Stochastically excited waves in a 1D random medium – the mean field and power spectrum

Scalar wave propagation is examined when both the wave source and the propagation speed are random. Results are derived for the mean field and the power spectrum using the second-order Born approximation. The results depend on whether the source S(x, t) and the propagation speed c(x, t) are correlated or not. When they are uncorrelated, the mean field is zero. When they are correlated, the mean field is non-zero only when the source is non-stationary. The power spectrum is incoherent to leading order. There is a transfer of energy from lower to higher frequencies owing to wave scattering. The corresponding frequency upshift of the power profile in the (k, ω) domain is mainly caused by the cross power between the direct and the twice scattered field, which represents a second-order incoherent power contribution. The results are confirmed using a numerical solution of the wave equation where the scattered field is expanded to fifth order.

Acknowledgements

I am grateful to my fellow physicists Krzysztof Murawski, Teresa Lynne Palmer, Trygve Sparr and the referee for providing excellent feedback.

Notes

^†The first contribution in this term is zero because ⟨ L ⟩ = 0. The second contribution involves the correlation between the one-time scattered field and L. This is zero because triple correlations average out to zero. This is also the reason for P _0,1 = 0.