34 



THEORY OF SEAKEEPING 



e 6 10 12 14 16 



Wave Period (sec) 



Fig. 33 Predicted wave spectra for various wind speeds 



for open sea and western coastal region (from Darbyshire, 



195 5*) 



cant period corresponds very closely with the period of 

 maximum amplitude of the analysis. 



To obtain an indejjendent check, the expressions for 

 significant height and period have been applied to the 

 wind and wave data listed by Sverdrup and Munk 

 (1947) and the predicted and observed wave heights and 

 periods are compared in Fig. 32. While the mean order 

 of magnitude is correct, the excessi^'e scatter of data 

 makes it cjuestionable whether the spectrum is correctly 

 expressed. 



Cox and Munk (1954) discuss the validity of Darby- 

 shire's spectrum on the basis of the spectriun of wave 

 slopes which they measured by means of the sun's 

 glitter (see Section 4.3 ) . Quoting from their work : "The 

 spectrum was obtained by frequency analysis of pressure 

 records of waves generated by local storms at Lands 

 End, England. Onlj' those st(_irms were analyzed for 

 which the fetch was sufficient to generate equilibrium 

 waves. The bottom-pressure transducer was usually 

 located at depth of 50 ft. This reciuires that for wave 

 periods less than 14 sec the usual hydrodyiiamic relation- 

 ship must be used to convert bottom pressure to surface 

 elevation, as has been done by Darbyshire. But for 

 periods less than 6 sec the bottom-pressure oscillations 

 are reduced to such a low level that this is no longer prac- 

 ticable. Apparently these short waves have been 



ignored by Darbyshire. But it will be shown that these 

 waves contribute about fi\'e sixths of the total wave 

 energy and that consequently Darbyshire's spectrum is 

 too narrow and the total energy too small. A pressure 

 recorder at 50 ft is evidently not a suitable instrument for 

 obtaining the energy spectrum." Indeed the .sample rec- 

 ord in Fig. 26 .shows no periods below G .«ec, in Fig. 28 

 below 9 sec, and in Fig. 29 (a) and (6) below 7 and 11 

 sec, respectively. 



6.11 The Darbyshire (1955) spectrum. The work de- 

 scribed in the previous section was based on the records 

 of a pressure gage installed at a depth of about 50 ft off 

 Land's End, England. In the .subsequent period 1953- 

 1955, Darbyshire collected a number of records from a 

 shipborne wave recorder (Tucker, 19526, 1956) which 

 consisted of a pressure gage at a depth of about 10 ft and 

 an integrating accelerometer for recording the heaving 

 motions of the ship. 



Quoting Darbj'shire: "It is now possible to investi- 

 gate wa\-e generation in deep water, for since February 

 1953 waves have been recorded by the ocean weather 

 ship Weather Explorer using shipborne wa\^e recorder 

 described l\v Tucker (1952). Records were taken eight 

 times daily while the ship was at sea. Most of them 

 when she was stationary. The .ship occupied, m turn, 

 the po.sitions 'India' (61° 00' N, 15° 20'W) and 'Juliet' 

 (52° 20'N, 20° 00' W). The sensitivity of the recorder is 

 nearly constant for wa\'e periods from 8 to 24s." 



"The records taken by the Weather Ex-plorer are of 7 to 

 10 min dvu'ation. . . . To avoid uncertainties about wave 

 atteiuiation, the study was confined to waves generated 

 by winds acting within about 100 miles of the ship's 

 position, and a large niunber of records were analyzed to 

 find forty-five w'hich showed no evidence of extraneous 

 swell from distant storms . . . The examples chosen 

 covered a fairly wide range, in which the fetch (the dis- 

 tance the wind acts on waves) varied from 50 to 400 

 miles. The previous paper showed that the effect of 

 fetch was not very marked after 100 miles, and this ap- 

 pears to be true for deep-sea storms also." 



The same anal.ytical method was used as before. A 

 much larger content of high-frequency waves was found 

 than before. This difference cannot be completely ex- 

 plained by the lesser depth of the pres.sure pick-up. It 

 was concluded that the wa^'es m the previous investiga- 

 tion were modified by the presence of the continental 

 shelf. 



The spectrum is now exjiressed as 



Hr = 0.0036(F"2 - 0.437) r^'-^ (65) 



The corresponding period of the highest energy content 



is 



Tn,,, = 1.64 F"- 

 "Equivalent" wave height is given as 

 H = 0.0038 Y- 



(66) 



(67) 



