always the case, since Equation (13) provides a more complex relation. Substituting the Taylor expansion for \(S\) and \(\lambda\), in principle the nonlinear dependence of \(c\) and \(\alpha\) could be derived with a Taylor expansion for small strains:  
\[c=c_{0}\left(1+\delta_{c}\epsilon^{2}+\gamma_{c}\epsilon^{4}\right)\] \[\alpha=\alpha_{0}\left(1+\delta_{\alpha}\epsilon^{2}+\gamma_{ \alpha}\epsilon^{4}\right) \tag{15}\]  
The analytical solution is very cumbersome and it is not reported here.  
Figure 1: Table of the nonlinear parameters used, for each different case, in the numerical simulations.  
A simpler procedure to estimate the coupling between modulus/damping and velocity/attenuation nonlinearities can be adopted. For given values of modulus/damping nonlinear parameters, \(c\) and \(\alpha\) can be derived using Equation (13) and plotted vs. strain. The resulting curves are fit with a polynomial function to derive numerically the coefficients of Equation (15). An example of derivation is shown in Figure 2.  
Results of the calculation for the various cases analysed in the following are reported in the last two columns of the table in Figure 1. We recall that in the cases considered in this paper we always have \(\gamma_{\alpha}\approx 0\), since the input nonlinear parameter \(\gamma_{\lambda}\) was always set equal to zero. Results indicate that neglecting, as often done, the effects of damping nonlinearity on velocity nonlinearity is not always correct, while even more critical is neglecting modulus nonlinearity effects on attenuation coefficient nonlinearity. Indeed:  
* For Cases 1 and 2, \(\delta_{c}=0.5\;\delta_{S}\). In Case 3, the relationship between \(\delta_{c}\) and \(\delta_{S}\) given by Equation (14) is still approximately valid, being the effect of \(\delta_{\lambda}\) on \(\delta_{c}\) about 1% only.
* In all cases, the effect of modulus nonlinearity \(\delta_{S}\) on attenuation nonlinearity \(\delta_{\alpha}\) is more significant (more than 3% in Case 3). Therefore, the assumption \(\Delta\lambda/\lambda_{0}=\Delta\alpha/\alpha_{0}\) is not accurate. The effect increases when increasing the modulus nonlinear parameter.
* In the case of materials with high damping, Cases C and D, a significant effect of damping nonlinearity \(\delta_{\lambda}\) on velocity nonlinearity is found and \(\Delta c/c_{0}\
eq 0.5\;\Delta S/S_{0}\).  
Figure 2: Derivation of the nonlinear parameters for velocity and attenuation, once given the nonlinear parameters for modulus and damping, for Cases 3 (cyan circles) and 4 (red circles). The nonlinear dependence on strain of \(S\) and \(\lambda\) is shown in the upper row. The derived values (using Equation (13)) of \(c\) and \(\alpha\) are plotted vs. strain in the lower row. Solid line corresponds to the best fitting obtained using a fourth-order polynomial of the curves, from which the nonlinear parameters for \(c\) and \(\alpha\) are derived using Equation (15).