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.

中文翻译：

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使用具有改进色散的Boussinesq方程对规则和不规则浅水波进行数学建模

建立了数学模型来分析沿海地区小振幅线性和非线性浅水波的行为。耦合的Boussinesq方程（BEs）是从Euler方程获得的，而速度是从距平均水位任意距离处测得的速度变量。与传统的BE相比，BEs改善了分散特性，并且适用于可变的水深，就深度平均速度而言。利用有限差分法（FDM）的Crank-Nicolson程序获得了具有运动学和动态边界条件的时变BE的解。此外，还针对线性BE对Crank Nicolson方案进行了冯·诺依曼稳定性分析。先前的研究和实验结果验证了规则波和不规则波在可变水深中传播的数值模拟。本数值模型可用于确定近岸区域的波浪特征，包括衍射，折射，浅滩，反射和非线性波浪相互作用。因此，当前模型为模拟港口或沿海地区的水浪提供了一个有力的工具，以供实际应用。