Static stress perturbation in magnetoelastic wave equations

The linear system ${\partial }_t{v}_i=-{\partial }_i\pi +{\partial }_j{m}_{ij}$, ${\partial }_t{m}_{ij}=v_A^2({\partial }_i{v}_j+{\partial }_j{v}_i)$ and $\nabla \cdot \mathbf{v}=0$ admits stationary stress as well as propagating waves. For instance, $\mathbf{v}=0$, $m_{ij}=f(\mathbf{r}){\delta }_{ij}$ and $\pi =f(\mathbf{r})$ solve it for any sufficiently small smooth $f$. Thus arbitrary initial magnetic second-moment tensor perturbations need not be oscillatory.