Abstract
We consider a Whitham equation as an alternative for the Korteweg–de Vries (KdV) equation in which the third derivative is replaced by the integral of a kernel, i.e., ηxxx in the KdV equation is replaced by ∫−∞∞Kν(x−ξ)ηξ(ξ,t)dξ. The kernel Kν(x) satisfies the conditions limν→∞Kν(x) = δ″(x), where δ(x) is the Dirac delta function and limν→0Kν(x) = 0. The questions studied here, by means of numerical examples, are whether adjustment of the parameter ν produces both continuous solutions and shocks of the kernel equation and how well they represent KdV solutions and solutions of the underlying hyperbolic system. A typical example is for resonant forced oscillations in a closed shallow water tank governed by the kernel equation, which are compared with those governed by a partial differential equation. The continuous solutions of the kernel equation associated with frequency dispersion in the KdV equations limit to the shocks of the shallow water equations as ν → 0. Two experimental problems are solved in a single equation. As another example, suitable adjustment of ν in the kernel equation produces solutions reminiscent of a hydraulic and undular bore.
Highlights
Shallow water theory is given by the nonlinear equation ηt
Observations have established that some shallow water waves do not break, e.g., a solitary wave as observed by Russell in 1844.1 The Korteweg–de Vries (KdV) equation ηt + c0(1 + 2h0 η)ηx + γηxxx = 0, (2)
The purpose of this paper was to construct a simple Whitham type equation that contains both continuous solutions and shocks that correspond to breaking due to nonlinear hyperbolicity
Summary
Shallow water theory is given by the nonlinear equation ηt + c0ηx 2c0 h0 ηηx = (1). Where η = η(x, t) is the displacement of the fluid surface about the √. Depth h0 and c0 = gh[0] is the phase speed. This theory predicts that all solutions carrying an increase of elevation break. By wave breaking is meant that the solution remains bounded, but its slope becomes unbounded in finite time. Observations have established that some shallow water waves do not break, e.g., a solitary wave as observed by Russell in 1844.1 The Korteweg–de Vries (KdV) equation ηt + c0(1 + 2h0 η)ηx + γηxxx = 0, (2). Which includes frequency dispersion through the term γηxxx, with γ c0 h20
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