A new method to predict the evolution of the power spectral density for a finite-amplitude sound wave

A method to predict the effect of nonlinearity on the power spectral density of a plane wave traveling in a thermoviscous fluid is presented. As opposed to time-domain methods, the method presented here is based directly on the power spectral density of the signal, not the signal itself. The Burgers equation is employed for the mathematical description of the combined effects of nonlinearity and dissipation. The Burgers equation is transformed into an infinite set of linear equations that describe the evolution of the joint moments of the signal. A method for solving this system of equations is presented. Only a finite number of equations is appropriately selected and solved by numerical means. For the method to be applied all appropriate joint moments must be known at the source. If the source condition has Gaussian characteristics (it is a Gaussian noise signal or a Gaussian stationary and ergodic stochastic process), then all the joint moments can be computed from the power spectral density of the signal at the source. Numerical results from the presented method are shown to be in good agreement with known analytical solutions in the preshock region for two benchmark cases: (i) sinusoidal source signal and (ii) a Gaussian stochastic process as the source condition.