Water wave propagation interaction patterns in forced Korteweg-de Vries using homotopy analysis method

Abstract

Water waves phenomenon with a forcing disturbance causing unsteady waves is considered as a complicated phenomenon. In this research, forced Korteweg-de Vries (fKdV) equation is found to explain the behaviour of unsteady waves over underwater obstacles. The forcing terms in fKdV non-linear equation are modelled and the approximate analytical solutions are found using Homotopy Analysis Method (HAM). Specifically, standard fKdV equation for three different choices of forcing term such as quadratic, sinusoidal and exponential are studied in this research. The ability of HAM in solving non-integrable soliton-type fKdV models are validated using Hirota’s Method with reference to Jun-Xiao and Bo-Ling works in 2009. The relationship between forcing term in fKdV equation and bottom topography with specific critical flow of an ocean are also investigated. Transcritical flow over a hole and a bump are examined using nonlinear shallow water fKdV equation. It is found that multi solitary waves exist and maximum elevation of waves occurs at the deepest hole of the seabed. The water wave exhibits solitary pattern when it flows over sloping region of a hole but no distinctive pattern on flattened based seabed. The transcritical flow over a bump consequently generates upstream and downstream flows. Meanwhile, flow over a flatten bump shows no activity on the flat part of bottom topography but the waves exhibit multi solitary interactions over positive and negative sloping region bump. Furthermore, water wave propagation interaction patterns over a moving bump is explored and it is found that the flow of water waves become subcritical and supercritical based on the critical parameter in the fKdV equation. Three different sloping shapes of Gaussian bump are analyzed as underwater disturbances. If the forcing slope is steep, then it triggers a high amplitude peaked waves. The water wave propagation interaction patterns are also observed when it travels over a flat bottom to inclination plane. In particular, at different degree of inclinations, water wave interaction patterns show higher amplitude at higher steeper planes. In summary, this study shows that steeper sloping underwater topography and types of criticality flow determine the nonlinearity of water wave propagation interaction pattern when it travels over some certain underwater topography.