Stokes Finite-Amplitude Wave TheoryPerturbation approach: ..... Cnoidal Wave TheoryIn the limit: Isobe, M... Developed by Dean 1965, 1974Describe wave through sine and cosine functions.
Trang 1Higher-Order Wave Theory
Higher-Order Wave Theory
• Stokes wave theory
• Cnoidal wave theory
• Solitary wave theory
• Stream-function theory
• Korteweg - de Vries equation
• Boussinesq equation
Trang 2Governing Equation and Boundary Conditions
0
( , , ) ( , , )
( , , ) ( , , )
z
x z t x L z t
x z t x z t T
1
( ) ( , ) 2
( , )
B p
gz C t z x t
z x t
governing equation
bottom BC
periodicity
DFSBC
KFSBC
Characterize Nonlinearity
/
/
H
L
d
L
d L d
Wave steepness
Relative water depth
Relative wave height
Trang 3Stokes Finite-Amplitude Wave Theory
Perturbation approach:
(velocity potential)
2 a H
ka
(perturbation parameter)
(surface elevation)
2
3
cos
2
2 cosh(4 / ) cos
8 sinh (2 / )
d L
Stokes 2nd Order Wave Theory
Wave Profile:
Trang 4Mass Transport in 2nd-Order Stokes Waves
2
2
cosh 4 ( ) /
( )
2 sinh (2 / )
z d L
U z
Mean drift velocity:
Cnoidal Wave theory
Periodic waves in shallow water (U R> 20).
Solution expressed in elliptic integrals (K)
and functions (cn):
2
cn 2 ( ) ,
s t
x t
L T
Trang 5Cnoidal Wave Theory
In the limit:
(Isobe, M 1985 ”Calculation and application of
first-order cnoidal wave theory,” Coastal Engineering, 9,
309-325)
Solitary Wave Theory
sech
4
o
o
o
H x Ct
u
h
gh
Wave form is entirely above the SWL.
(sech x = 1/sinh x)
Trang 6Developed by Dean (1965, 1974)
Describe wave through sine and cosine
functions Determine coefficient values of
each term so that the best fit in a
least-square sense is obtained with respect to
fulfilling the dynamic free surface boundary
condition
Boussinesq equation
Shallow water hydrostatic pressure
3 2 2
0 1 3
t t
continuity
momentum
2
Eliminate u
Korteweg-deVries Equation
3
3
0
o