113.42 Lecture:Ocean WavesSpring 2005Alexandra H. TechetMIT Ocean EngineeringOcean Waves2OCEAN WAVE GENERATIONWave and Sea State• Idea of sea state is vague since it does not indicate wave period.• However it is widely used so we deal...3World Meteorological Org. Sea State Codes01234567890 (meters)0-0.10.1-0.50.5-1.251.25-2.52.5-4.04.0-6.06.0-9.09.0-14.0> 14.00 (meters)0.050.30.8751.8753.255.07.511.5> 14.0Calm (glassy)Calm (rippled)Smooth (mini-waves)SlightModerateRoughVery RoughHighVery HighHugeSea State CodeRange MeanDescriptionSignificant Wave HeightThe highest winds generally occur in the Southern Ocean, where winds over 15 meters per second (represented by red in images) are found. The strongest waves are also generally found in this region. The lowest winds(indicated by the purple in the images) are found primarily in the tropical and subtropical oceans where the wave height is also the lowest. The highest waves generally occur in the Southern Ocean, where waves over six meters in height (shown as red in images) are found. The strongest winds are also generally found in this region. The lowest waves (shown as purple in images) are found primarily in the tropical and subtropical oceans where the wind speed is also the lowest. In general, there is a high degree of correlation between wind speed and wave height.45Wind Generated Waves• Wind blows over long distance and long period time before sea state is fully developed. • When wind speed matches wave crest phase speed the phase speed is maximized. Thus the limiting frequency is dependent on the wind speed due to the dispersion relationship. //wind pUC kgωω≈==/cwindgUω≈Limiting frequency:Wave development and decay• Fetch is the distance wind must blow to achieve fully developed seas (usually given in standard miles). • For a storm with wind speed Uwthe effects of the storm can be felt a distance away, R. • The number of wave cycles between the storm and the observation location is N = R/λ.• The amplitude of the waves decay exponentially as where()tat eγ−=24222/kgγν νω==(From Landau and Lifshitz)6Typical SpectrumBased on measured spectra and theoretical results, several standard forms have been developed.Limitations on Empirical Spectra• Fetch limitations• State of development or decay• Seafloor topography• Local Currents• Effect of distant storms (swells)7Wave Spectra• Many spectra are strictly valid for FULLY DEVELOPED SEAS. • Developing seas have a broader spectral peak. Decaying seas have a narrower peak. Pierson-Moskowitz SpectrumDeveloped by offshore industry for fully developed seas in the North Atlantic generated by local winds. One parameter spectrum.Mathematical form of S+(ω) in terms of the significant wave height, H1/3. (H1/3=ζ)8Spectrum Assumptions• Deep water• North Atlantic data• Unlimited fetch• Uni-directional seas•No swellBretschneider SpectrumReplaced P-M spectrum since need for fully developed seas is too restrictive. Two parameter spectrum.Significant wave height Modal frequency2910JONSWAP SpectrumJONSWAP spectrum was developed for the limited fetchNorth Sea by the offshore industry and is used extensively.Amplitude spectrum11For wave slope spectra these two do not match as wellOchi Spectrumλ determines the width of the spectrumΓ(x) = the gamma function of xOchi spectrum is an extension of the BS spectrum, allowing to make it wider, λ small, for developing seas, or narrower, λ larger, for swell. Three parameter spectrum.12Storm and Swell• Two spectra can be superimposed to represent a local storm and a swell. storm swellDirectionality in waves• In reality, waves are three-dimensional in nature and different components travel in different directions.• Measurements of waves are difficult and thus spectra are made for “uni-directional”waves and corrected for three-dimensionality.13Correction to uni-directionalityM(µ) spreads the energy over a certain angle contained within (- π/2, π/2) from the wind directionπ π/2/2Short Term Statistics• Short term statistics are valid only over a period of time up to a few days, while a storm retains its basic features• During this period the sea is described as a stationary and ergodic random process with a spectrum S+(ω) parameterized by (ωm, ζ).• Wave spreading and swell are two additional parameters of importance. Fetch also plays an important role.14Long Term Statistics• Over the long term the sea is not stationary.• We can represent long term stats as the sum of several short term statistics by piecing together a group of storms with different durations and significant wave heights.Storm Statistics• For each storm (i) we use the significant wave height and average period to construct a spectrum and then find the short term statistics.• For structural analysis the failure level is a large quantity compared to the rms value, so we use the rate of exceeding some level ao.15Observed Wave HeightsSea conditions reported by sailors estimating the average wave height and period. It was found that this is VERY close to the significant wave height.H1/3= 1.06 Hv(meters)Hogben and Lumb (1967)T = 1.12 Tv(seconds)H1/3= 1.68 (Hv)0.75(meters)Nordenstrom (1969)T= 2.83 (Tv)0.44(seconds)Tz= 0.73 Tv(seconds)Use