Theory of deep-water surface gravity waves derived from a Lagrangian

Abstract

An exact set of equations describing deep-water irrotational surface gravity waves, originally proposed by Balk (Phys. Fluids, vol. 8 (2), 1996, pp. 416-420), and studied in the case of standing waves by Longuet-Higgins (J. Fluid Mech., vol. 423, 2000, pp. 275-291) and Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 457 (2006), 2001, pp. 495-510), are analytically examined and put in a form more suitable for practical applications. The utility of this approach is its simplicity. The Lagrangian is a low-order polynomial in the Fourier coefficients, leading to equations of motion that are correspondingly of low degree. The structure of these equations is examined, and the existence of solutions is considered. To gain intuition about the system of equations, a truncated model is first examined. Linear stability analysis is performed, and the evolution of the fully nonlinear system is discussed. The theory is then applied to fully resolved permanent progressive deep-water waves and a simple method for finding the eigenvalues and eigenvectors of the normal modes of this system is presented. Potential applications of these results are then discussed.