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1/13/04
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
1/13/04
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
1/13/04
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
1/13/04
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
1/13/04
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
1/13/04
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
1/13/04
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
1/13/04
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)
1/13/04
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 )
1/13/04
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
1/13/04
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.
1/13/04
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+
=
1/13/04
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
1/13/04
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.