30 



THEORY OF SEAKEEPING 



0.26 



0.24 



0.22 



0.20 



0.18 



O.IG 



!o.l4 

 n 



0.12 

 0.10 

 0.08 

 0.06 

 0.0 4 

 0.0 2 



10' 



2-t m/sec , ;c ri ;e 



2 3 4 e 8 10^ 



gVu 



3 4 6 8 10^ 



Fig. 25 Height H of characteristic waves as function of wind duration / (from Neumann, 19526) 



number of pres.sure gages were iii.stalled on the .sea l)Ot- 

 tom near the shores of England and continuous record- 

 ings of tlie \va\'e-caused pressure fluctuations were ob- 

 tained. Visual examinations of these did not yield satis- 

 factory results, and, therefore, a frequency analyzer was 

 constructed (Barber, I'r.sell, Darl)v.shire and Tucker, 

 1946). Barber and Ursell (1948) pre.sented a theory for 

 tracing ^^arious Fourier components of the recorded 

 swell to their origin, often many hundreds of miles 

 away. In 1952 Darliyshire used such recordings and 

 analyses in formulating the shape of the spectrum and in 

 estimating the wave heights caused by the wind. The 

 mathematical details of the .spectral analysis will be pre- 

 sented in Section 8. At present, C[uotations from 

 Darbyshu'e and Neumann in the next two sections will 

 suffice. Several formuhitions of the spectra have been 

 developed .since 1952, and these will be described briefly 

 in the following .sections. 



6.1 Darbyshire's Wave Spectra. Quoting from Dar- 

 byshire (1952): "This paper describes an investigation 

 of the height and length of ocean wa\'es and swell in rela- 

 tion to the strength, extent and duration of the wind in 

 the generatmg area, and the subseqvient travel of the 

 swell through calm and disturbed water. The in- 

 vestigation is based (jn records of wa\'es made on the 

 North Coast of Cornwall, in the Irish Sea, and in Lough 

 Neagh. It is a practical continuation of the work of 

 Barber and Ursell (1948), who showed that the waves 

 leaving the generating area behave as a continuous 

 spectrum of component wave trains which travel inde- 



pendently with the group velocities appropriate to their 

 periods. The spectral distribution of energy in the 

 storm area is con.sidered, and the relati\'e amplitudes of 

 the different components are deduced empirically under 

 various wind conditions . . ." "A method of deriving the 

 wa\-e spectrum from a wave record is described by Bar- 

 ber, Ursell, Darbyshire and Tucker (1946)." . . ". "The 

 method of analysis . . . gives a Fourier analysis of a 20 to 

 30-minute record ; the analysis appears in the form of a 

 .series of peaks, each corresponding to a harmonic com- 

 ponent which is an exact .submultiple of the total length 

 of the record. .An example of such an analysis containing 

 wa-\^es due to a local storm and a band of swell from a 

 distant storm is shown in Ilg. 26. While it cannot be 

 implied that these discrete periodicities are actually pres- 

 ent in the sea waves, it is possible, for the diu'ation of the 

 record, to represent the pressure \'ariations at the point 

 of measurement by a combination of independent sine 

 waves with periods which are submultiplcs of the dura- 

 tion of the record and with amplitudes proportional to 

 the heights of the peaks on the spectrum, 



"The state of the sea can best be described in terms of 

 the wave energy. Assuming that for the duration of the 

 record, the wave pattern con.sists of a combination of in- 

 dependent .sine waves with periods and amplitudes cor- 

 responding to those of the peaks on the spectrum, it is 

 possible to evaluate the wave energy. Since the energy 

 per unit area of a single sine wave of height H is pgH-/8, 

 the total energy for all the wa^TS in the spectrum 

 = gp'ZH„-/8 where H„ corresponds to the height of the 



