The generation of capillary-gravity solitary waves by a surface pressure forcing
Introduction
In this paper, the generation of capillary-gravity waves by a moving pressure distribution is investigated. We observe that solitary waves can be generated by a subcritical forcing. A family of weakly nonlinear model equations for capillary-gravity water waves is derived from the potential flow equations. Model equations from this family can be characterized by the highest degree in their nonlinear term. We study the forced dynamics of a cubic member of this family.
Understanding forced water waves has been of interest for over a century. Steady flow past an obstacle in the potential flow equations was studied famously by Rayleigh [38] – more recent examples include [33], [42]. The dynamic, forced problem has been studied in a number of weakly nonlinear models which include long wave (or shallow water) assumptions, for example the KdV, Benjamin, Benjamin-Ono, and BBM equations [9], [6], [12], [23], [24], [29], [31]. Unforced capillary-gravity wave dynamics have been studied using weakly nonlinear models which do not include a long wave assumption – an overly restrictive assumption for waves at finite wavenumber [2], [3], [4], [8]. In this paper, the isotropic deep water capillary-gravity wave model of [3] is generalized to include cubic terms and a surface pressure distribution. The cubic model Eq. (2.5) is used to study the dynamic problem of flow in the wake of a surface pressure distribution, a simple model for the generation of water waves by wind.
One feature of capillary-gravity flows which is not present in flows without surface tension is the existence of traveling wavepacket solitary waves. The existence of these waves was first explained by Longuet-Higgins [27]. These waves, and their two-dimensional analogues, have been studied primarily in the context of potential flow, assuming an inviscid fluid, although some recent studies have begun to consider the effect of viscous damping [12], [15]. In the inviscid setting, these solitary waves have been predicted with asymptotic arguments [1], rigorously shown to exist in 1D [21] and 2D shallow water [19], and computed numerically [7], [10], [25], [35], [36], [41]. The dynamics of solitary waves – including stability, collisions, and generation by forcing – have been studied in a number of weakly nonlinear settings. On a two-dimensional fluid, shallow water studies of solitary wave dynamics focus on the KdV equation [5], [28], [31]. On a three dimensional fluid, shallow water dynamics have been studied with a Benney–Luke system, the KP equations, and the two-dimensional Benjamin equations [8], [23], [24]. Solitary wave dynamics have also been studied in deep water, both in two and three-dimensions [4], [2], [3]. The majority of time-dependent simulations have been restricted to quadratic weakly-nonlinear models, although some recent progress has been made in direct time-dependent simulations of the potential flow equations [18]. Weakly-nonlinear quadratic models neglect cubic nonlinear terms, which, due to the connection between envelope solitary waves and the cubic nonlinear Schrödinger equation, should play a role in solitary wave dynamics [1], [27].
A natural example of surface forcing is that due to wind [11], [40], where a number of recent experiments have focused on gravity-capillary wave generation. In 2005 Longuet-Higgins and Zhang experimentally observed transients resembling gravity-capillary solitary waves in the wake of a surface pressure distribution [26]. Similar waves have been observed in experiments in mercury [17] and in a “wind-ruffled” channel [44]. Here we present an inviscid numerical analogue of these experiments, including cubic nonlinear terms, and observe the generation of capillary-gravity solitary waves.
Section snippets
Derivation
In this section we derive a weakly nonlinear model for small amplitude water waves which are approximate solutions of the potential flow equation. To begin, recall the potential flow equations for a body of water, of average depth H, displacement η, conservative velocity field u = ∇ϕ, which acts under gravity g, atmospheric pressure P, and constant surface tension γ.The operators ∇x and Δx are the gradient and
Forcing and solitary waves
In this section we consider the forced problem (δ ≠ 0) for moving pressure distributions P : = P(x − ct). Solutions of Eq. (2.5), with a prescribed P(x, t), as for example in (3.9), depends on δ and c. The constant ϵ is a measure of the size of η. To understand the behavior of small solutions, we consider the expansionThe leading order solution η1 solves the linear problemwith solutionwhere
Conclusion
A weakly nonlinear model equation for deep water capillary-gravity waves is derived. This model approximates the potential flow equations to cubic order and includes the effect of surface pressure distributions. Asymptotic solutions for subcritical, near-resonant, traveling pressure distributions are explored using a Stokes-like expansion in the wave amplitude. Quartet interactions are shown to violate the perturbation hypothesis unless a slowly varying wave packet is included in the leading
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