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representing each ’storm’ period. Wave Scatter Diagram Sets of characteristic wave data values can be grouped and arranged in a table such as that given below based upon data from the northern North Sea. A ’storm’ here is an arbitrary time period - often of 3 or 6 hours - for which a single pair of values has been collected. The number in each cell of this table indicates the chance (on the basis of 1000 observations in this case) that a signi…cant wave height (in meters) is between the values in the left column and in the range of wave periods listed at the top and bottom of the table. Figure 5.43 shows a graph of this table. # Hsig ( m ) 4 -5 5 -6 6 -7 7 -8 8 -9 9 -10 10 -11 11- 12 12 -13 13 -14 P r ow >12 0+ 11 ,5- 12 ,0 0+ 1 11,0 -11 ,5 10,5 -11 ,0 10,0 -10 ,5 9 ,5 -10 ,0 0+ 1 0+ 2 9,0 -9 ,5 0+ 0+ 1 8,5 -9 ,0 1 1 1 1 4 8,0 -8 ,5 1 1 0+ 2 7,5 -8 ,0 0+ 1 3 1 1 6 7,0 -7 ,5 1 2 3 1 7 6,5 -7 ,0 1 5 3 1 0+ 10 6,0 -6 ,5 4 7 4 1 1 0+ 17 5,5 -6 ,0 2 5 7 3 2 19 5,0 -5 ,5 4 10 9 3 1 1 0+ 28 4,5 -5 ,0 1 11 18 3 1 1 0+ 0+ 36 4,0 -4 ,5 3 24 15 12 3 2 1 60 3,5 -4 ,0 8 29 20 9 4 1 1 72 3,0 -3 ,5 1 21 36 21 12 6 1 98 2,5 -3 ,0 2 37 37 22 7 2 1 1 109 2,0 -2 ,5 14 52 53 21 8 4 1 153 1,5 -2 ,0 1 28 49 36 21 13 6 1 1 156 1,0 -1 ,5 6 34 42 38 19 13 5 1 0+ 158 0,5 -1 ,0 1 8 15 16 11 5 1 1 0+ 58 0,0 -0 ,5 1 0+ 0+ 2 Pe riod s! 4 -5 5 -6 6 -7 7 -8 8 -9 9 -10 10 -11 11- 12 12 -13 c a lm : 1 Tota l: 1000 Wave Climate Scatter Diagram for Northern North Sea Note: 0+ in this table indicates that less than 0.5 observation in 1000 was recorded for the given cell. 5-56 CHAPTER 5. OCEAN SURFACE WAVES This scatter diagram includes a good distinction between sea and swell. As has already been explained early in this chapter, swell tends to be low and to have a relatively long period. The cluster of values for wave heights below 2 meters with periods greater than 10 seconds is typically swell in this case. A second example of a wave scatter diagram is the table below for all wave directions in the winter season in areas 8, 9, 15 and 16 of the North Atlantic Ocean, as obtained from Global Wave Statistics. W inter Data o f Areas 8 , 9 , 15 and 16 o f the North Atlant ic (G lob a l Wave Stat ist ics) T2 (s) 3.5 4 .5 5 .5 6 .5 7 .5 8 .5 9 .5 10 .5 11 .5 12 .5 13 .5 Tota l Hs (m ) 14 .5 0 0 0 0 2 30 154 362 466 370 202 1586 13 .5 0 0 0 0 3 33 145 293 322 219 101 1116 12 .5 0 0 0 0 7 72 289 539 548 345 149 1949 11 .5 0 0 0 0 17 160 585 996 931 543 217 3449 10 .5 0 0 0 1 41 363 1200 1852 1579 843 310 6189 9 .5 0 0 0 4 109 845 2485 3443 2648 1283 432 11249 8 .5 0 0 0 12 295 1996 5157 6323 4333 1882 572 20570 7 .5 0 0 0 41 818 4723 10537 11242 6755 2594 703 37413 6 .5 0 0 1 138 2273 10967 20620 18718 9665 3222 767 66371 5 .5 0 0 7 471 6187 24075 36940 27702 11969 3387 694 111432 4 .5 0 0 31 1586 15757 47072 56347 33539 11710 2731 471 169244 3 .5 0 0 148 5017 34720 74007 64809 28964 7804 1444 202 217115 2 .5 0 4 681 13441 56847 77259 45013 13962 2725 381 41 210354 1 .5 0 40 2699 23284 47839 34532 11554 2208 282 27 2 122467 0 .5 5 350 3314 8131 5858 1598 216 18 1 0 0 19491 Tota l 5 394 6881 52126 170773 277732 256051 150161 61738 19271 4863 999995 These wave scatter diagrams can be used to determine the long term probability for storms exceeding certain sea states. Each cell in this table presents the probability of occurrence of its signi…cant wave height and zero-crossing wave period range. This probability is equal to the number in this cell divided by the sum of the numbers of all cells in the table, for instance: Pr © 4 < H1=3 < 5 and 8 < T2 < 9 ª = 47072 999996 = 0:047 = 4:7% For instance, the probabilty on a storm with a signi…cant wave height between 4 and 6 meters with a zero-crossing period between 8 and 10 seconds is: Pr © 3 < H1=3 < 5 and 8 < T2 < 10 ª = 47072 + 56347 + 74007 + 64809 999996 = 0:242 = 24:2% The probabilty