  In this work, we solve the Euler's equations for periodic waves travelling
under a sheet of ice using a reformulation introduced in Ablowitz et al.
(2006). These waves are referred to as flexural-gravity waves. We compare and
contrast two models for the effect of the ice: a linear model and a nonlinear
model. The benefit of this reformulation is that it facilitates the asymptotic
analysis. We use it to derive the nonlinear Schrodinger equation that describes
the modulational instability of periodic travelling waves. We compare this
asymptotic result with numerical computation of stability using the
Fourier-Floquet-Hill method and show how well these agree. We show that
different models have different stability regimes for large values of the
flexural rigidity parameter. Numerical computations are used to go beyond the
modulational instability and show high frequency instabilities that are the
same for both models for ice in the regime examined.
