Mathematical Modeling of Regular and Irregular Shallow Water Waves Using Boussinesq Equation with Improved Dispersion

A mathematical model has been formulated to analyze the behaviour of small amplitude linear and nonlinear shallowwater waves in the coastal region. The coupled Boussinesq equations (BEs) are obtained from the Euler's equation in terms of velocities variable as the velocity measured from arbitrary distance from mean water level. BEs improves the dispersion characteristics and is applicable to variable water depth compared to conventional BEs, which is in terms of the depth-averaged velocity. The solution of the time-dependent BEs with kinematic and dynamic boundary conditions is obtained by utilizing Crank-Nicolson procedure of finite difference method (FDM). Further, the Von Neumann stability analysis for the Crank Nicolson scheme is also conducted for linearized BEs. The numerical simulation of regular and irregular waves propagating over the variable water depth is validated with the previous studies and experimental results. The present numerical model can be utilized to determine the wave characteristics in the nearshore region, including diffraction, refraction, shoaling, reflection, and nonlinear wave interactions. Therefore, the current model provides a competent tool for simulating the water waves in harbour or coastal regions for practical application.