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03 Waves, Design Condition and Breakers Ref: Shore Protection Manual, USACE, 1984

Coastal Engineering Manual, USACE, 2003 Basic Coastal Engineering, R.M. Sorensen, 1997 Applied Probability and Stochastic Processes, M.K. Ochi, 1990 Coastal Engineering Handbook, J.B. Herbich, 1991 Water Wave Mechanics for Engineers & Scientists, R.G. Dean and R.A. Dalrymple, 1991 Coastal, Estuarial and Harbour Engineers' Reference Book, M.B. Abbot and W.A. Price,

1994, (Chapter 29) Topics

Description of Ocean Waves Time Series Common Parameters Spectral Parameters and Standard Wave Spectra

Design Wave Estimation Based on measured Data

Extreme value analysis- Graphical and Asymptotic Approaches Return period and Risk of Encounter

Based on Wind Data Empirical Formulae Spectral Models Description of Hurricane Waves

Review of Linear Waves Wave Transformation in Coastal Waters

Shoaling and Refraction Diffraction Reflection Damping due to Bottom friction Wave Breaking

--------------------------------------------------------------------------------------------------------------------- Description of Ocean Waves

Irregular waves characterizing them requires statistical parameters may be time-series (wave height, period) based or frequency domain (energy spectrum) based Time-Series

Describe wave height & period for each individual wave using "zero up-crossing" analysis (i.e. when profile crosses the zero mean upward) Wave height from minimum and maximum on either side of up-crossing

Common parameters: 1. Significant wave (H1/3, T1/3) - average of heights and periods of highest 1/3 waves

of a given record

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2. Mean wave ( T ,H ) mean of a record 3. One-tenth wave (H1/10, T1/10) - average of heights

and periods of highest 1/10 waves of a given record H

F(H)66%

Havg = H1/3

90%

Havg = H1/10

4. Root mean square wave (Hrms, Trms) - ∑= 2

iN1

rms HH 5. Mean wave energy per unit surface area -

∑ρ= 2

iHN8gE

(recall for monochromatic waves 281 gHρ=E )

Assuming a Raleigh distribution for the wave height gives: ( ) avgs3/1 H63.0H63.0H H ===

3/110/1 H27.1H = , average of highest 10 percent of waves

3/110/1 H27.1H = , average of highest 1 percent of waves

3/1max H86.1H = , expected maximum in 500 waves

Generally, the extreme wave height He may be estimated at H1/10

** Note: multiplication factors are from the Shore Protection Manual, not the classic Raleigh distribution

Spectral (freq. domain) Parameters

22rms 8H σ= , where σ2 is the variance of the time series

recall:

∑==µ iHN1H mean

(∑ µ−=σ 2i

2 HN1 ) variance

using Rayleigh distribution assumption H1/3 = 1.416 Hrms = 4σ Peak energy period (Tp) period corresponding to the peak in the energy spectrum (i.e. the modal frequency)

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E(f)

f

fm

modal freq

Significant period: ss H86.3T = (occasionally used, T in sec, H in meters)

Standard Wave Distributions: 1. Pierson-Moskowitz (PM) Spectrum (one parameter, wind speed at 19.5 meters

above the surface, U in m/s)

( )

π−

π×

=− 4

54

23

f2Ug74.0exp

f2g1010.8)f(E , E(f) in m2-s

modal frequency for this spectrum is s21

m Hgπ=f

Requires U at 19.5 mor H1/3

convert to Hs vice U 21.0

gHU s= (narrow band spectrum assumed)

( ) ( )

π

−π×

=− 2

2s

54

23

f2Hg032.0exp

f2g1010.8)f(E

2. Bretschneider Spectrum (two parameters: ms f ,H )

4

m24m ff25.1

Requires H1/3 & fm

( )

−= s5 f25.1expH

f4fE

3. JONSWAP (Joint North Sea Wave Project), for fetch limited seas (i.e.

hurricane generated waves)

