What I explain in this section is valid for the homogenized equations coming from the c- or z-dispersive media or a combination of them. For simplicity I will stick with the z-dispersive scenario in the rest of the section.

Recall the homogenized system with one non-vanishing correction can be combined into the second order equation:

\(K_h^{-1}\rho_m\sigma_{tt}-\sigma_{xx}=\delta^2\left(\alpha_1+\gamma_1\right)\sigma_{xxxx}+O\left(\delta^4\right),\)

where:

\(\alpha_1+\gamma_1=\lambda^2\frac{\left(Z_A^2-Z_B^2\right)^2}{192K_m^2\rho_m^2}.\)

From here we can obtain the dispersion relation to be:

\(\omega=\pm\hat{c}k\sqrt{1-\Omega^2(\alpha_1+\gamma_1)k^2}\),

where \(\omega\) is the angular frequency, \(k\) is the wavenumber and \(\hat{c}=\sqrt{\frac{K_h}{\rho_m}}\).

Stability of the homogenized equations

For fixed material parameters \(\alpha_1+\gamma_1\) is fixed and then the equations become unstable for arbitrarily large wavenumbers (or arbitrarily short wavelengths).

Is the homogenized equation stable just for waves with wavelength larger than Omega?

Not necessarily, just if \(\alpha_1+\gamma_1\) is of order 1, but that may not be true. This term is independent on the wavelength and it varies just according to the material parameters so I can make it small enough to consider waves with wavelength much smaller than the material period and yet have stable homogenized equations (at least in the linear case). This may not be true with the nonlinear homogenized equations because when \(\alpha_1+\gamma_1\) is small the solution may lead to shock formation leading to wavenumbers arbitrarily large that end up breaking the homogenized system.

Take home message: the stability of the homogenized equations is not just linked with the wavelength considered but also with the size of the factor \(\alpha_1+\gamma_1\); i.e., with the amount of effective dispersion introduced.

Different options for choosing the small parameter

We have been choosing \(\delta=\Omega/\lambda\). Remember that in the final homogenized equations for each \(\delta^n\) there is a corresponding \(\lambda^n\) factor, this is stated in the last paragraph of section 2 of the paper ‘Two-dimensional wave propagation in layered periodic media’. Therefore, we obtain that the dispersive terms in the homogenized equation are proportional just to \(\Omega^n\).

We might as well take \(\delta=\Omega/L\), where \(L\) is the total length of the domain; this has been done before, see for instance TODO: add citation. The result would be the same: for each \(\delta^n\) in the homogenized equations there would be a corresponding \(L^n\) factor. Again the dispersive terms in the homogenized equations are proportional to just \(\Omega^n\).

The chose of \(\lambda\) or \(L\) in the definition of \(\delta\) is immaterial in the final result. Formally, we really just want to be sure that \(\delta\) is small. For this reason and considering the situation we are studying now where \(\lambda\) is small we should choose \(\delta=\Omega/L\). Again, this leads to the exact same result.

What is essentially different from choosing each option for delta?

Recall the idea of the homogenization. We introduce an small parameter that leads to a fast scale. This gives an scaled system. Later we formally expand the solution using the small parameter so we get something like: \(u=u_0+\delta u_1+\dots\) where \(u_0\), \(\dots\) are assumed to be periodic with respect to \(\lambda\) or \(L\), depending on the way \(\delta\) is defined. Afterwards, we take the average of \(u=u_0+\delta u_1+\dots\) with respect to \(\lambda\) or \(L\). So by choosing \(\lambda\) or \(L\) we just change the periodicity of the terms in the formal expansion but when we take the average we obtain that \(u\) is unaffected by this choice.

Does this mean that the homogenized equations should be valid for arbitrarily small wavelengths?

The answer is no, for fix material parameters. As explained before, the homogenized equations are unstable under certain conditions that depend not only on the wavelength but also on the impedance contrast.

For fixed material parameters, we can’t consider arbitrarily small wavelengths. But if we fix the wavelength (no matter how small it is) we can choose material parameters that lead to stable homogenized equations (again at least for the linear case).

The homogenized equations in the setting of (dispersive) shock formation

Leading order: By choosing \(\delta=\Omega/L\) there is no formal problem to obtain the homogenized equations. In the case of the leading order, the homogenized equations are stable for any wavelength and any impedance contrast so we can use them to study effective properties of shock formation and we don’t violate any assumption taken.

Corrections: If we want to make corrections to the leading order we get into problems because as the dispersion is small (and fixed) shocks may appear leading to wavenumbers large enough to make the homogenized equations unstable. In this case we would need to apply some high frequency homogenization, which is out of the scope of this work. I don’t know any reference for this in the nonlinear case. For the linear problem, there are some procedures that obtain ‘homogenized’ equations but yet with the solution depending on the fast scale, see for instance TODO: add citation.