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Third Order Cnoidal Wave Solutions in Shallow Water and Its Horizontal and Vertical Fluid Velocity Components

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Third Order Cnoidal Wave Solutions in Shallow Water and Its Horizontal and Vertical Fluid Velocity Components International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051 Third Order Cnoidal Wave Solutions in Shallow Water and its Horizontal and Vertical Fluid Velocity Components Parvin, M. S. Sultana, M. S. Sarker, M. S. A. Assistant Professor, Dept. of Applied Associate Professor, Dept. of Applied Assistant Professor, Dept. of Applied Mathematics, University of Rajshahi, Mathematics, University of Rajshahi, Mathematics, University of Rajshahi, Rajshahi-6205, Rajshahi-6205, Rajshahi-6205, , E-mail: E-mail: [email protected] E-mail: [email protected] [email protected] Abstract – Third order cnoidal wave solutions in shallow real wave in shallow water. Higher order solutions of water are developed where waves progress steadily without cnoidal waves have been derived by Laitone [15], who any change of form. Shallow water wave problems are solved provided a number of results, re-casting the series in terms at the bottom and at the free surface. The boundary of the wave height/depth. Tsuchiya and Yasuda [21] conditions are also taken from Navier-Stokes equation of obtained a third order solution with the introduction of motion. Using these boundary conditions, three nonlinear ordinary differential equations are derived which can be another definition of wave celerity based on assumptions solved using series expansion method. Taking Jacobi elliptic concerning the Bernoulli constant. Nishimura et al. [16] function, third order cnoidal wave solutions have been devised procedures for generating higher order theories for established from which wave elevation, horizontal wave both Stokes and cnoidal theories, making extensive use of velocity and acceleration due to gravity for cnoidal wave are recurrence relations. Nishimura et al. [17] continued the expressed. Assuming Taylor expansion for the stream work of Nishimura et al. [16] and presented a unified view function about the bed, the fluid velocity components are of Stokes and cnoidal theories. Karabut [12] solved an derived in terms of Jacobi elliptic function. ordinary quadratic nonlinear differential difference equation of the first order containing an unknown function Keywords – Navier-Stokes Equation, Jacobi Elliptic Function, Cnoidal Wave, Elliptic Parameter, Velocity under certain conditions. Halasz [10] discussed on higher Components. order corrections for shallow water solitary waves. Chen et al. [4] studied the cnoidal wave solutions of the I. INTRODUCTION Boussinesq systems in two different techniques using the Jacobi elliptic function series. Carter et al [3] discussed the In recent years, the capabilities for calculating shallow kinematics and stability of solitary and cnoidal wave water wave problems have advanced a great deal. The solutions of the Serre equations which are a pair of computation of higher order cnoidal wave solutions for strongly nonlinear, weakly, dispersive, Boussinesq type steady, incompressible flow is an important aspect in partial differential equations. They [3] also described the coastal engineering. Several higher order approximations model of the surface elevation and the depth averaged to irrotational water waves of constant form have appeared horizontal velocity of an inviscid, irrotational, in recent years, often based on Fourier series, but where incompressible shallow water. Jain et al. [11] derived convergence is slow, if at all, for shallow water. These coupled evolution equations for first and second order solutions are often numerical and of an inverse potentials using reductive perturbation method with formulation, and are generally of such higher order, that it appropriate boundary conditions. Oh and Watanabe [18] is difficult to obtain expressions for physical quantities as obtained the second order solution by taking into account functions of position for practical use. The presentation of the unstable and dissipative effects based on a cnoidal results has been limited to tables of integral quantities for wave solution to the KdV equation. Xu et al. [22] a range of wave lengths and heights. However, these calculated the cnoidal function in cnoidal wave theory methods have achieved real success in obtaining based on the precise integration method and also provided numerically exact solutions for the first time Schwartz a trigonometric function approximation for the cnoidal [20] and Cokelet [5]. A survey and comparison of the function. Parvin et al. [19] derived first and second order methods is given Cokelet [5]. cnoidal wave