Ship wave resonance in stratified seas

Waves of an unsteady disturbance in a 2-layer fluid

Oscillation of underwater vehicles (particularly submarines) in littoral zones, where water is more likely to be stratified, generates particular waves that can reveal information about size, speed and the depth of the vehicle. We studied the mechanism behind the generation of these waves and their interactions. For a steadily translating and sinusoidally oscillating disturbance in a two-layer fluid, up to eight waves may appear on the water surface/interface. Unlike in homogeneous fluid, nonlinear resonance can occur between these eight waves. We theoretically found, and numerically validated, that existence of stratification along with second-order nonlinearities can result in generation of a series of new waves that can travel away or toward the disturbance.

Schematic of the problem that we studied: A steadily translating and oscillating disturbance in a two-layer density stratified fluid generates up to six waves behind and up to two waves ahead of itself. Some pairs of these waves, under specific yet realistic conditions, can resonate a third free wave; which is not expected if nonlinearity is not taken into account. This phenomena is never seen in homogenous fluids (Bonus question: why?)

Artist rendering of disturbance wave resonance in a 2-layer density stratified fluid:
Assume our pirate is driving his (or her?) submarine in a strong stratified ocean well below the water surface. If the captain pirate does not know any math, he thinks that he is fully concealed from the police boats on the surface. But fortunately (or unfortunately!) at a specific speed situation changes. At that speed, it will send a signal (of course due to nonlinearity) to the surface…. you watch the fate of the poor captain pirate !!!


– Alam, M.R., Liu, Y. and Yue, D.K.P., “Waves due to an oscillating and translating disturbance in a two-layer density stratified fluid”,J. Engineering Mathematics, Volume 65 (2009), Issue 2 Pages 179–201. [PDF]

– Alam, M.R., Liu, Y. and Yue, D.K.P., “Resonance Interaction of Ship waves in Stratified Seas”, Submitted to J. Fluid Mechanics.