56 



THEORY OF SEAKEEPING 



(a) Weather Ship 



N. Atlantic Ocean 

 Feb 1953 -Jan 1954 



40 

 20 



40-50 -ft 



-1 r"^— 



±D^ 



10 



— Percentage Frequency ot 

 Occurrence 

 Mean Wave Period 



(b) Perranporth Cornwall 

 Feb 1946-Jan 1947 



FMAMJJ.ASONDJ F 



, ^^30-40tt 



20LI r^ n 



NU Mt^ 



. 10 



40 

 20 



20 



20 



-- TT 



H5-30ft 



Over 20 -ft 



20 



riT-L 



■_ 10 



F MAMJ J.A SOND J 



20-25 ft 



T^.-^ 



10- 



5-20tt 



-a 

 c 

 o 



20^3 

 IO"g 



40 

 20 



l5-20ft 





10-15 tt 



i^^ 



u 40 



^ 20 



O 



c 



I0-I5ft 



- -r ^. _r n — ' 



20 



a) 



20 tr 



10 -E 



ai 



20 : 



40 

 20 



40 

 20 



5- 10 ft 



20 



0-5 tt 



h20 

 10 



20 

 10 



FMAMJJ,ASONDJF 



Fig. 63 Monthly distribution 

 shire, 



FMAMJ J, ASOWDJ 



of wave heights (from Darby- 

 1952«) 



called "apparent height," or H. While described thus 

 with respect to a record, the apparent amplitudes can be 

 judged by direct visual observations, and the periods 

 measured with a^top watch. 



Both f and H are highly variable. A number, say 

 200, of consecutive apparent periods can be measured 

 and collected in a table. The entire content of this 

 table is then divided into classes of equal intervals, for 

 instance 0.5 sec, so that the classes are from 2 to 2.5 sec, 

 2.5 to 3 sec, 3 to 3.5 sec, and so on. The number of ob- 

 servations falling into each class is plotted as ordinate 

 A'ersus f as abscissa. The actual frequency of occurrence 

 of each class may be plotted, but it is preferable to plot 

 the ratios of these numbers to the total number of read- 

 ings. These percentages are usually referred to as the 

 "frequency distribution." Examples of such diagrams 

 which are often called "histograms," are shown in Fig. 



.Experimental Histogram 



Jheoretical Rayleigh 

 Distribution 



1.4 2.8 ^^ 4.2 ^ 5.S 



Double Wave A-nplitude', In, 



Fig. 64 (From Lewis, 3-1956) 



41. Similar diagrams are prepared to show the fre- 

 quenc.y distribution of wave heights. Fig. 62 shows the 

 wave height distribution in the Atlantic Ocean based on 

 visual observations reported by ten weather ships and 

 subdivided according to wind speed ranges on Beaufort 

 scale. The last number in the legend of each subdivision 

 indicates the number of wave reports used in making the 

 plot. Fig. 63 shows the comparative distribution of 

 wave heights as measured by Darbyshire with a ship- 

 borne wave recorder in the open ocean and with a pres- 

 sure instrument off the coast of England. Frequency 

 distributions of apparent wave periods and heights in a 

 wind flume test are shown in Fig. 12. 



Consider a histogram of the wave height and designate 

 by N the total nimiber of heights recorded, by H^ the 

 heights of individual waves, and by AN/N the percent- 

 age of points in a class of interval AH. If the intervals 

 are sufficiently small and a smooth cur\'e is drawn to re- 

 place the stepped histogram, the "probability density" of 

 //, can be defined as 



p{Hc) = lira (AN/N)/ AH (105) 



AHi--0 



The integral of this curve 

 P{H,) 



Jo 



piH,) dH 



(106) 



is known as the probability of //;. It defines the per- 

 centage of the total number of observations N (assumed 

 to be very large) which will have A'alues below a specified 

 wave height H^. 



The histograms shown in the figures have an ap- 

 parently wide range of shapes and appearances. How- 

 ever, there is cogent reason for bclic\'ing that the fre- 

 ((uency distribution of a large numljer of indi\idual meas- 

 urements of periods or amplitudes of waves follows a 

 well-defined law developed in the statistical theory. 



The following parameters are first defined: 



Mean amplitude H = 



N i 



N 



1 



Variance a- 



N - 



-- E (H, - ny 

 1 1=1 



(107) 



(108) 