( )

( )

σ

−−

γ

−

πα=

2m

2

2m

f2ff

exp4m

54

2

ff25.1exp

f1

2g)f(E

Requires U& X

( ) 22.0x076.0 −=α ,

where 2UgXx = , X is fetch length, U is mean wind speed

(α ≈ 0.008)

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fm = modal frequency, ( ) 33.0m x

Ug5.3f −=

σ = 0.07 for f ≤ fm σ = 0.09 for f > fm γ = shape parameter, γ = 3.3

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

50

100

150

200

250

300

350Pierson-Moskowitz & JONSW AP Spectra

Hs = 8.6mfm = 0.068 Hz

freq. (Hz)

E(f)

(m

2 -sec

)

P ierson-MoskowitzJONSW AP

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0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.220

10

20

30

40

50

60

70

80

90

100Bretschneider & Pierson-Moskowitz Spectra

Hs = 8.6mfm = 0.068 Hz

freq. (Hz)

E(f)

(m

2 -sec

)

B retschneiderPierson-Moskowitz

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Design Wave Estimation Required information varies with objective. 1. For sediment transport, need long term directional wave data 2. For structural integrity, need extreme design wave parameters

Based on Measured Data - Extreme value analysis/ return interval analysis

A. Graphical Approach (requires large data set, > 30)

Pr{X ≤ xn} = 1 - F(xn)= 1/n return interval )x(F11n)x(

nn −==T ,

F(xn) is the cdf

Case 1: Have ordered sample, y1 < y2 < … < yn ln(n)

ln(xn)

End of data set

ln(N)

ln(xN)

1. Plot ln(n) vs. ln(xn)

Extend plot to desired ln(N), e.g. N is a number of observations for a desired time period, (e.g. 100 years) (extrapolate with straight line)

i.e. m observations per year, let T = desired years (100) N = T x m

2. For desired N, find xN on plot

Case 2: Have random data, X with N = total number of observations 1. Build histogram with data set, X

gives freq. (f) and center bin, y1 < y2 < … < yn

2. tabulate associated distribution function N

f)x(F

n

1ii

n

∑== ( )nxF1

1n−

=

3. Plot ln(n) vs. ln(xn) and find yn for the desired T as above

B. Asymptotic Distribution Methods:

Basic idea is to determine a distribution function which fits the data. Given the data set {y1, y2, … yn} where y is the maximum (or minimum) value in a given time interval. Most coastal and ocean engineering problems are Type I or Type III Type I - exponential type ( )( )[ ]µ−α−−= nn yexpexp)y(F -∞ < yn < ∞

Type III - limited type

−−

−=vwywexp)y(F n

n -∞ < yn < w

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Type I

( )( )[ ]µ−α−−= nn yyF expexp)(

[ ]nyvar6π

=α , [ ] [ ]nn yyE var6γ

π−=µ , and the asymptotic values are

[ ] 6var π=∞y and y∞ = 0.5772 (Euler's number) Return Period and Risk of Encounter

The return period can be calculated from ( )nyFNmT

−==

11 , where T is the

return period or return interval in years (e.g. 100 years, 500 years, etc.), m is the number of observations per year (from data yn) and F(yn) is the c.d.f. Substituting the Type I Asymptotic Distribution gives

(([ ))]µ−α−−−=

nyN

expexp11 which can be solved for yn to determine the

extreme event for a given return period (alternative to the graphical method) It may be more desirable to evaluate the design event in terms of a Risk of Encounter. For example, what is the return interval of an event which has a 50% chance of occurring over the life time of the structure… the risk of encounter of this event is 50% Can calculate from

R = risk or probability of encounter (%) T = return period of an extreme event (years) L = design structural life (years)

( )

−−=<−=

LTxxR n exp1Pr1 ( )1001ln R

LT−−

= (1)

or (from SPM and Coastal, Estuarial and Harbour Eng.'s Ref. Book)

