1. Chapter 3.2 Finite Amplitude Wave Theory
Limitation of Weakly Nonlinear Wave Theory (WNWT):
1) When truncated at a relatively high order, it is burdened by tedious and extremely lengthy algebraic work.
2) It was discovered by Schwartz (1974) that the small wave steepness expansion is not convergent for deep or intermediate-depth water waves before reaching their breaking limits.
3) It cannot be applied to shallow water waves (Ursell number).
Finite Amplitude Wave Theory (FAWT)
was developed to overcome the above shortcomings.
Schwartz (1974), Cokelet (1977) and Hogan (1980)

2. Key Differences b/w WNWT and FAWT
The two free-surface boundary conditions are satisfied exactly at the free surface in FAWT, while they are satisfied at the still water level in WNWT.
FAWT can be applied to shallow water waves while Stokes expansion is limited to deep or intermediate-depth water waves.
A recursive relation between low-order coefficients and high-order Fourier coefficients has been derived in FAWT, which eliminates similar computational burden in WNWT.
FAWT is very powerful tool for computing waves, but it limited to 2-D periodic wave trains. On the other hand, WNWT can be applied to 3-D and Irregular Waves

3. Moving Coordinates X-Z
The fixed coordinates (x-z) and the coordinates (X-Z) moving at the phase velocity (C) of the periodic wave train.

4. Non-dimensional & Normalized Variables

5. Conformal Mapping from X-Z to s-n

6. Figure Conformal
Mapping

10. Substituting (3.2.11)-(3.2.14) into (3.2.10) and making use
of the orthogonal property of the cosine function, (3.2.10)
reduces to the following subsets of equations.
Rules for all summations, 1) the value of a summation
is taken to be zero if the lower limit exceeds the upper.
and 2), if the upper limit is not specified, it is defined as
a positive infinity.

11. Perturbation Schemes and Hierarchy Equations
Eq. (3.2.15a) and (3.2.15b) are a set of nonlinear algebraic equations governing the Fourier coefficients, aj , which can be solved by a perturbation technique.

12. Substitution of the expansions into (3. 2
Substitution of the expansions into (3.2.15a & b) leads the following recurrence relations.

13. Equ. (3.2.15a) Equ.(3.2.17a)
Equ. (3.2.15b) Equ.(3.2.17b)
Equ. (3.2.13) Equs.(3.2.17c) & (3.2.17d)
Equ. (3.2.14) Equ.(3.2.17e)

14. Three Choices of the Expansion Parameter

15. General rules for the procedure of obtaining these coefficients are similar but different in details w. r. t. the choices of the expansion coefficients.

16. The Pade Approximation Bender and Orszag (1978).

18. Computation of Wave Characteristics