Propagation of One-Dimensional Waves from a Source in Random Motion

The problem treated here consists in determining the simplest statistical properties of the solution of an initial-value problem for the wave equation with a nonhomogeneous term that involves the random motion of the source. There are several physical problems for which this is an appropriate mathematical model. A simple one that serves to fix ideas concerns a stretched string, initially at rest, which occupies the entire x axis in its undeflected state. At time t--O, a concentrated transverse force is suddenly applied to the string. As time increases, the magnitude of this force oscillates sinusoidally with a given frequency o•, while the position of force at time t is x= X(t), where X(t) is a stochastic