Effective wavelength of envelope waves on the water surface beneath an ice sheet: small amplitudes and moderate depths

We focus our attention on the comparison of wavelengths of envelopes, monochromatic waves, and speeds of so-called envelope solitary waves on the surface of water beneath an ice cover under an initial stress for water layers of moderate depths within two formulations. First, we use the Euler equations for the water layer of a finite depth, with the ice cover modeled by an elastic geometrically nonlinear Kirchhoff–Love plate (we refer to this formulation as the fully nonlinear one). Traveling envelope solitary waves (which we call envelope wave packets), for which the phase speed is equal to the group velocity, corresponding to the occurrence of a velocity minimum at a finite wave number in the dispersion curve and having small amplitudes, can be described asymptotically within this formulation using the center manifold reduction and the normal form analysis. Second, for low amplitudes and long waves, we can use a weakly nonlinear formulation and formally derive the nonlinear Schrödinger equation. Within these formulations, the wavelengths of the envelope, of the monochromatic wave, and of the phase speed of the wave are uniquely determined. We compare these parameters for envelope wave packets and find that they are close for moderate depths of water basins. We discuss the existence of a singular limit in the equations of the fully nonlinear formulation, when the flexural rigidity of the ice cower tends to zero and we formally obtain the gravity–capillary case. We also discuss the possibility to theoretically determine the wavelengths and wave speeds for nontraveling envelope solitary waves via the weakly nonlinear formulation.