7h 



THEORY OF SEAKEEPING 



60° 



Range of X 



Curves of 

 Constant 



Fig. 85. Energy spectrum for a sea (idealized) (from 

 St. Denis and Pierson, 1953) 



of Jines parallel to the x and j/-axes of a spatial picture of 

 the ocean. This method is essentially that derived 

 earlier and used in the observed sea spectrum detemiina- 

 tion by Chase, et al (1957) which will now be discussed. 



The first attempt to measure directional spectra at 

 sea is known as project SWOP, the complete designation 

 of which is Stereo Wave Observation Project (Chase, et 

 al, 1957). 



Several pairs of aerial stereophotographs of an open- 

 sea surface, were made from two airplanes flying upwind 

 in tandem 2000 ft apart at an altitude of 3000 ft. The 

 cameras in the two airplanes were simultaneously trig- 

 gered by a radio link. In all, each camera had taken 100 

 pictures. The scale of distances was provided by the 

 RV Atlantis, which towed a raft at a distance of 500 ft 

 from it, approximately in the middle of the photographed 

 area. RV Atlantis also took wave records by means of a 

 wave pole. 



After examining all photographs, two overlapping pairs 

 were chosen for analysis with rectangular usable areas of 

 1800 X 2700 ft. This area was subdivided into inter- 

 vals of 30 ft, giving a 60 X 90 grid. The grid intersec- 

 tion points were used both for stereophotogrammetric 

 measurements and for subsecjuent spectral analyses. 

 The use of the same points for both purposes was impor- 

 tant since the point height accuracy was estimated at 

 ±0.5 ft while the contours drawm at the 3-ft intervals 

 were accurate to ±2.0 ft. 



The spectral analyses were made in terms of wave- 

 number projections u, v, on the x and ?/-axes, as shown by 



eciuations (144) and (145). These equations represent, 

 of course, only the mathematical foundation and were 

 converted to summations of finite intervals for actual 

 use. Twenty lags were used in the analysis for each of 

 two stereo-pairs. 



A good deal of difficulty was experienced, because the 

 first analj'sis indicated a tilted sea surface. A large 

 amount of corrective calculations was thereby made 

 necessary. The warpage of the negatives also caused 

 difficulties. The completed spectrum, E{u, v), was con- 

 verted to an E{oi, 6)-spectrum and was integrated with 

 respect to 8 in order to obtain a scalar spectrum for com- 

 parison with the wa^'e-pole data and with the corre- 

 sponding Neumann spectrum. The final expression for 

 the directional operator is gi\'en by 



0.25(1 - 7) + (0.50 



0.40 7) cos^e -I- 1.288 cos^e 



(146) 



where 



7 = exp[- {u^U/gy/2] 



Quoting from Pierson (Chase, et al, 1957, p. 240): 

 "If CO is small, the angular term in (146) becomes 



0.04 cos- e + 1.28 cos^ d 



(147) 



which shows that the spectrum is more peaked at low 

 frequencies than has been assumed in (136). Con- 

 versely if oj is large, the angular term in (146) becomes 



0.25 -f 0.50 co; ^ d 



(148) 



which shows that the spectnmi is more e^'enlj^ spread out 

 at high frequencies than has been assumed previously." 

 Fig. 85 shows an idealized directional spectrum. Fig. 86 

 shows the actual directional spectrum obtained by the 

 SWOP measurements. 



The SWOP report gives valuable data on one particu- 

 lar set of sea measurements. It is the only instance to 

 date in which the directional spectrum was measured in 

 an open sea. The organization of the measuring expedi- 

 tion involved a ship and two airplanes; joint efforts of 

 seA'eral organizations and of many persons were involved. 

 Up to the time of writing the final report, 33 months' 

 time was spent in planning, data collecting and data re- 

 duction, and presimiably, the enterprise was very ex- 

 pensive. There is little question that the methods of 

 SWOP leave much to be desired from the economics 

 (time, effort, money) point of \'iew. It is hoped that 

 the methods of Cartwright and of Barber, which are far 

 simpler in nature, can be made to 3'ield more information 

 on the directional parameter of a mnd-generated sea. 



8.72 Miscellaneous Investigations of the directional 

 spectrum. Alarks (1954) outlined a mathematical 

 method for evaluating the directional spectrum from 

 stereo-movies of the sea surface. A fast, low-flying air- 

 craft, photographs the sea surface with a stereopho to- 

 graphic movie camera which effectively records the three- 

 dimensional wave form of a narrow path through the 



