THE INFLUENCE OF WAVE-NOISE ON WAVE SPEEDS AND AMPLITUDES OF SURFACE-GRAVITY WAVES

Abstract

We have analytically examined surface-gravity waves which propagate in space- and time-dependent random velocity fields. Using a perturbative method, we have derived a dispersion relation which is solved for the case of wave-noise whose spectrum E(k,ω) ∼ E(k)δ(ω−c_rk), where δ is Dirac’s delta-function and c_r is the random phase speed. We have found that for a dispersionless noise resonance occurs when c_r is equal to the group velocity cg of the surface-gravity wave. In this resonance the real part of the wave frequency is finite, but its imaginary part exhibits the characteristic 1/x singularity. The wave-noise interacts with a packet of the surface-gravity waves in such a way that the waves are attenuated for c_r < c_g and are amplified for c_r > c_g. As the real part is positive for high values of k, the surface-gravity waves are accelerated by the wave-noise.