# Wave equation in a medium PDE

Suppose there is a string in a medium that applies a resistant force per unit length proportional to the velocity of the string. How do you write the equation of string vibrations?

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If the displacement of the string is $u(x,t)$ then the velocity is $u_t$ the acceleration is $u_{tt}$ and the curvature is $u_{xx}$. How are the acceleration and curvature related when there is no resistive force? –  Chris Taylor Feb 14 '12 at 16:39

The wave equation with damping is $$u_{tt}=c^2\,u_{xx}-k\,u_t, \quad c>0,\quad k>0.$$ The dumping term is $-k\,u_t$, $k>0$, proportional to the velocity of the string and in the oposite direction.