6.1. Introduction

Nonlinear surface gravity waves may be analyzed by many analytical and numerical methods. The modest review given here is limited to perturbation and semi-numerical methods. In Sec. 2, the classical Stokes theory of the method of successive approximations is reviewed for its historical perspective. In Sec. 3, the Lindstedt-Poincare perturbation method is reviewed to 4th order in the perturbation parameter ?=kA. There are two methods that may be applied in the Lindstedt-Poincare perturbation method to suppress resonant forcing at the higher perturbation orders. One is to expand the wave celerity C in a perturbation expansion and the second is to expand the radian wave frequency ? in a perturbation expansion. Both methods are applied in this review. The wave celerity C is expanded in the traditional Stokes wave expansion and the radian wave frequency ? is expanded in the weakly nonlinear planar wavemaker theory in Sec. 7. Included in the Lindstedt-Poincare perturbation method review is the derivation of Stokes drift in Sec. 6.3.1 where both Eulerian and Lagrangian derivations are reviewed as well as Stokes drift in a 2D wave channel. In Sec. 4, the modern method of multiple scales (MMS) is reviewed to the 3rd order of approximation. In Sec. 5, the semi-numerical stream function theory is reviewed. In Sec. 6, progressive wave breaking is reviewed. In Sec. 7, a weakly nonlinear planar wavemaker theory is reviewed to only 2nd order where the complete 2nd order theory accurately computes the Stokes...