Stability of periodic traveling waves for the hydroelastic Whitham equation

In this work, we investigate the stability of hydroelastic periodic traveling waves within a Whitham-type equation framework. The Whitham equation is well known in the literature as a relatively simple model that nevertheless captures rich nonlinear phenomena such as short waves and breaking. Periodic traveling waves are computed numerically, and their stability is analyzed by evaluating the spectrum via the Fourier–Floquet–Hill method. We show that for small values of the flexural rigidity coefficient, small-amplitude periodic traveling waves are unstable; however, as the amplitude increases beyond a critical threshold, we first observe stabilization (not complete); subsequently, the spectrum bifurcates, and the traveling waves become increasingly unstable. In contrast, when the flexural rigidity coefficient is large, periodic traveling waves remain stable for all amplitudes. For moderate elasticity, two scenarios may occur: either (i) the maximal instability growth rate exhibits a monotonic dependence on the wave height, or (ii) complete stabilization is achieved for sufficiently large heights within numerical tolerance.