In high-order analysis and simulation of surface waves using truncated Hamiltonian equations, a key procedure is to obtain the surface vertical velocity from a boundary value problem using perturbation analysis. For a multi-scale wave field, this perturbation series contains large terms proportional to the product of short-wave wavelength and long-wave amplitude. Due to this issue, there has long been doubt on the convergence of the series, resulting in an insufficient understanding of the long-short wave interactions.

We provided the first theoretical proof, using mathematical induction, on the cancellation of large terms at arbitrary order of the perturbation for a multi-scale wave field. This resolves the historical issue and lays foundation on the numerical simulation of long-short wave interactions. The study also sheds light on the perturbation solution of boundary value problems in different physical contexts.

Publication:

Pan, Y., Liu, Y. and Yue, D.K.P. 2018, On the high-order perturbation expansions on the study of long-short wave interactions, Journal of Fluid Mechanics, 846, 902-915. (PDF)