Features of double-frequency triad interactions in the nonlinear response to a moving load on a floating ice sheet

We study nonlinear resonant triad interactions among flexural-gravity waves generated by a steadily moving load on a floating ice sheet. Of the many possible triad interactions involving at least one load-produced wave, we focus on the double-frequency case where the wavenumber of the leading wave is double that of the trailing wave. This case stands out because resonant interactions can occur with or without the presence of an ambient wave. Using multiple-scale perturbation analysis, we obtain amplitude evolution equations governing the leading-order, steady-state response. We complement the theoretical predictions with direct numerical simulations of the initial–boundary value problem using a high-order spectral method accurate to arbitrary order. Our results show that the double-frequency interaction can cause the trailing wave amplitude to decay with distance from the load, with its energy transferred to its second harmonic which radiates forwards to coherently interfere with the leading wave. Depending on the length and orientation of the load, the resonant interaction can in some cases cause the wave drag to become vanishingly small, or in other cases nearly double the maximum bending strain compared to the linear prediction. We also consider the effect of a small ambient wave that can initiate a resonant interaction in the leading wave field in addition to the trailing wave field interaction. This can result in a steady, localised wave packet containing two mutually trapped wave components, leading to vanishing wave drag. This work has potential implications for defining safe operating profiles for vehicles travelling on floating ice.