42 



THEORY OF SEAKEEPING 



;> f 



Fig. 44 Wave spectrum in fully arisen sea with a significant range of periods between /„ and ji,. 

 Wave spectrum in non-fuUy developed state is shown schematically by dashed area, where upper 

 limit of significant periods is given by/i — Df (from Neumann, 1953) 



examples. Additional figures for different ranges of 

 wind velocity will be found in Neumann (1953) and in 

 Pier.son, Neumann and James (H). 



The value of E for the total spectrum, i.e., the value 

 indicated hy the intercept on the LH A'ertical scale of 

 these figures, characterizes a fully arisen sea. In prac- 

 tice it was found convenient to neglect the waves of x^vy 

 high and very low fre()ucncies which contribute little to 

 the total energy. Quoting from Neumann (195.'-)): "As 

 a rule, by empirical evidence, about 5% of the total E 

 value in the fvdly dex'elojied state of wind generated sea 

 can be cut off at the upper ]iart of the cnr\-es. At the 

 lower part it is 3% of the total E value. For example, 

 the total E walue at a wind speed of 30 knots is read off 

 at the ordinate in Fig. 42, E = 58.5 (feet)-. 



5% of E = 2.9 



thus, 55.6 on the E scale intersects the CCS curve at/ = 

 0.0(3 or T" = 16.7 seconds. This is the upper limit of the 

 significant range of periods in a fully de\'eloped sea with 

 a 30 knot wind. If it is wanted to determine a lower 

 limit of the significant range of periods, 3% of E = L76 

 is the ordinate \'alue which intersects the CCS curve at 

 / = 0.213, or T = 4.7 seconds. Therefore, the signifi- 

 cant range of periods is between 4.7 and 16.7 seconds . . ." 

 Table 9 shows the range of significant periods in a fully 

 arisen sea ut different wind speeds. The co-cnmulative 

 curves in Figs. 42 and 43 are intersected by lines of wind 

 duration in hours on the first and lines of fetch in nautical 

 miles on the second. The position of these lines were 

 deri^-ed from the data of I'lgs. 22 to 25. 



Quoting Neumann (1953, p. 32); "The intersection 

 points of the CCS curves with the duration or fetch lines, 

 respectively, show the limit of the development of the 

 composite wave motion at the given duration or fetch. 

 I^hysically, it means that the state of de\'elopment is 

 limited by a certain maximum amount of total energy 



which the waye motion can absorb from the wind with 

 the given conditions. The E value of the ordinate of 

 each intersection point is a practical measure of the 

 total energy accumulated in the wa\'e motion of the 

 no [t] -fully arisen state, limited either by the fetch or 

 duration. 



"Under actual conditions, both fetch and duration 

 may be limited, and the E \-alue for any given situation in 

 most cases will be different for the fetch and duration. 

 It is easily seen that in such cases the smaller of the two 

 E values has to be taken. 



"From the E \'alue, the wa^-e height characteristics 

 can be computed, as in the case of a fully arisen sea. 



"The upper limit of significant periods in not-fully 

 arisen sea is approximately determined by the 'fre- 

 ciuency of intersection', /,, that is, the frequency of the 

 intersection point between the CCS curve of a given 

 wind speed and the gi\-en fetch or duration line, re- 

 spectively. By this, theoretically, the wave spectrum is 

 cut off abruptly at a given maximum period, T, (or 

 minimum frecjuency, /;), without considering possible 

 \va\'e components with periods a little longer than T^ = 

 1/fi, which are just in the beginning stage of develop- 

 ment (Fig. 44). These wave components probably have 

 a small amplitude, and contribute so small amount of 

 enei-gy to tlie total wave energy, that they may be neg- 

 lected in most practical cases of wa\'e forecasting." 



6.23 Roll, Fischer and Walden's modification of Neu- 

 mann's spectrum. Neumann's spectrum has often been 

 referred to as theoretical but this description has also 

 often been oljjected to on the grounds that the spectrum 

 deri\'ation really represented an ingenious treatment of 

 empirical data. In particular, two crucial steps in the 

 deri\'ation appear to be intuitive: 



a) The transition from the apparent wave steepness 

 distribution to the distribution of ftpcctral component 

 waves. 



