alternatives to sinusoidal waves ce a676 coastal...

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



• 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



• 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



– 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



• Asymmetrical motion, as with u and w

• Asymmetry increases with wave steepness

axH

2

2

cosh k d z( )[ ]


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( )[ ]


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( )[ ]


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( )[ ]


3 H2

16 L

sinh 2 k d z( )[ ]

sinh k d( )4

cos 2 k x t( )[ ]



• 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[ ]



• 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



• 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



• 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



• 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)

alternatives to sinusoidal waves ce a676 coastal...

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