solutions using Jacobi elliptic function. Fenton [6] presented a fifth order cnoidal wave theory Khater et al. [13] used a suitable ansatz and Jacobi elliptic where boundary condition comes from Bernoulli’s function expansion method to construct new exact cnoidal equation which was both apparently complicated, wave solutions of the modified fifth order KdV equation requiring the presentation of many coefficients as and generalized fifth order KdV equation which included unattractive floating point numbers and also gave poor as special cases, some well known equations. Fu et al. [8] results for fluid velocities under high waves. However, in applied Jacobi elliptic function in Jacobi elliptic function a later work, Fenton [7] showed that instead of fluid expansion method to construct the exact periodic solutions velocities being expressed as expansions in wave height, of nonlinear wave equations. In this paper, shallow water the original spirit of cnoidal theory were retained. Cnoidal wave problems have been solved using boundary theory obtained its name in 1895 when Korteweg and_de conditions, at the bottom Y = 0 and at the free Vries [14] obtained their eponymous equation for the surface Y = η(X ). Also the boundary conditions from propagation of waves over a flat bed. The cnoidal solution Navier-Stokes equation of motion generate third order shows the familiar long flat troughs and narrow crests of Copyright © 2016 IJISM, All right reserved 164 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051 cnoidal wave solutions. Then horizontal and vertical fluid ∂U ∂U ∂ 2U ∂ 2U velocity components have been established in terms of U +V = υ + on free ∂ ∂ ∂ 2 ∂ 2 Jacobi elliptic function. X Y X Y surface Y = η (X ) ,pressure p = 0 , (3) II. MATHEMATICAL FORMULATION and ∂ ∂ ∂ 2 ∂ 2 V + V = +υ V + V (4) Consider the wave as shown in figure-1, with a U V g 2 2 ∂X ∂Y ∂X ∂Y stationary frame of reference (x, y) , x in the direction of We assume a Taylor expansion for ψ about the bed of propagation of the waves and y vertically upwards with the following form the origin on the flat bed. The waves travel in the x d d Y 3 d 3 Y 5 d 5 ψ ()X ,Y = −sin Y f (X ) = − Y + − + ...... f ()X direction at speed c relative to this frame. Consider also a 3 5 dX dX !3 dX !5 dX frame of reference (X ,Y ) moving with the waves at (5) velocity c , such that x = X + ct , where t is time df = as in Fenton [7], where is the horizontal velocity on and y Y . The fluid velocity in the (x, y) frame is dX (u,v) and that in the (X ,Y ) frame is (U,V ) . The the bed. Now, the velocity components anywhere in the fluid are velocities are related by u = U + c and v = V . ∂ψ d In the (X ,Y ) frame, all fluid motion is steady and U = = −cos Y f ′()X ∂ dX consists of a flow in the negative X direction, roughly of Y (6) the magnitude of the wave speed, underneath the ∂ψ = d ′() stationary wave profile. The mean horizontal fluid velocity V = − sin Y f X ∂X dX in this frame, for a constant value of Y over one (7) wavelength λ is denoted by -U . It is negative because From Eqs. (3) and (4), we get the apparent flow is in the - X direction. For the convenience of our calculation, the velocities in this frame d d ′ ′′ are used to obtain the solutions. cos Y f (X) cos Y f (X) dX dX λ Y d ′ d ′′ +sin Y f (X)sin Y f (X) = 0 dX dX H c (8) and X d ′ d ′′ −cos Y f (X)sin Y f (X) dX dX y h d ′ d ′′ +sin Y f (X)cos Y f (X) = g x d dX dX (9) At the free surface = η Eqs. (2), (8) and (9) Fig. 1. Wave train, showing important dimensions and Y (X ) , coordinates. become d For irrotational flow, stream function satisfies Laplaces sin η f (X ) = Q (10) ∂ 2ψ ∂ 2ψ dX equation + = 0 (1) ∂X 2 ∂Y 2 d ′ d ′′ cos η f (X)cos η f (X) The boundary conditions at the bottom Y = 0 is a dX dX stream line on which ψ (X ,Y ) is constant and at the free d ′ d ′′ surface Y = η (X ) is also a stream line. +sin η f (X)sin η f (X) =0 dX dX ∴ψ (X )0, = 0 (Taking zero constant) and ψ (X ,η(X )) = −Q (2) (11) and where Q is the volume flux underneath the wave train per d ′ d ′′ unit span. The negative sign is for the flow which is in the −cos η f (X)sin η f (X) dX dX negative X-direction, such that the wave will also propagate in the positive X- direction. d ′ d ′′ (12) For two components, Navier-Stokes equation of motion +sin η f (X)cos η f (X) = g for steady incompressible flow is dX dX Copyright © 2016 IJISM, All right reserved 165 International Journal of Innovation in Science and Mathematics Volume 4, Issue 5, ISSN (Online): 2347–9051 Differentiating Eq. (10) and then substituting this value IV. THIRD ORDER CNOIDAL WAVE SOLUTION in Eqs. (11) and (12), we have d ′ d ′′ To derive third order cnoidal wave solutions, we need cos η f (X)cos η f (X) dX dX some values from Parvin et al.
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