We have seen here that, without loss of generality, one can normalize the material parameters such that \(\rho_A=K_A=1\). Therefore, from the four material parameters we just have to consider \(\rho_B\) and \(K_B\). We can control these parameters depending on the desired sound speed and impedance contrast. Let

\(\begin{align} r_z&=\frac{\sqrt{K_B\rho_B}}{\sqrt{K_A\rho_A}}=\sqrt{K_B\rho_B},\\ r_c&=\frac{\sqrt{\frac{K_B}{\rho_B}}}{\sqrt{\frac{K_A}{\rho_A}}}=\sqrt{\frac{K_B}{\rho_B}}. \end{align} \)

Then the material parameters \(\rho_B\) and \(K_B\) are defined by \(r_c\) and \(r_z\):

\(\begin{align} K_B &= r_c r_z, \\ \rho_B &= \frac{r_z}{r_c} \end{align} \)

Note that we don’t loss any generality by doing this. For any arbitrarily \(\rho_B\) and \(K_B\), we can express the material coefficients as functions of the sound speed and impedance ratio. For instance, \(\rho=\begin{cases} 1 &\forall x\in\Omega_A \\ r_z/r_c & \forall x\in\Omega_B \end{cases}\).

Influence of sound speed on z-dispersion

Consider a z-dispersive situation; e.g., wave traveling transverse to the material heterogeneity. Assumes the initial condition is smooth and its wavelength is large enough that the first order homogenized equation holds. The homogenized equation can be seen as the second order system in the discussion of the linear_c-dispersion paper. This equation predicts that if \(\alpha_1+\gamma_1=0\), then there is no effective dispersion and the solution develop shocks. The coefficient is:

\(\begin{align} \alpha_1+\gamma_1=\frac{(Z_A^2-Z_B^2)^2}{192K_m^2\rho_m^2} \end{align} \)

We see imediately that no changes in impedance leads to shocks. What happens when we vary the sound speed considering a fix change in impedance? This can be answered by considering the (material parameter) setting at the beginning of this section. Under this setting we have:

\(\begin{align} \alpha_1+\gamma_1=\frac{1}{12}\left[\frac{r_c(1-r_z^2)}{(1+r_cr_z)(r_c+r_z)} \right]^2 \end{align} \)

Now we can see that if \(r_z\) is fixed, \(\alpha_1+\gamma_1\rightarrow 0\) as \(r_c\rightarrow\infty\); hence, the dispersion vanishes and the solution develops shocks. This is true for any fixed value of \(r_z\).

Stegoton formation

Then, to develop stegotons having an impedance contrast is not enough, the sound speed contrast can’t be arbitrarily large.

Shock stability

From KL we know:

\(\begin{align} s_{eff}^{(R)}>c_h(\sigma_r):=\left( \int_\Omega\left( \frac{\sigma_r^\prime (x)}{\rho(x)} \right)^{-1/2} dx \right)^{-1}, \end{align} \)

If we consider a right-going shock with zero right state, the setting for the material parameters used in this section and the exponential nonlinearity we have:

\(\begin{align} s_{eff}=2r_c\left[\frac{r_z}{(1+r_cr_z)(r_c+r_z)}\right]^{1/2}\sqrt{\frac{\sigma_l}{\log(\sigma_l+1)}}>c_h(\sigma_r)=\frac{2r_c}{1+r_c}. \end{align} \)

Since \(\sqrt{\frac{\sigma_l}{\log(\sigma_l+1)}}>1\) and is monotonically increasing, for any fixed impedance contrast \(r_z\), this condition is satisfied as \(r_c\rightarrow\infty\); i.e., for any impedance contrast, a shock is always stable as the sound speed contrast increases. This agrees with the conclusion from the homogenized equations.

Influence of impedance on c-dispersion

Consider now a purely c-dispersive situation; a wave traveling parallel to the material heterogeneity. If the initial condition is smooth and its wavelength is large enough, the first order homogenization holds, see the discussion of the linear_c-dispersion paper. If the coefficient \(\alpha_2+\beta_2=0\) there is no dispersion and the solution will develop shocks. This coefficients is given by:

\(\begin{align} \alpha_2+\beta_2=\frac{(c_A^2-c_B^2)^2}{192K_m^2}\rho_m\rho_h. \end{align} \)

We see that if the sound speed is constant, the coefficient vanishes leading to shock formation. We can fix the sound speed and change the impedance contrast. To do so, consider:

\(\begin{align} \alpha_2+\beta_2=\frac{(1-r_c^2)^2}{48(1+r_zr_c)^2}\frac{r_z}{r_c} \end{align} \)

Therefore, for any fixed \(r_c\), \(\alpha_2+beta_2\) vanishes as \(r_z\rightarrow\infty\).

Diffracton formation

To obtain diffractons we need a change in sound speed, but also that the change in impedance is not arbitrarily large.

Shock stability

We saw from the homogenized equations that, for any fixed sound speed contrast, we obtain shocks if the impedance contrast increases. We expect to predict this behavior with the “effective Lax-entropy condition”. We suggested that

\(\begin{align} s_{eff}^{(R)}>c_m(\sigma_r):=\int_\Omega\left( \frac{\sigma_r^\prime (x)}{\rho(x)} \right)^{1/2} dx, \end{align} \)

which in the setting used in this section means:

\(\begin{align} s_{eff}=\left[\frac{r_c(r_c+r_z)}{1+r_cr_z}\right]^{1/2}\sqrt{\frac{\sigma_l}{\log(\sigma_l+1)}}>c_m=\frac{1+r_c}{2} \end{align} \)

Fix the sound speed contrast and let the impedance contrast \(r_z\rightarrow\infty\). Our condition becomes:

\(\begin{align} \sqrt{\frac{\sigma_l}{\log(\sigma_l+1)}}>\frac{1+r_c}{2} \end{align} \)

which can easily be violated for any \(r_c>1\). However, from the homogenized equations we expected to obtain a shock for any fixed \(r_c\). Therefore, this effective shock condition must be wrong. Indeed, it appears to be true for the particular case when \(r_z=1\), see these results, but not in general.

We expect the shock condition to be true for any \(r_c\neq 1\) when \(r_z\rightarrow\infty\). However, \(c_m=\frac{1+r_c}{2}\) is not affected by changes in the impedance contrast.