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For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.

My question is...why is the above PDE so named? That is, how does $a$ represent a damping mechanism?

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Multiply by $u_t$, integrate and see what happens. –  Michael Renardy Nov 29 '12 at 2:28
    
You will likely to get a good answer if you ask this at MathStackExchange. –  timur Nov 29 '12 at 3:57
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It's much the same as how the ODE $\dfrac{d^2 y}{dt^2} + a \dfrac{dy}{dt} + b y$ for $a,b > 0$ represents a damped oscillator. In fact, if you take $u(x,t) = u(t) X(x)$ where $X$ is an eigenfunction of the Laplacian, this is exactly what you get. –  Robert Israel Nov 29 '12 at 6:29
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