SEAWAY 



35 



Fig. 33 shows the spectra U>v t'cnir wincl speeds. The 

 upper spectrum in each case is the one given above for 

 the open ocean, and the second for the coastal region 

 (Darbyshire, 1952). The wind speeds are for "gradient 

 wind," and the "anemometer height" wind can be taken 

 on an average as 2<j of it. 



6.2 Neumann's Wave Spectrum. This section covers 

 the spectrum of waxes in a (•(inii^icx seaway as it was de- 

 rived by Neumann (1953, 1954) and descritied by Pier- 

 son, Neumann, and James (H). The material is clearly 

 presented in these readily available publications. Only a 

 brief outline will be given here in order to make clear 

 subsecjuent developments. 



Quoting from Neumann (1953): "The wind never 

 raises well defined wave trains with uniform heights, 

 periods or wave lengths. When waves are being gener- 

 ated by wind, the energy supplied from wind to waves is 

 distributed more and moi'e o\'er a certain range of wave 

 lengths with different heights as the sea grows. The 

 spectrum of ocean wa\Ts is being formed, with wa\'e 

 components ranging from ripples to large billowing 

 waves in a storm sea. Therefore, it is extremely dif- 

 ficult to describe the wind-generated wave pattern by 

 means of only two single parameters such as 'wave 

 height' and 'wa\'e period' of some kind of a fictitious 

 single wa\'e train. A general method of wave forecast- 

 ing and describing the sea must embrace more than 

 this ... At the present stage of our knowledge it seems 

 that further progress is only possible by taking the en- 

 tire wave spectrum into account with the assumption of a 

 continuous distribution of component wave trains." 



"Since the wave energy per unit area of sea surface for 

 a component wave with the period T is proportional to 

 the square of the wave height /;,-, the spectral height 

 distribution may be gi\Tn by //,.- as a function of T, 

 where A/ij.- = Ht'-AT.-^ Fig. 34 shows such a hypotheti- 

 cal wave spectrum by the dashed cur\-e. 



"Assume a continuous distribution of an infinite num- 

 ber of wave components in the complex wind generated 

 wave motion, and let all wave periods between and qj be 

 possible in the spectrum of fully arisen sea. Like for 

 spectra resulting from temperature radiation, only an 

 energy interval AfV can be specified, corresponding to a 

 prescribed range of periods AT around the average 

 period T. That is, only the ratio has a finite value, and 

 a 'wave' can be defined approximately by a spectral 

 height, Hj-, as related to Al'j. for the period interval 

 AT. (For a 'sharp wav'e,' that is, for a wave with a 

 'sharp' period T, where AT —*■ 0, also AUj.-^ and hj- — »- 

 0.) 



"In Fig. 34 the spectral i^and concentrated around the 



''The reader is referred to Darbyshire (1952, see Section 6.1) 

 for a similar definition of the "equivalent wave." Attention is 

 called to tlie fact that only wave energies or the squares of wave 

 heights are Hnearly .'^uperposable. Wave heights are nonlinearly 

 connected with energies, and therefore cannot lie superposed 

 without violating tlie princijjle of conservation of energy. For 

 example, two wave trains of amplitudes Oi and Qj have total 

 energy proportional to ar + 02'. Direct superposition of ampli- 

 tudes would correspond to an amount of energy (ai + a->)- which 

 exceeds the initially available energy by 2aia2. 



.^T^v 



T, 



T, 



Hypothetical Wave Spectrum 



Fig. 34 Hypothetical wave spectrum (from Neumann, 

 1953) 



period Tj, would correspond to a \'ery young sea, whereas 

 the .spectral band around T-, would correspond to a swell 

 if separated from the original wind generated wave 

 spectrum. 



"The spectral wa\'e energy At ',■ for the ax-erage jjcricjd 

 T m the interval AT is proportional to the sc[uare of the 

 spectral wave height in the same inter\'al. This sjjectral 

 wave height can be defined for an infinitely small spec- 

 tral band 



H "t 



inn 



A/r, 

 ~AT 

 A 7 



dh'-j. 

 ^T' 



(68) 



►0 



where hj-- is the square of the heights of indix'idual waxes 

 m the spectrum. Let the spectral energy density be=* 



Wr = dUr/dT, erg cm"- sec"' 

 ^^'ith sufficient acciu'acy for our purposes 



(69) 



ALL 



WrAT 



1 d^, 



8^ 57 



f dT, erg cm-= (70) 



can be defined as the mean energy per unit area of the 

 sea surface of waves whose periods lie between T — 

 AT/2 and T + AT/2:'' 



"The total energy in the wind generated wave pattern 

 will be given by 



U 



4-r't^'- 



"The spectral wave height not only will be a function 

 of the wave period (or wave length), but it also will de- 



-^ The symbol Wt will not be used in other sections of this 

 monograph outside of Section 6.2 on Neumann's spectrum, and 

 therefore is not inckided in the list of symbols on page 90. 



^* The reader can compare this discussion based on the assiuup- 

 tion of a continuous spectrum with the one given liy Darbyshire 

 (1952, see Section 6.1) based on the summation of Fourier com- 

 ponents residting from an harmonic analysis of a record of finite 

 length. 



