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in 1967 and the 12th International Towing Tank Conference in 1969 as a standard for seakeeping calculations and model experiments. This is reason why this spectrum is also called the ISSC Wave Spectrum or the ITTC Wave Spectrum. The original One-Parameter Pierson-Moskowitz Wave Spectrum for fully developed seas can be obtained by using a …xed relation between the signi…cant wave height and the average wave period in the Bretschneider de…nition: T1 = 3:86 ¢ p H1=3 and T2 = 3:56 ¢ p H1=3 (5.125) In reality a measured spectral form di¤ers from theoretical formulations which give only a mean distribution. Figure 5.35 shows a comparison between a measured wave spectrum and the corresponding Bretschneider (Pierson-Moskowitz) wave spectrum during a storm in the Atlantic Ocean on 4 February 1979. JONSWAP Wave Spectra In 1968 and 1969 an extensive wave measurement program, known as the Joint North Sea Wave Project (JONSWAP) was carried out along a line extending over 100 miles into the North Sea from Sylt Island. Analysis of the data - in which the TU Delft participated - yielded a spectral formulation for fetch-limited (or coastal) wind generated seas. The following de…nition of a Mean JONSWAP wave spectrum is advised by the 17th ITTC in 1984 for fetch limited situations: S³(!) = 320 ¢H21=3 T 4p ¢ !¡5 ¢ exp ½¡1950 T 4p ¢ !¡4 ¾ ¢ °A (5.126) 5.4. WAVE ENERGY SPECTRA 5-45 0 1 2 3 4 0 0.25 0.50 0.75 1.00 1.25 Measured wave spectrum Bretschneider wave spectrum Wave Frequency (rad/s) S pe ct ra l D en si ty (m 2 s ) Figure 5.35: Comparison of a Measured and a Normalised Wave Spectrum with: ° = 3:3 (peakedness factor) A = exp 8<:¡ Ã ! !p ¡ 1 ¾ p 2 !29=; !p = 2¼ Tp (circular frequency at spectral peak) ¾ = a step function of !: if ! < !p then: ¾ = 0:07 if ! > !p then: ¾ = 0:09 Taking °A = 1.522 results in the formulation of the Bretschneider wave spectrum with the peak period Tp. For non-truncated mathematically de…ned JONSWAP spectra, the theoretical relations between the characteristic periods are listed below: T1 = 1:073 ¢ T2 = 0:834 ¢ Tp 0:932 ¢ T1 = T2 = 0:777 ¢ Tp 1:199 ¢ T1 = 1:287 ¢ T2 = Tp (5.127) Sometimes, a third free parameter is introduced in the JONSWAP wave spectrum by varying the peakedness factor °. Wave Spectra Comparison Figure 5.36 compares the Bretschneider and mean JONSWAP wave spectra for three sea states with a signi…cant wave height, H1=3, of 4 meters and peak periods, Tp, of 6, 8 and 5-46 CHAPTER 5. OCEAN SURFACE WAVES 10 seconds, respectively. 0 2 4 6 0 0.5 1.0 1.5 2.0 JONSWAP spectrum Bretschneider spectrum H1/3 = 4.0 m Tp = 10 sec Tp = 8 sec Tp = 6 sec Wave Frequency (rad/s) S pe ct ra l D en si ty (m 2 s ) Figure 5.36: Comparison of Two Spectral Formulations The …gure shows the more pronounced peak of the JONSWAP spectrum. Directional Spreading A cosine-squared rule is often used to introduce directional spreading to the wave energy. When this is done, the unidirectional wave energy found in the previous sections is scaled as in the following formula: S³(!; ¹) = ½ 2 ¼ ¢ cos2 (¹¡ ¹) ¾ ¢ S³ (!) (5.128) with: ¡¼ 2 · (¹¡ ¹) · +¼ 2 in which ¹ is the dominant wave direction. A comparison of a measured wave directionality with this directional spreading is given in …gure 5.37. Because the directionality function in this theory is a scalar, the form of the spectrum along each direction is the same; only its intensity varies as a function of direction. At sea, this distribution depends on the local weather situation at that moment (for the wind waves or sea) as well as on the weather in the whole ocean in the recent past (for any swell component). Deviations from the theoretical distribution will certainly appear when, for instance, when sea and swell travel in quite di¤erent directions. 