Effects of viscosity and elasticity on the nonlinear resonance of internal waves

Authors:
  • DHD. F. Hill
    D. F. Hill
  • MFM. A. Foda
    M. A. Foda
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Abstract
It is well established that waves propagating through a viscoelastic medium experience both attenuation and frequency modulation. For the case of infinitesimal waves, linear theory may be utilized to solve the boundary value problem for either a complex wavenumber or a complex frequency, the imaginary components corresponding to exponential decay in space and time, respectively. Recent contributions to the body of literature on weakly nonlinear resonant interactions have demonstrated that, in an inviscid two‐layer system, internal waves can be parametrically excited by surface waves. Exponential 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 two effects is demonstrated. It is found that viscoelasticity reduces the exponential growth rate of the internal waves. Sufficiently large viscoelasticity is found to completely suppress the destabilizing effects of the nonlinearity. General results as well as results for conditions characteristic of an estuarine environment are presented.

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