Short wind waves on the ocean: Wavenumber‐frequency spectra

Dominant surface waves on the ocean exhibit a dispersion relation that confines their energy to a curve in a wavenumber‐frequency spectrum. Short wind waves on the ocean, on the other hand, are advected by these dominant waves so that they do not exhibit a well‐defined dispersion relation over many realizations of the surface. Here we show that the short‐wave analog to the dispersion relation is a distributed spectrum in the wavenumber‐frequency plane that collapses to the standard dispersion relation in the absence of long waves. We compute probability distributions of short‐wave wavenumber given a (frequency, direction) pair and of short‐wave frequency given a (wavenumber, direction) pair. These two probability distributions must yield a single spectrum of surface displacements as a function of wavenumber and frequency, F( k ,f). We show that the folded, azimuthally averaged version of this spectrum has a “butterfly” pattern in the wavenumber‐frequency plane if significant long waves are present. Integration of this spectrum over frequency yields the well‐known k −3 wavenumber spectrum. When integrated over wavenumber, the spectrum yields an f −4 form that agrees with measurement. We also show that a cut through the unfolded F( k ,f) at constant k produces the well‐known form of moderate‐incidence‐angle Doppler spectra for electromagnetic scattering from the sea. This development points out the dependence of the short‐wave spectrum on the amplitude of the long waves.