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Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 1
Alternatives to sinusoidal waves
CE A676 Coastal Engineering
• Stokes 2nd‐order finite amplitude solutions• Cnoidal wave tables and applications• Other waves theories
• References– Sorenson, 2006. Basic Coastal Engineering (text), Springer
– Sorenson, 1993. Basic Wave Mechanics for Coastal and Ocean Engineers, Wiley Interscience
– Dean and Dalrymple, 1991. Water Wave Mechanics for Engineers and Scientists, World Scientific
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 2
• All theories require simplifying assumptions and approximations
• H/L << 1: “small amplitude,” “linear” – leads to sinusoidal solution
– Also H/d < 1
• H/L is not necessarily so small → “finite amplitude” solutions
– H/d can approach 1
• Power series – perturbation approach 1. Consider a mean value, ̅, plus a departure
(perturbation, ) from the mean ̅• Perturbation term, , is proportional to H/L
2. Expand dependent variables as power series (for example): ∅ ∅ ∅ ⋯
• Stokes first order terms – equivalent to sinusoidal
• Second order terms – results in profile closer to nature• Readily programmed equations
• Refinements to particle kinematics useful for some situations
• Higher order – mainly research applications
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 3
• Same governing equation:
• BBC (same): ∅
0 at z = ‐d
• Lateral BBC’s (same): periodic in x and in t• Free surface boundary conditions – non‐linear
– DFSBC:∅ ∅ ∅
• at ,
– KFSBC:∅ ∅
at ,
• Same Dispersion Relation of T, L, and C
– Same deepwater
• Celerity unaffected by wave steepness
• Higher‐order Stokes solutions: celerity increases with wave steepness
• Stokes considered inapplicable for ⁄ 0.1
2
g k tanh k d( ) C
k
g
ktanh k d( )
g T
2 tanh
2 d
L
Lg T
2
2 tanh
2 d
L
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 4
– Shorter, higher crests; longer, flatter troughs
– Unrealistic bumps in trough when steepness over
H
2cos k x t( )
H2
k
16
cosh k d( )
sinh k d( )3
2( cosh 2 k d( )[ ] cos 2 k x t( )[ ]
0 200 400 600 800 1 103
1
0.5
0
0.5
1
x
H
L
sinh k d( )3
cosh kd 2 cosh 2 k d( )( )[ ]
• First term same as sinusoidal
• Net “u” results in mass transport velocity
uH
2
cosh k d z( )[ ]
sinh k d( ) cos k x t( )
3 H( )2
4 T L
cosh 2 k d z( )[ ]
sinh k d( )4
cos 2 k x t( )[ ]
wH
2
sinh k d z( )[ ]
sinh k d( ) sin k x t( )
3 H( )2
4 T L
sinh 2 k d z( )[ ]
sinh k d( )4
sin 2 k x t( )[ ]
unet
2H
2
2 T L
cosh 2 k d z( )[ ]
sinh k d( )2
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 5
• Asymmetrical motion, as with u and w
• Asymmetry increases with wave steepness
axH
2
2
cosh k d z( )[ ]
sinh k d( ) sin k x t( )
3 3
H2
T2
L
cosh 2 k d z( )[ ]
sinh k d( )4
sin 2 k x t( )[ ]
azH
2
2
sinh k d z( )[ ]
sinh k d( ) cos k x t( )
3 3
H2
T2
L
sinh 2 k d z( )[ ]
sinh k d( )4
cos 2 k x t( )[ ]
• (last term) increases with time – Particle paths are not closed
H
2
cosh k d z( )[ ]
sinh k d( ) sin k x t( )
H2
8 L sinh k d( )2
1
3 cosh 2 k d z( )[ ]
2 sinh k d( )2
sin 2 k x t( )[ ]
H2
4 L
cosh 2 k d z( )[ ]
sinh k d( )2
t
H
2
sinh k d z( )[ ]
sinh k d( ) cos k x t( )
3 H2
16 L
sinh 2 k d z( )[ ]
sinh k d( )4
cos 2 k x t( )[ ]
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 6
• Same as sinusoidal
E1
8 g H
2 average energy per unit surface area
Pn E
TE Cg n E C n
1
21
2 k d
sinh 2 k d( )
• Time‐average dynamic pressure is not zero
– as with sinusoidal wave theory
• Last term varies with “z”, only (not “t”)
p g z g H
2
cosh k d z( )[ ]
cosh k d( ) cos k x t( )
3 g H2
4 L sinh 2 k d( )
cosh 2 k d z( )[ ]
sinh k d( )2
1
3
cos 2 k x t( )[ ]
g H2
4 L sinh 2 k d( )
cos 2 k d z( )[ ] 1[ ]
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 7
• Developed to improve shallow water predictions
– Stokes not best for ⁄ 0.1
• Deepwater: equivalent to sinusoidal
– Can begin with
• Shallow water limit: solitary wave
• Results apply Jacobian elliptical “cn” functions
• Has first and higher‐order solutions
– 1st‐order most commonly used
•
– Used throughout wave literature
– Compares wavelength, wave height, & depth
– Cnoidal theory best for 25– Stokes best for 10– Either okay for 10 25
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 8
• Shoaling: find H, given Ho, T, d
• Example: Ho= 1 m, T = 14 sec, d = 4 m
– 1.56305.8 ;
– d/Lo= 0.013; Cnoidal OK (< 0.1)
– Ho/Lo = 0.0033
• Table 3: H/Ho = 1.75; H = 1.8 m
• Example: T = 14 sec, H = 1.8 m, d = 4 m; find L & C
– 1.56 305.8– H/d = 0.45
– ⁄ 22
• Table 2: L/d = 24.7– L = 99 m
– C = L/T = 7.1 m/s
• Sinusoidal: – L = 86 m; C = 6.2 m/s
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 9
• Cnoidal waves: period, T, is not constant with depth– For H = 1.8m, L = 99m, d = 4m,
– 275 (>25)
– A = 0.581
– 1
7.0 /
– 14.1
• Displacement all above SWL
• Water particles move in direction of propagation
• Infinite period and wave length
– → ∞– Limit of Cnoidal solution
•
• 10 20 40 60 80 100
0
0.5
1
1.5
2
Solitary wave: H = 2 m & d = 3 m at t = 7 sec
x (m)
prof
ile
(m)
C = 7.2 m/s
Module 4 ‐Non‐linear wave theories 2/1/2016
CE A676 Coastal EngineeringOrson P. Smith, PE, Ph.D., Instructor 10
• Developed to better accommodate digital computer analysis of higher‐order solutions
, ,2
sinhcosh
cos
• Solve 0with Stokes‐equivalent BC’s• Taylor series solution expansion evaluated by matrix computer routines
• Has been used to evaluate nearly breaking waves and extreme wave forms
• Each theory has limits
• Sinusoidal stretched in practice
• Graph widely published
– Fig. 4.8 (Sorenson)
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