Steady hydroelastic cnoidal wave on a thin flexural plate floating on a shallow water surface

• KdV-type equation for steady nonlinear hydroelastic waves on a thin elastic plate floating in shallow water, is derived. • Analytical approaches are taken to obtain solutions to the derived equation, by using Jacobiʼs elliptic functions and trigonometric functions. • The predictions from the analytical approaches are compared with numerical simulations using steam function method originally proposed by Fenton. • Nonlinear hydroelastic waves can take depressed wave shape under certain conditions, which is contrary to cnoidal waves on free surface. This paper investigates steady nonlinear hydroelastic waves propagating along a thin elastic flexural plate floating on a shallow water surface. In a previous study, some of the present authors developed a numerical method for describing nonlinear hydroelastic waves in shallow water, following Fenton’s stream function approach. They predicted the existence of inverted cnoidal waves characterized by flattened crests and peaked troughs. In the present study, a Korteweg–de Vries (KdV)-type equation is derived under the assumption of wave periodicity to describe the characteristics of these nonlinear hydroelastic waves, accounting for plate elasticity and the inertia term. Two types of analytical solutions are pursued: one by using Jacobi’s elliptic functions and the other by trigonometric functions. After confirming the validity of these analytical solutions through comparison with the numerical predictions, the behavior of nonlinear waves on the floating plate is examined. The study reveals that the wave profiles on the floating plate are asymmetric with respect to the still-water line due to nonlinear effects, and that the wave shape is characterized by flattened crests and peaked troughs that coincide with the numerical results. The propagation speed of hydroelastic waves tends to decrease with an increase of wave height, because of nonlinear shallow-water effects.