Hydroelastic waves in a channel covered with a free ice sheet

The solution of the problem of flexural-gravitational wave propagation in a channel of rectangular cross-section coveredwith an ice sheet is constructed and studied. It is assumed that the ice plate thickness is constant and small compared with the channel depth, while its edges along the channelwalls are free. The ice cover deflection is described within the framework of the linear theory of elastic plates, the fluid flow under the ice is assumed to be potential, and the profile of the wave running along the channel is taken to be sinusoidal. The problem is solved by expansion in the oscillation modes of the ice plate. The dispersion equations are derived and the profiles of the flexural-gravitationalwaves transverse to the channel are found, together with the extension strains in the ice cover. An approximate formula for the dispersion equation is proposed, which is at a high accuracy valid for the first three modes of the hydroelastic waves under consideration. The elastic and hydrodynamic parameters are compared for the cases of the free ice cover and the cover iced to the channel walls. It is shown that the waves in the channel with a cover iced to the banks propagate more rapidly and have a greater frequency than in the channel with the free ice cover. For long waves and the first two modes of hydroelastic oscillations a minimum wavelength, starting from which the ice cover effect on the traveling waves can be neglected, is established. For the higher modes the ice cannot be excluded from the consideration for even very long waves.