Effects of viscosity and elasticity on the nonlinear resonance of internalwaves

It is well established that waves propagating through a viscoelastic medium experience both attenuation and frequency modulation. For the case of infi nitesimal waves, linear theory may be utilized to solve the boundary value problem for either a complex wavenumber or a complex frequency, the imagina ry components corresponding to exponential decay in space and time, respect ively. Recent contributions to the body of literature on weakly nonlinear r esonant interactions have demonstrated that, in an inviscid two-layer syste m, internal waves can be parametrically excited by surface waves. Exponenti al growth, rather than decay, of the internal waves has been predicted and conclusively verified in the laboratory. The two mechanisms are considered together in the current paper. By considering a two-layer system possessing both weak nonlinearity and viscoelasticity, the competition between the tw o effects is demonstrated. It is found that viscoelasticity reduces the exp onential growth rate of the internal waves. Sufficiently large viscoelastic ity is found to completely suppress the destabilizing effects of the nonlin earity. General results as well as results for conditions characteristic of an estuarine environment are presented.