Higherorder Boussinesq equations for twoway propagation of shallow water waves
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Standard perturbation methods are applied to Euler's equations of motion governing the capillarygravity shallow water waves to derive a general higherorder Boussinesq equation involving the smallamplitude parameter, α = a / h0, and longwavelength parameter, β = (h0 / l)2, where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ / ρ g h02, where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1)  frac(1, 3)  τ  ≫ β, and (2)  frac(1, 3)  τ  = O (β). Case 1 leads to the classical (illposed and wellposed) fourthorder Boussinesq equations whose dispersive terms vanish at τ = frac(1, 3). Case 2 leads to a sixthorder Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159207] as a dispersive regularization of the illposed fourthorder Boussinesq equation. The relationship between the sixthorder Boussinesq equation and fifthorder KdV equation is also established in the limiting cases of the two small parameters α and β. © 2006 Elsevier SAS. All rights reserved.
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Boussinesq Equations

Perturbation Analysis

Shallow Water Waves

Solitary Waves

Twoway Wave Propagation
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