( )L

n TxxR

−−=<−=

111Pr1 ( ) LR

T /1100111

−−= (2)

R (%) 10 20 30 40 50 60 70 80 90 T from eqn. (1) 475 224 140 98 72 55 42 31 22 T from eqn. (2) 475 225 141 98 73 55 42 32 22 T in years

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Based on Design Wind Information Empirical formulae Wave energy spectrum method Hurricane wave description

Definitions

Fetch - region in which wind speed and direction is relatively constant Fetch Limited - seas build to the maximum possible for the wind blowing over the given distance (i.e. if the wind blows longer, the seas don't increase) Duration Limited - seas do not reach the maximum possible for the distance

Empirical Wave Prediction Models (from the SPM)

Sverdrup-Munk-Bretschneider (SMB) Model H,T = f(wind speed, fetch, duration) (requires stationary system, uniform wind field, deepwater… but it's simple)

Speed variation should be < ±2.5 m/s (5 knots) Direction variation should be < ±45 degrees (accuracy deteriorates for > ±15deg) Procedure:

1. Determine wind fetch (usually due to geographic constraints) 2. Determine wind duration 3. Determine if seas are Fetch Limited or Duration Limited 4. Compute significant wave parameters

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Determine if conditions are Fetch-Limited or Duration-Limited

2/3

A2A

t

Ugt015.0

UgF

=

Ft < F (actual or available fetch) Duration Limited Ft > F (actual or available fetch) Fetch Limited

t = wind duration Ft = fetch corresponding to t UA = wind stress factor , U in m/s 23.1

A U71.0U =

Compute Significant Wave Parameters (for deep water) 2/1

2A

32A

s

UgF106.1

UgH

×= −

3/1

2A

1

A

s

UgF107.2

UgT

×= −

use small of available fetch or Ft for F shallow water corrections (constant depth, d):

= 4/3

2A

2/1

2A

4/3

2A

2A

Ugd530.0

UgF00565.0

tanhUgd530.0tanh283.0

UgH

= 8/3

2A

3/1

2A

8/3

2AA

Ugd833.0

Ugf0379.0

tanhUgd833.0tanh54.7

UgT

for sloped bottom use "equivalent water depth"

Spectral Wave Models - based on development of the wave energy spectrum. Numerical models include spatial and time varying input wind fields; dissipation and energy transfer mechanisms (e.g. wave breaking, bottom friction, wave-wave interaction, etc.).

Description of Hurricane Waves

Estimate deep water wave conditions at the point of maximum wind due to hurricanes

∆

α+=

4700pRexp

UV29.0103.5H

R

F3/1 and

∆

α+=

9400pRexp

UV145.016.8

R

F3/1T

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H1/3 in meters T1/3 in seconds R = radius of maximum wind in kilometers UR = sustained wind speed at R in m/s VF = forward speed of storm in m/s ∆p = pamb - pcenter in millimeters of Hg α = resonance factor, depends on VF, slowly moving hurricane α = 1.0 ** once significant wave height for the point of maximum wind is determined, can obtain approximate deepwater Hs for other areas by constructing isolines of wave height based on ∆p.

Ranks of waves (i.e. highest to lowest)

nNH707.0H 3/1n = , N is the total number of waves passing a point

during the storm: 3/1FTV

RN =

H1 = most probable maximum wave, H2 = second most probable, etc. (generally just use H1 and H2 for design)

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Review of Linear Wave Terms and Equations (see waves handout + following notes)

Pressure Field for a progressive wave: ( ) ( )tkxcoskh cosh

zhkcosh2Hggzp ω−

+ρ+ρ−=

Progressive vs. Standing Waves

Progressive wave (propagating) ( )tkxcos2H

ω=η m

(-) indicates wave propagating in the (+) x direction

Standing wave (phase oscillates while x location of peak is the same)

t cos kx cos2H

ω=η

Wave Energy and Wave Energy Flux

Wave energy propagates at group velocity, cg cg = nc

where

+=

kh2 sinhkh21

21n ,

deep water: kh ∞ but sinh 2kh ∞ faster, so n ½ shallow water: sinh 2kh 2kh, so n 1

Wave Set-up and Set-down

Changes in momentum as wave approach shore due to shoaling and wave breaking result in a force imbalance which is offset by a variation in mean water level know as set-up and set-down ( η ).

