nwhm/homogenized equations

A detailed derivation of these equations is on the repo. The homogenized equations up to one non-vanishing correction is:

\(\begin{eqnarray*} K_h^{-1}\sigma_t-\left(\sigma+1\right)\left(u_x+v_y\right) & = & \alpha_1\left[\left(\sigma+1\right)\left(u_{xyy}+v_{yyy}\right)+2\sigma_y\left(u_{xy}+v_{yy}\right)\right]\\ & & +\alpha_2\left[\left(\sigma+1\right)\left(u_{xxx}+v_{xxy}\right)+2\sigma_x\left(u_{xx}+v_{xy}\right)\right]\\ & & +\alpha_3\sigma_y\left(u_{xy}+v_{yy}\right),\\ \rho_h u_t-\sigma_x & = & \beta_1\sigma_{xyy}+\beta_2\sigma_{xxx},\\ \rho_m v_t-\sigma_y & = & \gamma_1\sigma_{yyy}+\gamma_2 \sigma_{xxy}, \end{eqnarray*}\)

where:

\(\begin{eqnarray*} \alpha_1 & = & \frac{\left(K_A-K_B\right)}{192K_m^2}\cdot\frac{\left(Z_A^2-Z_B^2\right)}{\rho_m},\\ \alpha_2 & = & \frac{\left(K_A-K_B\right)}{192K_m^2}\cdot\frac{\left(c_A^2-c_B^2\right)}{\rho_m^{-1}},\\ \alpha_3 & = & \frac{\left(\rho_A-\rho_B\right)^2}{192\rho_m^2},\\ \beta_1 & = & \frac{-\left(\rho_A-\rho_B\right)}{192K_m}\cdot\frac{\left(Z_A^2-Z_B^2\right)}{\rho_m^2},\\ \beta_2 & = & \frac{-\left(\rho_A-\rho_B\right)}{192K_m}\cdot\left(c_A^2-c_B^2\right),\\ \gamma_1 & = & \frac{\left(\rho_A-\rho_B\right)}{192K_m}\cdot\frac{\left(Z_A^2-Z_B^2\right)}{\rho_m^2},\\ \gamma_2 & = & \frac{\left(\rho_A-\rho_B\right)}{192K_m}\cdot\left(c_A^2-c_B^2\right). \end{eqnarray*}\)

Situation 1. Initial condition homogeneous in \(x\)

If the initial condition is homogeneous in \(x\) one may expect to have the same situation as in 1D but for a wave propagating in \(y\). In this case all \(x\) derivatives in the homogenized equations vanish, so we get:

\(\begin{eqnarray*} K_h^{-1}\sigma_t-\left(\sigma+1\right)v_y & = & \alpha_1\left(\sigma+1\right)v_{yyy}+\left(\alpha_3+2\alpha_1\right)\sigma_y v_{yy},\\ \rho_m v_t-\sigma_y & = & \gamma_1 \sigma_{yyy}. \end{eqnarray*}\)

Situation 2. Initial condition homogeneous in \(y\)

If the initial condition is homogeneous in \(y\), then all \(y\) derivatives in the homogenized equations vanish, then we get:

\(\begin{eqnarray*} K_h^{-1} \sigma_t-\left(\sigma+1\right)u_x & = & \alpha_2 \left[\left(\sigma+1\right)u_{xxx}+2\sigma_x u_{xx}\right],\\ \rho_h u_t-\sigma_x & = & \beta_2 \sigma_{xxx}. \end{eqnarray*}\)

We could also consider initial conditions that vary only in a particular direction, but with the direction arbitrary (not just the x- or y-direction). What do the equations tell us in that case?