Time-varying spectrum of the random string*

We consider the response of a finite string to white noise and obtain the exact time-dependent spectrum. The complete exact solution is obtained, that is, both the transient and steady-state solution. To define the time-varying spectrum we ensemble average the Wigner distribution. We obtain the exact solution by transforming the differential equation for the string into the phase space differential equation of time and frequency and solve it directly. We also obtain the exact solution by an impulse response method which gives a different form of the solution. Also, we obtain the time-dependent variance of the process at each position. Limiting cases for small and large times are obtained. As a special case we obtain the results of van Lear Jr and Uhlenbeck and Lyon. A numerical example is given and the results plotted.