78 



THEORY OF SEAKEEPING 



waves. Equispaced lines are drawn parallel to the direc- 

 tion of travel and points are read on these lines as if the}^ 

 were wave records made as a function of time at a fixed 

 point. The«e records are averaged and a mathematical 

 filter is constructed which depends on the number of 

 original space records, the distance between them, and 

 the point reading interval. The filter is then tuned 

 angularity through the averaged space record represented 

 by the original component space records. This method 

 has never been put into practice. 



Cartwright (1956) attempted to measure the direc- 

 tional distribution of waves in the open sea using a .ship- 

 borne wa\'e recorder on the RRS Discovery II. The in- 

 strument ga\'e a wave record which could be analyzed to 

 obtain a scalar spectrum. Directional sensitivity was 

 obtained by operating the ship so that each wave com- 

 ponent (co, 6) was subjected to a "Doppler shift." The 

 ship was steered round a regular dodecagonal circuit at a 

 constant speed of 7 knots and wa\'es were recorded for 

 about 12 min on each of the 12 courses. Cartwright 

 presented his analytical theory and samples of two sets 

 of observations. Theoreticall.y, a complete evaluation of 

 the directional spectrum E(io, d) is possible, but actually 

 the statistical variabilit_v in the spectrum estimates is too 

 great (12-min records are too short) to make such an 

 evaluation practical. The situation is simplified if the 

 spectrum contains some isolated peaks at well-separated 

 values of w and d which are nf>t lost in the short samples. 

 Then the mean directions of the dominant wave systems 

 can be determined. 



St. Denis (1957) proposed another method, based on 

 records to be obtained from several probes arranged along 

 a straight line. The row of probes is to be oriented alter- 

 nately in several directions. St. Denis' anal.ysis leads 

 to the suggested solution of a 48th-order matrix which 

 can be solved on a large electronic computing machine. 

 The proposed method has not yet been tried experi- 

 mentally. 



N. F. Barber (1954) measured the directional wave 

 spectrum by means of a row of detectors, in connection 

 with a mechanical frequency filter (a resonant pendu- 

 lum). Quoting Barber: "In the experiment, each de- 

 tector was a pair of parallel vertical copper strips, partly' 

 immersed in the sea and fed with 2 volts (mains fre- 

 quency) from a transformer. The current that passed 

 was proportional to the depth of immersion, varj'ing 

 with the waves, and a corresponding voltage signal was 

 conveyed to the analyzer. Two such instnmients were 

 hung from a pile wharf at separation of 2, 9, 18 and 27 

 feet in succession." 



The theory of the analysis was based on the correlation 

 coefficient 



R.„ = Jj^i"' '■) e^P f^'('«- + t'y)] du dv (149) 



— CO 



where x, // are the components of the distance between 

 two points at which water-surface elevations are meas- 

 ured and M, V are the projections of the wave mmiber k 

 on the X and y-a,xes. The average of expression (149) 



was to be taken o\'er a long period of time. In the ex- 

 periment a 15-min record was taken for each pair of de- 

 tectors in succession. Since a very narrow frequency 

 filter was used, equation (149) applies to a single wave 

 number, k = ko. 



Quoting further from Barber (1954): "It was desired 

 to restrict the wa\'e measurements to discrete points 

 upon a line taken as the ;i--axis. Such results can give E 

 so long as attention is restricted to waves very near some 

 frequencjr coo with corresponding wave length 2ir/ko. It 

 then appears, to a sufficient approximation, that equa- 

 tion (149) inverts to a Fourier series: 



^(A-o, e) = Eo sin d 



+ H c„ cos (unD — 6„) 



(150) 



where 6 = cos~'m/A-o, the direction of wave progress rela- 

 tive to the line of recorders, £"(^0, d)8d is the energy of 

 waves with number Ao and direction between 6 and d + 

 56, and c,„ e„ are the modulus and phase of the complex 

 correlation coefficient between wave elevations at points 

 separated by distance 7iD. The distance D should not 

 exceed tt/Ao or half a wave-length. The expression is 

 valid so long as no waves have directions outside the 

 range 0°-180°. The factor sin 9 is an approximation the 

 error of which causes an underestimate of the wave 

 energy in directions near 0° and 180°." 



"A pendulum of period near 2 sec was used to select a 

 narrow frequencj'-bantl and to produce the pictures in 

 Fig. 87. Its free motions in all azimuths had ec}ual 

 period and damping, and it was electrically driven 

 through amplifiers fed with the wave signals. The 

 motion of the pendulum was recorded optically as a spot 

 of light moA'ing o\-er a photographic plate, an exposure of 

 15 min being required for each of the pictures in Fig. 87." 



The electrical connections were such that combined 

 signals from two wa\'e detectors cau.sed a spot of light to 

 move on a straight line, ^'ariations of the amplitude 

 and direction of irregular waves of the filtered frequency 

 coo caused the exposure of an approximately elliptical 

 area. The major axis of this area was inclined at an 

 angle e/2, and the modulus C was given in terms of the 

 principal diameters A and B, 



C = (.4= - B-)/iA'- + B'-) 



(151) 



The theorj' leading to these results, the design of the 

 analyzer-pendulum, and the electi'ical network were de- 

 scribed more completely in later papers by Barber and 

 Doyle (1956) and Barber (1957). 



The foregoing outline can be completed by two addi- 

 tional quotations from Barber (1954): "Some prelimi- 

 nary experiments ha\'e been made in the Waitemata Har- 

 bor, Auckland, measuring wave direction by the correlo- 

 gram. The photographs in Fig. 87 show statistical rela- 

 tions between wind wa\'es of period near 2 sec at points 

 separated by distances of 2, 9, 18 and 27 ft along a fixed 

 line. From them may be obtained the curve in Fig. 88 

 showing how the energy is distributed among the various 

 directions of travel. The wind was about 15 knots and 

 2-sec waves were dominant; but because the fetch in the 