5.4.4 Transformation to Time Series Especially when solving non-linear problems - such as in chapters 9 or 12, for example - one often needs a deterministic time record of water levels which has the statistical properties 5.4. WAVE ENERGY SPECTRA 5-47 Figure 5.37: Wave Spectra with Directional Spreading one would associate with a known spectrum. This is often referred to as the inverse problem in wave statistics. (It is at moments such as this that one is sorry that the phase information from the Fourier series analysis - which yielded the spectrum in the …rst place - has been thrown away.) Luckily, there is no real need to reproduce the input time record exactly; what is needed is a record which is only statistically indistinguishable from the original signal. This means that it must have the same energy density spectrum as the original. This is done by …lling in all the necessary constants in equation 5.107 which is repeated here for convenience. ³(t) = NX n=1 ³an cos(knx¡ !nt+ "n) The desired output - whether it is the water surface elevation as in this equation or anything else such as the horizontal component of the water particle velocity 17 meters below the sea surface - depends upon three variables for each chosen frequency component, !n: kn, "n and an amplitude such as ³an . Successive!n values are generally chosen at equally spaced intervals¢! along the frequency axis of the given spectrum. Note that with a constant frequency interval, ¢!, this time history repeats itself after 2¼=¢! seconds. The amplitudes, ³an , can be determined knowing that the area under the associated seg- ment of the spectrum, S³ (!) ¢¢! is equal to the variance of the wave component. Equation 5.109 is adapted a bit for this so that:¯¯¯¯ ³an = 2 q S³ (!) ¢¢! ¯¯¯¯ (5.129) The wave numbers, kn, can be computed from the chosen frequency !n using a dispersion relationship. 5-48 CHAPTER 5. OCEAN SURFACE WAVES When obtaining the wave spectrum S³(!) from the irregular wave history, the phase an- gles "n have been thrown away. New "n-values must be selected from a set of uniformly distributed random numbers in the range 0 · "n < 2¼: While "n is needed to generate the time record - and they may not all be set equal to zero!- the exact (randomly selected) "n do not in‡uence the record’s statistics. Looked at another way: By choosing a new random set of "n values, one can generate a new, statistically identical but in detail di¤erent time record. This procedure is illustrated by extending …gure 5.32 as shown below in …gure 5.38. Note that the original (on the left in the …gure) and the newly obtained wave history (on the right hand part of the …gure) di¤er because di¤erent phase angles have been used. However, they contain an equal amount of energy and are statistically identical. Figure 5.38: Wave Record Analysis and Regeneration Directional spreading can be introduced as well by breaking each frequency component up into a number of directional components. One should not underestimate the (extra) computational e¤ort however. 5.5 Wave Prediction and Climatology In 1805, the British Admiral Sir Francis Beaufort devised an observation scale for measuring winds at sea. His scale measures winds by observing their e¤ects on sailing ships and waves. Beaufort’s scale was later adapted for use on land and is still used today by many weather stations. A de…nition of this Beaufort wind force scale is given in …gure 5.39. The pictures in …gure 5.40 give a visual impression of the sea states in relation to Beaufort’s scale. Storm warnings are usually issued for winds stronger than Beaufort force 6. 5.5. WAVE PREDICTION AND CLIMATOLOGY 5-49 Figure 5.39: Beaufort’s Wind Force Scale 5-50 CHAPTER 5. OCEAN SURFACE WAVES Figure 5.40: Sea State in Relation to Beaufort Wind Force Scale 5.5. WAVE PREDICTION AND CLIMATOLOGY 5-51 5.5.1