2-10-00, Environmental Factors in the Coastal Region, Waves
Ref:
Shore Protection Manual, USACE, 1984
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 timeseries (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
2. Mean wave ( H , T ) mean of a record
Havg = H1/10
H
Havg = H1/3
3. One-tenth wave (H1/10, T1/10) - average of heights
and periods of highest 1/10 waves of a given record
4. Root mean square wave (Hrms, Trms) -
H rms  N1  H i2
5. Mean wave energy per unit surface area g
2
E
 Hi
8N
(recall for monochromatic waves E  18 gH 2 )
66% 90%
F(H)
Assuming a Raleigh distribution for the wave height gives:
H1/ 3  H s   0.63H  0.63H avg
H1/ 10  1.27 H1/ 3 , average of highest 10 percent of waves
H1/ 100  1.67 H1/ 3 , average of highest 1 percent of waves
H max  1.86 H1 / 3 , expected maximum in 500 waves
2
1.26
H rms 
H
H1 / 3


Spectral (freq. domain) Parameters
2
H rms
 8 2 , where 2 is the variance of the time series
recall:
1
  H   H i  mean
N
1
2
 2   H i     variance
N
using Rayleigh distribution assumption
H1/3 = 1.416 Hrms = 4
Peak energy period (Tp) period corresponding to the peak in the energy spectrum
(i.e. the modal frequency)
E(f)
fm
modal freq
Significant period:
f
Ts  3.86 H s (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)
4

 g  
8.10 10 3 g 2
  , E(f) in m2-s
E( f ) 
exp  0.74
4
5
U
2

f
2 f


 
modal frequency for this spectrum is f m  21 g H s
gH s
(narrow band spectrum assumed)
0.21
2

 g Hs  
8.10  10 3 g 2
 
E( f ) 
exp  0.032
2 
24 f 5

 2f   
convert to Hs vice U  U 
2. Bretschneider Spectrum (two parameters: H s , f m )
4

 fm  
1.25 f m4 2
E f  
H s exp  1.25  
4 f5

 f  
3. JONSWAP (Joint North Sea Wave Project), for fetch limited seas (i.e.
hurricane generated waves)
 f  fm 
4

 f m   exp   2 2 f m2
g2 1
E( f )  
exp  1.25   
24 f 5 
 f  

  0.076x 
0.22
2



,
gX
, X is fetch length, U is mean wind speed
U2
(  0.008)
g
0.33
fm = modal frequency, f m  3.5 x 
U
 = 0.07 for f  fm
 = 0.09 for f > fm
where x 
 = shape parameter,  = 3.3
Pierson-Moskowitz & JONSWAP Spectra
350
Pierson-Moskowitz
JONSWAP
300
E(f) (m2-sec)
250
200
Hs = 8.6m
fm = 0.068 Hz
150
100
50
0
0.02
0.04
0.06
0.08
0.1
0.12 0.14
freq. (Hz)
0.16
0.18
0.2
0.22
Bretschneider & Pierson-Moskowitz Spectra
100
Bretschneider
Pierson-Moskowitz
90
80
E(f) (m2-sec)
70
60
Hs = 8.6m
fm = 0.068 Hz
50
40
30
20
10
0
0.02
0.04
0.06
0.08
0.1
0.12 0.14
freq. (Hz)
0.16
0.18
0.2
0.22
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)  return interval T ( xn )  1
, F(xn) is the cdf
1  F ( xn )
Case 1: Have ordered sample, y1 < y2 < … < yn
ln(N)
ln(n)
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)
End of data set
ln(xN)
i.e. m observations per year, let T = desired
years (100)  N = T x m
ln(xn)
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
n
2. tabulate associated distribution function F ( xn ) 
 fi
i 1
N
 n
1
1  F xn 
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
F ( yn )  exp exp   yn  
Type I - exponential type
- < yn < 
Type III - limited type
  w  y n 
F ( y n )  exp  

  w  v 
- < yn < w
Type I
F ( yn )  exp  exp   yn  

6
 6
 var  y n  , and the asymptotic values are
,   E y n  

var  y n 
var y   
6 and y = 0.5772 (Euler's number)
Return Period and Risk of Encounter
1
, where T is the
1  F  yn 
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.
The return period can be calculated from mT  N 
Substituting the Type I Asymptotic Distribution gives
1
N
which can be solved for yn to determine the
1  exp  exp   y n   
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)
 T
R  1  Pr x  xn   1  exp   
 L

T
L
ln 1  R 100
(1)
or
(from SPM and Coastal, Estuarial and Harbour Eng.'s Ref. Book)
 1
R  1  Pr  x  xn   1  1  
 T
R (%)
T from eqn. (1)
T from eqn. (2)
T in years
10
475
475
20
224
225
30
140
141
L
40
98
98

T
50
72
73
60
55
55
1
(2)
1/ L
1  1  R 100
70
42
42
80
31
32
90
22
22
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
Determine if conditions are Fetch-Limited or Duration-Limited
3/ 2
 gt 
gFt

