Nonlinear Kelvin wakes and exponential asymptotics

Abstract

A notable feature of Kelvin surface-wave patterns due to submerged sources at small Froude numbers (F≪1) is that the wave amplitudes are exponentially small with respect to F. Moreover, as F→0 the assumption of linearization is not appropriate. Thus, the role of nonlinear effects in Kelvin wake patterns at small F hinges on exponential (‘beyond-all-orders’) asymptotics. This outstanding theoretical issue is addressed here in the context of a simple partial differential equation with a quadratic nonlinear term, controlled by a parameter ɛ, and a locally confined forcing term in two spatial dimensions. The linear far-field wave response of this model equation is akin to a Kelvin wake whose wave amplitudes are exponentially small with respect to a parameter μ, analogous to F in water waves. To understand how nonlinearity affects this Kelvin-like pattern for ɛ, μ≪1, we develop an exponential asymptotics technique motivated by our earlier analysis in the wavenumber domain of the fKdV equation. Results are presented for three forcing terms that mimic a submerged source, a submerged doublet and a surface pressure. While the linear wave-crest geometry remains intact, the wave amplitudes in the Kelvin wake are significantly enhanced/suppressed by nonlinearity, depending on the type, strength and sign (polarity) of the forcing term. The asymptotic predictions generally are supported by direct numerical computations for various ɛ and μ.