Basic Wave Theory Review Graham Warren Bureau of Meteorology Australia

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 1.515 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

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

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

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

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

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

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

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

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

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