 0.015
2
UA
U A 
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 A  0.71U 1.23 , U in m/s
Compute Significant Wave Parameters (for deep water)
 gF 
gH s
 1.6  10 3  2 
2
UA
U A 
1/ 2
1/ 3
 gF 
gTs
 2.7 10 1  2 
UA
U A 
use small of available fetch or Ft for F
shallow water corrections (constant depth, d):
1/ 2

 gF  
 0.00565 2  
3/ 4

 gd  
gH

U A  
 0.283 tanh 0.530 2   tanh 
2
3/ 4 
UA

 gd 
 U A  


 0.530 U 2 

 A


1/ 3

 gf  
 0.0379 2  
3/8

 gd  
gT

U A  
 7.54 tanh 0.833 2   tanh 
3/8 
UA

 gd  
 U A  

 0.833 U 2  
 A 

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
 0.29VF 


 exp  Rp  and T1 / 3  8.61  0.145VF  exp  Rp 
H1 / 3  5.031 


U R 
U R 
 9400 
 4700 


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
 = resonance factor, depends on VF, slowly moving hurricane  = 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)
N
, N is the total number of waves passing a point
H n  0.707 H1 / 3
n
R
during the storm: N 
VF T1 / 3
H1 = most probable maximum wave, H2 = second most probable, etc.
(generally just use H1 and H2 for design)
Review of Linear Wave Terms and Equations (see waves handout + following notes)
Pressure Field for a progressive wave:
p  gz  g
H cosh k h  z 
coskx  t 
2 cosh kh
Progressive vs. Standing Waves
H
coskx  t 
2
(-) indicates wave propagating in the (+) x direction
Progressive wave (propagating)

Standing wave (phase oscillates while x location of peak is the same)
H
  cos kx cos t
2
Wave Energy and Wave Energy Flux
Wave energy propagates at group velocity, cg
cg = nc
1
2kh 
where n  1 
,
2  sinh 2kh 
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 (  ).
Break point
Set-down
Set-up
MWL

h


H 2  2kh 


16h  sinh 2kh 
3 2 8
hb  h  , where b and hb are
set-up after break point:   b 
1  3 2 8
values at the break point and   H b hb  0.78 (note: set-up and set-down are
set-down up to break point:

2
H
equal at the break point, so b  b
16hb
 2kh 

 )
 sinh 2kh  b
Wave Transformation in Coastal Waters (i.e. shallow water effects)
Shoaling and Refraction (see Dean & Dalrymple, pp. 104-112)
o
bo
lo
Wave Ray 1
Wave Ray 2
b1
Depth Contours
sin  sin  o


c
co
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
Snell's Law:
l 1 = lo

b2
l2 = l o
Both shoaling and refraction result
in changes in wave height as the
wave approaches the shore.
lo
Use wave energy balance to evaluate E1b1  E 2b2 , where E = energy flux,
assumes no generation and dissipation
further assuming no energy flux across wave rays and no reflection
E1b1  E 2b2  EnC 1 b1  EnC 2 b2
since E  18 gH 2  H 2  H 1
c g1
cg 2
c go
b1
 Ho
b2
cg 2
bo
b2
where Ho, bo, cgo and o and are deepwater values;
let H  H o K s K r ,
the shoaling coefficient, K s 
the refraction coefficient, K r 
c go
cg 2
bo

co
2c g 2
b2
1/ 4
bo
cos  o  1  sin 2  o 



the wave ray geometry gives K r 
b2
cos  2  1  sin 2  2 
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
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
y
Geometric illuminated zone
Geometric shadow zone
Diffracted wave crests
impermeable barrier
(breakwater)
x
Incident wave
When not taken into account, diffraction can cause greatly exaggerated
calculations of the distributions of wave energy.
2F 2F

 k 2F  0,
x 2 y 2
F is a "surface wave potential" from the velocity potential,
x, y, z, t   Z z F x, y e it (with separation of variables)
Computed using Helmhotz equation:
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),
Kd  H
, where Hi is the incident wave height (similar to shoaling
Hi
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.
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:
K R2  KT2  fraction of energy lost  1
KR = reflection coefficient, K R  H R
KT = reflection coefficient, K T  H T
Hi
Hi
KR and KT usually determined by experiment
, HR = reflected wave height
, HT = transmitted wave height
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:
 D   b ub  12 fub2 ub
(overbar indicates time mean)
D = 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:
f
ub max 3  f  H 
D 
6
6  2 sinh kh 
note: dissipation increases as depth decreases
3
solve for wave height decay over distance for a flat bottom
dEcg
dx
  D 
H x  
1
2
1
dH
f
3
gc g

H3
3
8
dx 48 sinh kh
Ho
1
Ho
f
3 2kh  sinh 2khsinh kh
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 =  hb ,
Hb = breaking wave height
hb = breaking depth
 = 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 =  hb where  = 0.78, constant
2. Miche (1944)
H b  Lb 0.142 tanh k b hb  , steepness limited to 1/7
3. LeMehaute & Koh (1967)
H
Hb
 0.76m 1 / 7  o
Ho
 Lo
4. Collins & Weir (1969)
Hb =  hb where   0.72  5.6m
5. Weggel (1972)
1 / 4
b
Hb =  hb where  
1 a





hb
gT 2
Hb
T2
ba

a  43.8 1  e 19m , b  1.56 1  e 19.5m

1
SI units
6. Battjes & Jensen (1978)
0.88
 

Hb 
tanh 
k b hb ,    a " slightly adjustable coeff. (~ 0.83)
kb
 0.88

7. Svendsen (1987)
Hb =  hb where   1.9
Sb
and
1  2S b
1 / 2

 Ho  
L
 
Sb  m   m2.30
L
 h b

 o  
8. Hansen (1990)
Hb =  hb where   1.05S 0.2 and
0.45


 Ho 
 L


S  m   m2.87
L
 h b


 o 
9. Smith & Kraus (1991)
Hb =  hb where   b  a



Ho
;
Lo
a  5 1  e 43m , b  1.12 1  e 60m

1
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.