Wave fields under the influence of a random-driven force: The Burgers equation

In this work, we examine the classical Burgers equation and investigate the effects of a random force on the wave field. Two scenarios are considered: the impact of a random force on different wave fields within the viscous Burgers equation and the effect of a periodic random force in the inviscid Burgers equation. For the first case, we demonstrate that the random force primarily causes wave fronts to increase or decrease depending on the dispersion parameter. For an initially deformed sinusoidal wave, the external force causes the mean wave field to spread out and dampen over time. The Cole-Hopf transformation is also used to obtain asymptotically the averaged wave field in certain regimes. For the inviscid problem, we assume the random force to be periodic with random phase to show that the mean wave field corresponds to the solution of the classical inviscid Burgers equation without external forces.