566 



numbers to about 0° for the lowest wave numbers (see 

 also Figure 17) . The energy of the higher wave num- 

 bers travels more or less in the wind direction but 

 the main direction of the peak of the spectrum ap- 

 pears to be about 10° relative to true North; that 

 is about 60° from the wind direction and almost 

 parallel to the coast. This seems to be the most 

 remarkable feature of this spectrum as one would 

 expect to find a uniform main direction of 70°, con- 

 sidering the wind direction and the effects of non- 

 linear interactions [Hasselmann et al. (1976)]. 

 Again, as with the spectrum off Sylt, it is felt 

 that the observed phenomenon is due to the asym- 

 metry of the coastline around the wind direction. 



To substantiate this preliminary conclusion 

 qualitatively, a simplified "hindcasting" model was 

 implemented for homogeneous, stationary wind fields, 

 arbitrary coastlines, and deep water. In this model, 

 which is basically the same as suggested by Seymour 

 (1977) , the wave components from different direc- 

 tions are decoupled. In this version the parameter 

 relationships from JONSWAP [Hasselmann et al. (1973)] 

 were taken and the suggestions of Mitsuyasu et al. 

 (1975) were used for the directional distribution 

 function. When applied to the situation of the 

 first and third sortie it did produce two-dimensional 

 f,e-spectra which at least qualitatively agreed with 

 the so far unexpected main directions in the observed 

 k-spectra. 



This seems to be .in contradiction with the con- 

 clusions of Hasselmann et al. (1976) that the shape 

 of the spectrum is fairly insensitive to variations 

 in the wind field due to the non-linear interactions 

 in the spectrum. It should be noted however that 

 the distance to the coast, in terms of wave lengths, 

 seems to be rather short for the lower wave numbers 

 in the two spectra so that non-linear interactions 

 may not have been sufficiently effective to over- 



e„ 



Noordwijk 760323 



frequency spectrum 

 from stereo doto 



FIGURE 17. The mean direction of the waves relative 

 to true North, as function of wave number. 



come the influence of the geometry of the coastline. 

 For the higher wave numbers the distance to shore 

 is relatively long and the non-linear interactions 

 may have produced the observed directional distribu- 

 tion functions which indeed seem to be hardly af- 

 fected by the asymmetry of the coastline. The ob- 

 servations therefore may still be consistent with 

 the theory of non-linear interactions and the con- 

 clusions of Hasselmann et al . (1976) if the relevant 

 space and time scales are considered. 



In an "ideal" generation case the directional 

 distribution of the wave energy is often approximated 

 with a simple unimodal function. The observed situa- 

 tions are distinctly multi-modal, but one such func- 

 tion, given in Eq. 14, has been fitted to the data. 

 This was done mainly to compare the results with the 

 piiblished data. 



D(e) 



1 r(s + 1) 



2/Tr r(s + 



COS 2s (- 



(14) 



In this expression s is the spreading parameter and 

 9ni is the mean direction, both of which may vary 

 with k. The values of 0^^^ and s have been computed 

 using a least-squares technique. The results for 

 6in as a function of wave number are given in Figure 

 15. Noise in the spectra (see Appendix) did influ- 

 ence these results and outliers had to be identified. 

 As a criterion for acceptation, the rate of change 



along the wave number axis has been chosen. 



of 



An accepted value of 9 should be within 30° of its 

 neighboring values on the wave number axis.* This 

 is equivalent to a rate of change of approximately 

 0.0024 m for the first sortie, 0.0033 m for the 

 second sortie, and 0.0031 m for the third sortie. 

 This allows for slow but significant variations in 

 8ni which is required, for instance, in the spectrum 

 of the third sortie. The resulting set of accepted 



values of 9 is also indicated in Figure 17. 



The 



030 0,40 



frequency (Hz) 



values of s at the corresponding values of the wave 

 number have been plotted in Figure 18 in a format 



FIGURE 16. Spectrum inferred from stereo data of ob- 

 servation off Noordwijk, March 23rd, 1976. 



The value of 30° was chosen arbitrarily. 



