This form uses the classical theory of plates on elastic foundation, where the plate rests on a subgrade, also called a Winkler foundation, behaving as a bed of independent springs, the soil reaction being proportional to the deflection y of the plate. This model is also exactly applicable to plates floating on a liquid.
The elastic foundation is here referred to as '(bi)elastic', to highlight the distinction with respect to a (mono)elastic foundation, that is treated elsewhere, where the subgrade reacts only when compressed, and not if the plate goes upwards.
The constant k relating the deflection to the reacting pressure is called the modulus of the foundation, and has units of a force over a volume.
In the case of a plate floating on a liquid, this modulus equals the unit weight of the liquid.
Typical values of the modulus in daN/cm³ are as follows (this is only a suggestion, actual values will depend also on the size of the loaded area and other parameters):
-dry or wet sand: 1.3 (loose) to 4.1 (medium) to 16 (compact)
-submerged sand : 0.8 (loose) to 2.5 (medium) to 9.6 (compact)
-compact clay : 2.4 (compr.strength 1-2 daN/cm²) to 4.8 (2-4 daN/cm²) to over 9.6 (>4 daN/cm²)

The solution is based onto a polynomial approximation formula obtained from strain energy minimization (hit the 'Formulas' button to see).