Discretization of Linearized Water Waves

Abstract

The concept of linearized water wave theory is fundamental in fluid dynamics and extensively utilized for studying wave propagation in various aquatic environments. Water waves are crucial in many engineering and scientific fields like ocean and coastal engineering, ship hydrodynamics, and offshore engineering. However, the complexity of nonlinear wave dynamics has limited the accuracy of traditional numerical models, highlighting the need for a simpler yet robust approach. Linearized water wave theory offers a promising solution by assuming small-amplitude waves, which simplifies the governing equations and provides an efficient tool for wave analysis. The numerical simulation of linearized water wave theory holds significant importance across a spectrum of engineering and scientific disciplines, spanning from coastal engineering to oceanography. This paper focuses on discretizing the Euler equation to facilitate precise and efficient numerical simulations of linearized water wave phenomena. The Euler equation, which governs the dynamics of inviscid fluid flow, undergoes linearization to simplify the mathematical formulation while preserving crucial wave dynamics. Discretization methods, including finite difference, finite element, or spectral techniques, are employed to approximate the continuous equations on a discrete computational grid. Subsequently, these discretized equations are numerically solved using iterative or time-stepping methods to forecast the evolution of water waves over time. The accuracy and stability of the numerical scheme are evaluated through convergence studies and validation against analytical solutions or experimental data.