1. Basic Wave Theory Review
Graham Warren
Bureau of Meteorology
Australia

2. Why Forecast Waves? SOLAS Shore Protection Surf
Oil and gas exploration
17 June, 2003

3. Wave Characteristics Some simple definitions Dispersion relation
Deep water waves
Wave Spectrum
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

4. Definitions
Wind (or sea) waves - generated by the local prevailing wind
Swell waves - the regular longer period waves that were generated by the winds of distant weather systems. There may be several sets of swell waves travelling in different directions, causing a confused sea state.
Sea state is the combination of wind waves and swell.
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

5. Properties of Waves Wavelength  (metres)
Height H (=2x amplitude) (metres)
Period T (seconds)
Phase velocity c
 = c T
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

6. Total wave height Height of the wind waves = Hw
Height of Swell waves = Hsw
Total wave height = (Hw2 + Hsw2)1/2
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

7. Dispersion Dispersion is the variation of wave speed with wavelength
Define
Dispersion relation is
deep water:
shallow water:
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

8. Group Velocity
Phase velocity is the speed at which a particular phase of the wave propagates
Group velocity
Velocity at which a group of waves travel
Velocity of propagation of wave energy
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

9. Deep Water Waves Applies when depth of water >  /4 c2k2=gk
Phase Velocity : c = g/=gT /2
 = cT = gT2/2 = 1.56T2 m (T in secs)
c=1.56T (m/sec)
Group velocity: cg= gT /4 = c/2 = 0.78 T m/sec
Thus: Longer waves travel faster
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

10. The Wave Spectrum Fourier Analysis of wave trains:
Variance of the wave record is obtained by averaging the squares of the deviations of each of the wave components from the mean - gives wave spectrum (Energy spectrum)
Frequency
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

11. Wave Growth Basic concepts Manual forecasting techniques Changing Wind
Swell Forecasting
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

12. Wave Heights, Wind and Fetch
Energy from the wind is transferred to waves
Waves lose energy
Whitecapping
Interaction with sea floor etc
The greater the wind speed, the higher the waves
The longer the duration of the wind, the higher the waves
The greater the distance over which the wind blows (the FETCH) the higher the waves.
Wave height depends on a balance between energy in and energy out
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

13. Wind Wave Growth Growth usually explained by shear flow instability
Airflow sucks at crests and pushes on troughs
Rate of growth is exponential as it depends on the existing sea state and wave age
Empirical formulae have been derived from large data set
Curves developed for manual forecasting
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

14. Characteristic Height and Period of Deep Water Waves
Empirical Studies show:
t = duration of wind
X=fetch
u = wind speed
g = 9.8m/s2
Duration limited
Fetch limited
ht, hx, pt and px are dimensionless functions.
They all tend to a limit as the parameter (gt/u or gX/u2) increases to ~ 105

15. Wave Height and Period
hx()
ht()
px()
pt() 
= gt/u
or gX/u2

16. Wave Height and Period for General Conditions
Need to take the fetch and duration (time for which the wind is blowing) into account
Can use the general curves based on non-dimensional parameters
simple diagram, “complicated” calculation
OR use a more complicated set of curves
Complicated diagram, no calculation
May need to take into account varying wind conditions (changes in direction and/or speed)
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

17. Manual Wave Forecasting Diagram (Gröen and Dorrestein, 1976
Need fetch >80km
2.8m
5.8s
Fetch=25km
1.8m 4s

18. Range of Wave Heights and Periods
Wave heights can range from 0 to 2Hc
The factor of 2 relates to the maximum wave likely to be observed in a period of a few hours, not the absolute maximum possible. The value depends only weakly on the length of time.
Most waves have periods in the range 0.5Tc to 1.5Tc
Important when forecasting swell
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

19. Wave Heights with Changing Wind Conditions
Change in wind direction
If wind direction changes by < 30°, calculate waves conditions as if no change in direction has occurred
If wind direction changes by > 30°, treat existing waves as swell waves, and start calculation for new wind direction from scratch.
As a rule of thumb, swell will decrease in height by 25% over period of 12 hours
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

20. Wave Heights with Changing Wind Conditions
2. Increasing wind speed (direction change <30°)
New wind speed is V2
Take wave height at time of increase = H1
Calculate the duration required to achieve H1 given the new wind speed (=T1)
If the new speed lasts for time T2, calculate wave conditions assuming duration = T1 + T2 and speed = V2.
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

21. Example of Increasing Wind
An 8 m/s wind has blown for 6 hours, fetch 100km
The wind gradually increases to 16m/s over a 6 hour period.
Estimate Hc and Tc at the end of the period
For a quick calculation, when wind speed increase is gradual from v1 to v2 over a period, use speed = v2 – (v2-v1)/4 as the speed in the calculation.
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

22. Wave Heights with Changing Wind Conditions
3. Slackening wind speed
When wind drops below speed needed to maintain height of existing waves*, the waves turn into swell.
As a first approximation, swell height may be reduced by 25% every 12 hours.
* The minimum wind speed that will produce the existing wave height at the specified fetch

23. Here we develop some simple, first approximations
Swell Forecasting
For distant storms, regard the source of the swell as a point
For nearby storms the situation is more complicated
Questions:
When will the swell arrive?
Which wavelengths are involved?
What is the height of the swell?
Here we develop some simple, first approximations

24. Swell Length and Arrival Time
Longest wavelengths travel fastest, so they arrive first
Range of periods is T~ 0.5Tc to 1.5Tc
Other periods exist, but the energy in them is small
 = 1.56T2 m (T in secs)
Speed is T knots (T in secs)
Longest waves arrive after time:
Time ~ distance (NM)/(1.5*1.5Tc) hrs
Shortest waves take 3 times as long to arrive.
Eg: Tc=6secs, distance = 600 nm, min time = 44 hours
maximum swell length = 126m

25. Swell Height Height of swell depends on
Height of waves in source region, and extent of source region
Speed dispersion (longer waves and shorter waves have different speeds – don’t arrive together)
Angular spreading of the waves (height decreases with distance as wave energy spreads over larger areas)
Angle between wind direction and direction to storm
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

26. Angular Spreading of Swell from a Storm
Extent of storm
Wind direction in storm
Swell calculated here
Factor =0.15
Distance to storm/extent of storm
% spreading factor for energy
Take square root for swell height
Eg: Swell = 0.15 * Hc

27. Wave Measurements Visual observations Instruments for measuring waves
Buoys
Sub-surface pressure sensors
Laser
Remote sensing
Radar Altimeter
Synthetic Aperture Radar 
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

28. Visual Observations
Guide only as visual observations are not generally reliable
Observations of height tend to approximate to the significant wave height
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

29. Instruments Wave buoys Wave staff
Vertical acceleration measured – can be converted to wave height
Wave staff
Attached to platforms – wave height measured by change in resistance or capacitance of the wave staff
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

30. Instruments (2) Pressure sensors Laser
Mounted from platforms below surface – change in pressure is measure of wave height
Laser
Attached to platforms – pointing downward
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

31. Remote Sensing Waves from ERS-2 Radar Altimeter
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

32. Remote Sensing (2) Synthetic Aperture Radar
Successive radar observations made along satellite track
Optical or digital processing produces high grade imaging of the longer waves
Wave directional spectrum (with 180o ambiguity) obtained by analysis of image
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003

33. Finally….
The accuracy of any wave forecast is dependant on the accuracy of the wind forecast.
Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
17 June, 2003