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Basic Wave Theory Review Graham Warren Bureau of Meteorology Australia

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

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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.

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries Total wave height Height of the wind waves = H w Height of Swell waves = H sw Total wave height= (H w 2 + H sw 2 ) 1/2

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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 E

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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

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14 Characteristic Height and Period of Deep Water Waves Empirical Studies show: h t, h x, p t and p x are dimensionless functions. They all tend to a limit as the parameter (gt/u or gX/u 2 ) increases to ~ 10 5 t = duration of wind X=fetch u = wind speed g = 9.8m/s 2 Duration limitedFetch limited

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15 Wave Height and Period h x ( ) h t ( ) p x ( ) p t ( ) = gt/u or gX/u 2

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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)

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17 Manual Wave Forecasting Diagram (Gröen and Dorrestein, 1976 Need fetch >80km 2.8m 5.8s Fetch=25km 1.8m4s

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries Range of Wave Heights and Periods Wave heights can range from 0 to 2H c –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 0.5T c to 1.5T c –Important when forecasting swell

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries Wave Heights with Changing Wind Conditions 1.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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries Wave Heights with Changing Wind Conditions 2. Increasing wind speed (direction change <30°) New wind speed is V 2 Take wave height at time of increase = H 1 Calculate the duration required to achieve H 1 given the new wind speed (=T 1 ) If the new speed lasts for time T 2, calculate wave conditions assuming duration = T 1 + T 2 and speed = V 2.

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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 H c and T c 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.

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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

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23 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

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24 Swell Length and Arrival Time Longest wavelengths travel fastest, so they arrive first Range of periods is T~ 0.5T c to 1.5T c –Other periods exist, but the energy in them is small – =1.56T 2 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: T c =6secs, distance = 600 nm, min time = 44 hours maximum swell length = 126m

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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 – dont 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

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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

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

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries Instruments Wave buoys –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

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

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

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17 June, Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries 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 180 o ambiguity) obtained by analysis of image

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

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