Periodic traveling interfacial hydroelastic waves with or without mass

Abstract

Abstract. We study the motion of an interface between two irrotational, incompressible fluids, with elastic bending forces present; this is the hydroelastic wave problem. We prove a global bifurcation theorem for the existence of families of spatially periodic traveling waves on infinite depth. Our traveling wave formulation uses a parameterized curve, in which the waves are able to have multi-valued height. This formulation and the presence of the elastic bending terms allows for the application of an abstract global bifurcation theorem of "identity plus compact" type. We furthermore perform numerical computations of these families of traveling waves, finding that, depending on the choice of parameters, the curves of traveling waves can either be unbounded, reconnect to trivial solutions, or end with a wave which has a self-intersection. Our analytical and computational methods are able to treat in a unified way the cases of positive or zero mass density along the sheet, the cases of single-valued or multi-valued height, and the cases of single-fluid or interfacial waves.

Solitary interfacial hydroelastic waves

“…An interesting extension of this work would be to include heavy elastic plates and investigate if solitary waves can exist in this setting. Very recently, Akers et al [14] have computed travelling periodic waves when a heavy elastic plate is separating two fluids.…”

“…The authors have used a vortex sheet formulation and both cases of a heavy and a massless elastic plate were considered. Results on the existence of periodic travelling interfacial hydroelastic waves in infinite depth [14] have been obtained by proving a global bifurcation theorem and some numerical computations of periodic waves were also presented.…”

Stability of periodic traveling flexural‐gravity waves in two dimensions

“…Using a variational approach, Groves et al . show the existence for hydroelastic solitary waves and Akers et al . use bifurcation theory for the existence of periodic waves in two dimensions.…”

“…Several works discuss the existence of solutions to equations describing hydroelastic waves; for example, the work of Toland 7 discusses the existence of solutions as an optimization of the Lagrangian formulation for traveling waves. Using a variational approach, Groves et al 8 show the existence for hydroelastic solitary waves and Akers et al 9 use bifurcation theory for the existence of periodic waves in two dimensions.…”

“…The fully nonlinear model for ice was considered in Guyenne and Părău 16,17 when computing solitary waves and by Gao and Vanden-Broeck 18 for both periodic and solitary waves. A more general discussion considering periodic interfacial waves with and without mass is seen in Akers et al 9,19 where a different parameterization of the problem was considered. Work on computing solutions for the threedimensional problem for flexural-gravity waves also exists, but will not be discussed here.…”