Second-order random wave solutions for internal waves in a two-layer fluid

The equations describing the random displacement of the density interface and the associated velocity potentials in a constant depth, two‐layer fluid with a rigid lid were solved to second order using an expansion technique analogous to that used to study random surface waves by Longuet‐Higgins [1963] and Sharma and Dean [1979]. As expected, the first‐order solutions represent a linear superposition of many waves with different amplitudes, wave numbers and frequencies. The second‐order solutions describe the second‐order wave‐wave interactions. The solutions derived from the present work include as special cases those obtained by Thorpe for progressive internal waves [Thorpe, 1968a] and standing internal waves [Thorpe, 1968b]. Moreover, the solutions reduce to those derived for random surface waves by Sharma and Dean [1979] when the density of the upper fluid is taken as zero.