for storms exceeding a certain signi…cant wave height is found by adding the numbers of all cells with a signi…cant wave height larger than this certain signi…cant wave height and dividing this number by the sum of the numbers in all cells, for instance: Pr © H1=3 > 10 ª = 6189 + 3449 + 1949 + 1116 + 1586 999996 = 0:014 = 1:4% Note that the above scatter diagram is based exclusively on winter data. Such diagrams are often available on a monthly, seasonal or year basis. The data in these can be quite di¤erent; think of an area in which there is a very pronounced hurricane season, for example. Statistically, the North Sea is roughest in the winter and smoothest in summer. 5.5. WAVE PREDICTION AND CLIMATOLOGY 5-57 5.5.3 Statistics Just as with waves in an individual storm, one often wants to estimate the storm intensity (wave characteristics in this case) that one should associate with some chosen and very low chance of exceedance. To do this one will usually have to extrapolate the wave data collected as in the scatter diagrams above. Two statistical distribution can be used to do this; they are described individually here. Semi-Logarithmic Distribution It has been found empirically that storm data - such as that presented in the wave scatter diagram for the North Sea, above, behave quite nicely if the wave height is plotted versus the logarithm of the chance of exceedance. The processing of the data in that scatter diagram is very analogous to that for waves in a single storm given much earlier in this chapter. The plot resulting from the North Sea is given in …gure 5.43. The graph is pretty much a straight line for lower probabilities, that would be needed to predict extreme events by linear extrapolation. Obviously this segment of the graph can be extrapolated to whatever lower probability one wishes to choose. This can be done using the formula: log f(P (H )g = 1 a H in which a is related to the slope of the curve; see …gure 5.43. Figure 5.43: Logarthmic Distribution of Data from Northern North Sea Weibull Distribution A generalization of the exponential distribution given above - the Weibull distribution - is often used, too. In equation form: 5-58 CHAPTER 5. OCEAN SURFACE WAVES Figure 5.44: Histogram and Log-Normal and Weibull Distributions P (H) = exp ( ¡ µ H ¡ c a ¶b) (5.130) in which a is a scaling parameter (m), b is a …tting parameter (-) and c is a lower bound for H (m). Figure 5.44-a shows a histogram of wave heights. A log-normal and a Weibull …t to these data are shown in …gures 5.44-b and 5.44-c. In many cases the value of b in the Weibull equation above is close to 1. If - as is usually the case with waves - c = 0 and b is exactly 1, then the Weibull distribution reduces to the log-normal distribution given earlier. Chapter 6 RIGID BODY DYNAMICS 6.1 Introduction The dynamics of rigid bodies and ‡uid motions are governed by the combined actions of di¤erent external forces and moments as well as by the inertia of the bodies themselves. In ‡uid dynamics these forces and moments can no longer be considered as acting at a single point or at discrete points of the system. Instead, they must be regarded as distributed in a relatively smooth or a continuous manner throughout the mass of the ‡uid particles. The force and moment distributions and the kinematic description of the ‡uid motions are in fact continuous, assuming that the collection of discrete ‡uid molecules can be analyzed as a continuum. Typically, one can anticipate force mechanisms associated with the ‡uid inertia, its weight, viscous stresses and secondary e¤ects such as surface tension. In general three principal force mechanisms (inertia, gravity and viscous) exist, which can be of comparable impor- tance. With very few exceptions, it is not possible to analyze such complicated situations exactly - either theoretically or experimentally. It is often impossible