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

hydrodynamic fundamental, small amplitude wave

hydrodynamic fundamental, three conservation equations