Probability distribution of the eigenvalues of systems governed by the stochastic wave equation

This paper studies the random eigenvalue problem of systems governed by the one dimensional wave equation. The mass and stiffness properties of the system are taken to vary spatially in a random manner. The probability distribution function of the natural frequencies is characterized in terms of the solution of a first order nonlinear stochastic differential equation. Analytical solutions are obtained based on the method of stochastic averaging. The effect of the mean and autocorrelation of the mass and stiffness processes and also the uncertainty in specifying the boundary conditions are included in the analysis. The theoretical predictions are compared with digital simulations