Equations for 1D waves on the surface of deep water
Abstract
We apply a canonical transformation to a water wave equations to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation from natural Hamiltonian variables η,ψ to new complex normal variables c,c∗ explicitly uses the fact of vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. The new variable is the sum c(x,t) = c+(x,t) + c-(x,t) of two analytic functions: c+(x,t) - is analytic in the upper half-plane, c-(x,t) - is analytic in the lower-plane. We obtained system of two coupled differential equations for c+ and c- which is very suitable for analytical studies and numerical simulations: [ ] ∂c+-+ iωˆc+ = ∂+ i(|c+|2 - |c- |2)c++ c+ˆk (|c+|2 - |c- |2)- ic+c- c-∗- c-∗ˆk(c+c- ) , ∂t x[ x x ] ∂c-+ iωˆc- = ∂- i(|c- |2 - |c+|2)c- c- ˆk (|c- |2 - |c+ |2)- ic- c+c+∗+ c+∗ˆk (c+c- )(1) ∂t x x x Here ωˆ and ˆk are correspond to the multiplication by √gk- and |k| in the Fourier space, ∗ denotes complex conjugation, the subscript x is the derivative with respect to the variable x, the differentiation operators ∂x+ and ∂x- are ikΘ(k) and ikΘ(-k), where Θ(k) is the Heaviside step function. Physical variables η(x,t) and ψ(x,t) can be restored from complex variable c(x,t). The system (1) has the simple solution: c+ = AeikAx- iωAt, c- = Be -ikBx-iωBt, where 2 2 2 2 2 ωA = ωkA + kA2(|A|2- |B|)2 - kAkB|B|2+ kA|kA - kB||B|2 ωB = ωkB +kB (|B |- |A| )- kAkB|A| + kB|kA - kB ||A| We performed numerical simulation of system (1) for water waves moving in opposite directions.
- Publication:
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EGU General Assembly Conference Abstracts
- Pub Date:
- April 2018
- Bibcode:
- 2018EGUGA..2013067K