Wave equation in a medium PDE

Question

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?

Answer 1

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.