Dispersion of plane harmonic waves in an elastic layer interacting with a one- or two-sided Winkler foundation is analyzed. The long-wave low-frequency polynomial approximations of the full transcendental dispersion relations are derived for a relatively soft foundation. The validity of the conventional engineering formulation of a Kirchhoff plate resting on an elastic foundation is investigated. It is shown that this formulation has to be refined near the cutoff frequency of bending waves. The associated near cutoff expansion is obtained for both cases. A simple explicit formula demonstrating veering of bending and extensional waves is presented for a one-sided foundation.

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