η

MWL

h

Break pointSet-down

Set-up

• set-down up to break point:

=η

kh2sinhkh2

h16H2

• set-up after break point: ( hh831

83b2

2

b −κ+

κ+η=η ) , where bη and hb are

values at the break point and 78.0hH bb ≈=κ (note: set-up and set-down are

equal at the break point, so bb

2b

b kh2sinhkh2

h16H

=η )

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Wave Transformation in Coastal Waters (i.e. shallow water effects)

Shoaling and Refraction (see Dean & Dalrymple, pp. 104-112) θo

bo

lo

lo

b1

l1 = lo

b2

θ

l2 = lo

Depth Contours

Wave Ray 2Wave Ray 1

Snell's Law: o

o

csin

csin θ

=θ

wave direction tends to decrease as the wave shoals (i.e. enters water of decreasing depth) tends to make wave approach the shore normally

process is known as refraction Both shoaling and refraction result in changes in wave height as the wave approaches the shore.

Use wave energy balance to evaluate , where = energy flux, assumes no generation and dissipation

2211 bEbE && = E&

further assuming no energy flux across wave rays and no reflection

2211 bEbE && = ( ) ( ) 2211 bEnCbEnC =

since 281 gHE ρ=

2

o

2g

goo

2

1

2g

1g12 b

bcc

Hbb

cc

HH ==

where Ho, bo, cgo and θo and are deepwater values; let H , rso KKH=

the shoaling coefficient, 2g

o

2g

gos c2

cc

cK ==

the refraction coefficient, 2

or b

bK =

the wave ray geometry gives 4/1

22

o2

2

o

2

or sin1

sin1coscos

bbK

θ−θ−

=θθ

==

NOTE: Kr is always < 1, i.e. perpendicular spacing between rays always becomes greater as the wave shoals. NOTE: if wave propagate perpendicular to contours bo = b and the wave height change is due only to shoaling. When waves propagate at an angle refraction

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Diffraction due to Structures (see Dean & Dalrymple, pp. 116-122)

Diffraction is the process by which energy spreads laterally perpendicular to the dominant direction of wave propagation. Incoming waves interrupted by a barrier such as a breakwater or a groin tend to curve around the barrier and spread into the shadow zone as shown

impermeable barrier(breakwater)

x

y

Diffracted wave crests

Geometric shadow zoneGeometric illuminated zone

Incident wave

When not taken into account, diffraction can cause greatly exaggerated calculations of the distributions of wave energy.

Computed using Helmhotz equation: 0FkyF

xF 2

2

2

2

2

=+∂∂

+∂∂ ,

F is a "surface wave potential" from the velocity potential, ( ) ( ) ( ) tiey,xFzZt,z,y,x ω=φ (with separation of variables)

Theoretical solutions are available for limited simple geometries (i.e. semi-infinite breakwater). Otherwise, numerical solutions are available. Results is a change in wave height (or a wave height distribution over a given area). Solutions develop a 2D plot for a diffraction coefficient (i.e. contours),

id H

HK = , where Hi is the incident wave height (similar to shoaling

and refraction coefficients)

** Refraction and Diffraction often occur simultaneously. Approximations, model equations and numerical models can be used to solve problem. Crudest approach (and most often used in practice) is to assume diffraction dominates within several wave lengths of the structure and refraction dominates further away.

