Addressing the Disturbance Potential through the Application of Fourier Transforms and Complex Analysis

Water dynamics in finite depth are more complex and interesting than in infinite depth, even under simple conditions with zero surface pressure and steady motion.

Consider the effect of a pressure source located within the interval $[−a,a]$ acting on a free surface and with a flow velocity of U, examine the relationships between $\phi$ and $\eta$.

This post primarily introduces The Variational Approach and demonstrates how to employ the method to derive the Euler-Lagrange equations for the water wave equations

incompressible flow momentum equation and Surface of water waves and boundary condition and Variational Formulation

dispersion relation and solutions of the water wave equation incorporating surface tension

Asymptotic expansion of analytic functions and contour integral

asymptotic expansions for integrals

asymptotic expansions and Abel's lemma

hydrodynamic fundamental, small amplitude wave