Is there a physical model described by this equation? (vectorial damped wave equation)

Question

Consider the wave equation $\partial_t^2 u\left(t,x\right) - \Delta u\left(t,x\right) = 0$ where $t\in\mathbb{R}$ is the time variable and $x\in\Omega$ (a nice open subset of $\mathbb{R}^n$) is the space variable. If $\partial \Omega$ the boundary of $\Omega$ is not empty (ie if $\Omega \notin \mathbb{R}$) we add the boundary condition $u_{|\partial\Omega}=0$. When $u$ is real (or complex) valued this is the standard wave equation of a drum or a vibrating string.

If I fix a smooth function $a: \Omega \to \mathbb{R}_+$ then the equation

$$ \begin{array}{ccc} \partial_t^2 u\left(t,x\right) - \Delta u\left(t,x\right) +a\left(x\right)\partial_t u\left(t,x\right) & = & 0 \\ % {\left. u \right|}_{\partial \Omega} & = &0 \end{array} $$ is called the damped wave equation. Indeed if we define the energy of a solution $u$ as kinetic energy plus potential energy $\left( E(u,t)=\frac{1}{2}\int_\Omega \left|\partial_t u \left(t,x \right) \right|^2 + \left|\nabla u(t,x) \right|^2 \mathrm{d}x \right)$ an integration by part shows that the variation of energy is given by the formula $$\frac{\mathrm{d}}{\mathrm{d}t}E\left(u,t\right)~=~-\int_{\Omega}\left< a\left(x\right)\partial_t u\left(t,x\right),~ \partial_tu(t,x) \right> \mathrm{d}x$$ and so the term $a$ do play the role of a damping mechanism.

Now the equation i'm interested in is the vectorial version of the damped wave equation. If I take $u\left(t,x\right)=\left(u_1\left(t,x\right),\ldots,u_n\left(t,x\right)\right)$ to be vector valued and $a=(a_{i,j})$ a matrix valued function my equation becomes the following system of equations $$\partial_t^2 u_i(t,x)-\Delta u_i(t,x)+\sum_{j=1}^{n}a_{i,j}(x)\partial_t u_j(t,x)=0,\;\; i=1,\ldots, n$$ or equivalently, using the matrix product and applying $\partial_t$ and $\Delta$ component by component $$\partial_t^2 u\left(t,x\right) - \Delta u\left(t,x\right) +a\left(x\right)\partial_t u\left(t,x\right) ~=~ 0 \,.$$

If we add the condition that $a(x)$ must be hermitian positive for every $x$ then using the formula for the variation of energy we see that the energy is still decreasing.

MY QUESTION: Is there a physical model described by this equation? I know that the wave equation and the scalar damped wave equation can describe the vibrations of a string or a drum. I also know that the seismic waves do vibrate in several dimension but they are modeled by an other equation, the same goes for the electromagnetic waves that are modeled by the Maxwell equations. So, is my equation modeling something real?

My equation comes from a purely mathematical generalization of the scalar damped wave equation but i was wondering if it was corresponding to something existing in physics.

Answer 1

The isotropic case (when the matrix $a$ is a multiple of the identity) describes many examples, for instance the motion of a string in a medium causing friction or the electric current in a transmission line (as mentioned in the comments).

The anisotropic case is the one you're interested in, if I understand correctly. An example would be a medium with anisotropic conductivity. The electromagnetic wave equation in presence of sources is $$\frac{1}{c^2}\frac{\partial^2 \mathbf E}{\partial t^2}-\nabla^2 \mathbf E=-\frac{1}{\epsilon_0}\nabla \rho-{\mu_0}\frac{\partial \mathbf J}{\partial t}\tag{1}$$ An anisotropic conductor has $\mathbf J = \sigma \mathbf E$ where $\sigma$ is the conductivity tensor, so $(1)$ reduces to your equation minus a forcing term. I think this equation is used among other things to describe wave propagation in a plasma.