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Wave Reflection Waves impinging on a structure may be reflected or transmitted or both (some energy is reflected and some transmitted) and/or absorbed (i.e. the energy absorbed by the structure) Balance the energy flux across a structure: 1lost energy of fractionKK 2

T2R =++

KR = reflection coefficient, i

RR H

H=K , HR = reflected wave height

KT = reflection coefficient, i

TT H

HK = , HT = transmitted wave height

KR and KT usually determined by experiment

Wave Damping Due to Bottom Friction

Energy will be dissipated by interaction with the bottom and the wave height will be attenuated (without breaking). Effect is small for short distances, but accumulates. Quadratic energy dissipation equation:

b2b2

1bbD uufu ρ=τ=ε (overbar indicates time mean)

εD = rate of energy dissipation f = bottom friction coefficient ub = bottom velocity outside the BL

for flat bottom, linear wave with turbulent BL, averaging over a wave period gives:

( )3

3max bD khsinh2

H6

fu6

f

ω

πρ

=π

ρ=ε

note: dissipation increases as depth decreases

solve for wave height decay over distance for a flat bottom

Dg

dxdEc

ε−= 33

3

g Hkhsinh48

fdxdHgc

81 ω

πρ

=ρ

( )( ) khsinhkh2sinhkh2

H3f1

H21xH

o

o

+π+

=

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Wave Breaking As a wave shoals (and wave height increases), it will eventually become unstable and break, dissipating its energy in turbulence and work against bottom friction. Design of structures which may be inside the surf zone requires prediction of the location of the breakerline. Various types of breakers, which depends on the nature of the bottom and characteristics of the wave.

Generally: spilling breakers - (mildly sloping beaches) forward face of wave becomes

unstable and water-air mixture slides down the slope surface roller that travels with the wave; most common in deep water

plunging breakers - (steeper beaches) crest curves forward and plunges into the trough in front, penetrating the water column

surging breakers - (very steep beaches) wave front steepens without breaking, turbulence forms at the toe and the wave rushes up the beach in a bore-like fashion; very short surf zones & high reflection

(collapsing breakers combine characteristics of plunging and surging breakers)

Determining the location of the breakerline (very empirical) Basic Equation from McCowan (1894): Hb = γ hb ,

Hb = breaking wave height hb = breaking depth γ = breaker index, = Hb/hb

T = wave period m = bottom slope Ho = deep water wave height Lo = deepwater wave length kb = wave number at breaking depth Lb = wave length at breaking depth

Breaker Index Formulae (Linear Wave Theory) 1. McCowan (1894) Hb = γ hb where γ = 0.78, constant

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2. Miche (1944) ( )bbbb hkLH tanh142.0= , steepness limited to 1/7

3. LeMehaute & Koh (1967) 4/1

7/176.0−

=

o

o

o

b

LH

mHH

4. Collins & Weir (1969) Hb = γ hb where m6.572.0 +=γ

5. Weggel (1972) Hb = γ hb where 2

21 THab

gTha

b b

b−=

+=γ

( ) ( ) units SI 156.1 ,18.43 15.1919 −−− +=−= mm ebea

6. Battjes & Jensen (1978)

0.83)(~ coeff. adjustableslightly " a ,88.0

tanh88.0=′

′

= γγ

bbb

b hkk

H

7. Svendsen (1987) Hb = γ hb where b

b

SS21

9.1−

=γ and

=

=

− 2/1

30.2o

o

bb L

HmhLmS

8. Hansen (1990) Hb = γ hb where and 2.005.1 S=γ

=

=

− 45.0

87.2o

o

b LHm

hLmS

9. Smith & Kraus (1991) Hb = γ hb where o

o

LHab −=γ ;

( ) ( ) 16043 112.1 ,15 −−− +=−= mm ebea

Use the refraction and shoaling formula to determine where the wave height will reach Hb, and the beach slope or profile to determine the distance hb is from shore.