nwhm/symmetry

The following is based on the homogenized equations, see Homogenized equations.

Z-dispersion

The linear homogenized system for Z-dispersion up to one non-vanishing correction is given by:

\(\begin{eqnarray*} K_h^{-1}\sigma_t-u_x & = & \delta^2\alpha_1u_{xxx},\\ \rho_m u_t-\sigma_x & = & \delta^2\gamma_1 \sigma_{xxx}. \end{eqnarray*}\)

Combine both equations to get:

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

However, in the homogenized process we neglected terms of order \(O\left(\delta^4\right)\); therefore, we must neglect \(\delta^4\alpha_1\gamma_1\sigma_{xxxxxx}\). Doing so we get:

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

The coefficient \(\alpha_1+\gamma_1\) can be written as:

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

c-dispersion

The linear homogenized system for c-dispersion up to one non-vanishing correction is given by:

\(\begin{eqnarray*} K_h^{-1}\sigma_t-u_x & = & \delta^2\alpha_2u_{xxx},\\ \rho_h u_t-\sigma_x & = & \delta^2\beta_2 \sigma_{xxx}. \end{eqnarray*}\)

Combine both equations, and drop terms of order \(O\left(\delta^4\right)\) to get:

\(K_h^{-1}\rho_h\sigma_{tt}-\sigma_{xx}=\delta^2\left(\alpha_2+\beta_2\right)\sigma_{xxxx}+O\left(\delta^4\right).\)

The coefficient \(\alpha_2+\beta_2\) can be written as:

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

Lajos has worked out other expressions for these coefficients, but there doesn’t appear to be a precise symmetry.