THEORY OF SEAKEEPING 



p'-t 



where tj- is the mean square surface displacement, p- the 

 mean sciuare turbulent pressure on water surface, t the 

 elapsed time, Lc the convection speed of the surface 

 pressure fluctuations, and p the water density. ..." 



"We are now in a position to see rather more clearly 

 the probable reas(jn for the failure of Eckart's theory to 

 predict the magnitude of the wave height generated b.y 

 the wind. His less precise specification of the pressure 

 distribution has 'smoothed oft' the resonance peak of 

 the response of the water surface, and it is the wave 

 numbers near the peak that can contribute largely to the 

 wave spectrum at large durations." 



Application of ecjuation (36) reciuires knowledge of the 

 mean pressure fluctuation p-. Quantitative data on the 

 turbulence in the boundary layer of the wind at the sea 

 surface are meager and uncertain. How'ever, Phillips 

 used certain plau.sible data and evaluated p^ and iq- as 

 functions of the elapsed time. He was thus able to 

 demonstrate excellent agreement of wave-height gro^\th 

 versus time with the data of Sverdrup and JMunk (.see 

 Section 5.1). The author believes, however, that this 

 comparison is premature and has little meaning, .since 

 dissipation of the energy in waves has not been con- 

 sidered. It is evident that I'hillips made a major con- 

 tribution to the subject of wave generation bj^ wind. 

 He has abl.v treated, however, only one facet of the 

 problem. This mu.st be combined with other aspects 

 (wave-correlated pressures, energj^ dissipation) before a 

 comparison with obser\-ed waves can be meaningful. 

 Phillips' results may be directly applicable to the initial 

 formation of small ripples, at which time the energy 

 dissipation depends on the molecular visco.sity and is 

 small, and the drag coefficient Ca and therefore the 

 wave-correlated pressures are also small. The applica- 

 tion of Groen and Dorre.stein's (1950) and Bowden's 

 (1950) results .showing that energy dissipation grows 

 with wave height and length may limit the indicated 

 wave growi;h and eliminate the need of uncertain ref- 

 erence to nonlinearities. 



In his 1958 work, Phillips defined ])y dimensional 

 reasoning the theoretical shape of the high-frecjuency 

 end of a wave spectrum. This definition was based on 

 the observed occurrence of sharp-crested waves, the 

 physical definition of a sharp crest by the vertical water 

 acceleration ij = —g, and the mathematical-statistical 

 definition of a discontinuous function expressing the 

 water surface elevation. Phillips found the spectrum '- 

 to be 



E{co) = ag~ 0)-= (36a) 



where a is a constant, g the acceleration of gravity and 

 CO the circular frequency. 



4.3 Statistics of the Sea Surface Derived from Sun 

 Glitter (Cox and Munk 1954a, b). This work, describ- 



'^ The reader is referred to Sections 6 and 8 for the discussion 

 of wave spectra. 



ing the method and the results of observations at sea 

 based on the statistical theory and outlining certain im- 

 portant relationships of this theory, .serves as one of two 

 basic con.stituent parts of the work of Munk (1955), to be 

 discussed in the next section. 



The following resume is abstracted from Cox and 

 JNIunk (1954a) : If the sea surface were absolutely calm, 

 a single mirror-like reflection of the sun would be seen 

 at the horizontal specular point. In the usual case there 

 are thousands of "dancing" highlights. At each high- 

 light there must be a water facet, possibly quite small, 

 which is so inclined as to reflect an incoming ray from 

 the sun towards the observer. The farther the facet is 

 from the horizontal specular point, the larger must be its 

 slope in order to reflect the sun's rays back to the ob- 

 server. The distribution of the glitter pattern is there- 

 fore closely related to the distribution of .surface slopes. 



In order to exploit this relationship plans were laid in 

 1951 for co-ordination of aerial photographs of glitter 

 from a B-17G plane with meteorological measurements 

 from a 58-ft schooner, the Reverie. One of the objects 

 of this investigation was a study of the effect of surface 

 slicks. In the methods adopted oil was pumped on the 

 water, . . . With 200 gal of this mixture, a coherent 

 slick 2000 by 2000 ft could be laid in 25 min, provided 

 the wind did not exceed 20 mph. Two pairs cf aerial 

 cameras, mounted in the plane, were wired for S3'n- 

 chronous exposure. Each pair consisted of one vertical 

 and oiTC tilted camera with some overlap in their fields of 

 view. One pair gave ordinary image photographs for 

 the purpose of locating cloud .shadows, slicks, and ves- 

 sels; this pair also gave the position of the horizon and 

 the plane's shadow (to correct for the roll, pitch, and yaw 

 of the plane). The other pair of cameras, with lenses 

 removed, provided photogrammetric photographs. 



The method consists e.s.sentially of two phases. The 

 first identifies, from geometric considerations, a point on 

 the sea surface (as it appears on the photograph) with the 

 particular slope rec|uired at this point for the reflection of 

 sunlight into the camera. This is done by suitable grid 

 overlays. Lines of constant a (radial) give the azimuth 

 of ascent to the right of the sun; lines of constant /3 

 (closed or circumferential curves) give the tilt in degrees. 



The second phase interprets the average brightness of 

 the sea surface (darkening on the photometric negative) 

 at various a-0 intersections in terms of the frequency 

 with which this particular slope occurs. On the density 

 photographs the glitter pattern appears as a round blob 

 with a bright core (on the po.sitive print) and a gradually 

 diminishing intensity to the outside. The density of the 

 blob on the negative is then measured with a densitom- 

 eter at points which correspond to the intersection of 

 appropriate grid lines. 



The results are expressed as the mean of squares of 

 wave slopes in up-down wind direction a J, and in cross 

 wind direction o-/. The data on the observed waves and 

 on measured mean squares of slopes are given in Table 

 5. This table represents an abstract of data from Cox 

 and ]Munk (1954a), Table 1, with columns of X/H and 



