Chapter 3.2 Finite Amplitude Wave Theory

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

1. 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.
2. FAWT can be applied to shallow water waves while
   Stokes expansion is limited to deep or
   intermediate-depth water waves.
3. A recursive relation between low-order
   coefficients and high-order Fourier coefficients
   has been derived in FAWT, which eliminates
   similar computational burden in WNWT.
4. 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 3.2.1 Conformal Mapping
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
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 (3.2.15a 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).
17
(No Transcript)
18
Computation of Wave Characteristics