Wave equation on a string

A stretched uniform string obeys y_tt=c^2 y_xx with wave speed equal to the square root of tension over density.

Zero values at both endpoints of a length-$L$ string allow the D'Alembert formula to be written with opposite travelling waves and a $2L$-periodic profile. This represents reflection with sign reversal at each fixed endpoint. For positive-time boundary conditions, periodicity concerns the arguments visited by the string solution; unused profile values can be defined by periodic extension.

With fixed endpoints, $y_{tt}-y_{xx}+y_t=0$ separates into sine modes whose amplitudes satisfy

Critical damping occurs when a second-order mode has a repeated real characteristic root. It returns to equilibrium without oscillating and as quickly as possible among nonoscillatory regimes.

For initial sine coefficient $a_n$ and zero initial velocity, put ${\omega }_n^2=n^2{\pi }^2-1/4$. The modal amplitude is

For fixed endpoints andintegration by parts and the damped wave equation give