74 



THEORY OF SEAKEEPING 



waA'e record. The strongest justification for computing 

 wave spectra however lies in the description of the ship 

 motions induced by the waves. This sul)ject will be dis- 

 cussed in detail in Chapter 3, Section 3. 



Before the subject of wave (and seakeeping) statistics 

 is closed, some ciualifying remarks on their validity, in 

 view of the confidence bounds discussed in Section 8.42 

 should be made. The height statistics given in (128) 

 depend only on E as given in (127). The record from 

 which E is computed may be regarded as a sum of sinu- 

 soids whose amplitudes and phases are unrelated. Each 

 sinusoid may be in turn considered as a chance sample 

 from a population whose \-ariance depends on the true 

 energy density at that particular frequency. The total 

 energy, 7?, is the sum of the energies in all of the com- 

 ponent sinusoids in the spectrum. The de\'iation of the 

 measured spectral densities from the true spectral 

 densities is given by the degrees of freedom (12(j) and by 

 Table 12. Since the spectral densities are independent 

 estimates, their deviations from the true estimates are 

 random and their addition is a much better estimate of 

 the total energy than the error associated with the in- 

 dividual spectral densities. Consequently, confidence 

 bands resulting from say 50 degrees of freedom which 

 may appear to exhibit wide \ariation in the estimate of 

 the true spectrum. Fig. 71. actually give a very good 

 estimate of the total energy in the population from which 

 the sample was drawn. The remaining statistics, (129) 

 to (132), depend on moments associated with E and ac- 

 curacies shoidd be of the same f)rder of magnitude as for 

 E. 



The proportional standard error (see the example. 

 Tucker, 1957) of the variance (total energy) of a record in 

 terms of the squares of the amplitudes in the harmonic 

 components (S,,-) is given by 



1 (I.S,, 



\ 



/■'> 



2.S'„ 



Froni this, it is seen that the proportional standard error 

 is inversely proportional to the square root of the length 

 of the record. 



8.7 The Directional Sea Spectrum. In the words of 

 Marks (1954): "The sea surface is l)elie\-ed to l>e made 

 up of a sum of an infinite number of infinitesimally high 

 sine wa\'es traveling in different directions with different 

 frequencies added together in random phase (Pierson 

 1952). This surface, at any point, has a certain amount 

 of energy associated with it, and this energy is chstributed 

 among the component waves according to frec[uency and 

 direction. It is this distribution of energy and the par- 

 ticular direction of tra\-el of each of the component wa\'es 

 which defines the two-dimensional energy spectrum 

 /?(a), 9) where w = 'Iw'T is the spectral frequency and d 

 is the direction of propagation. This is the property of 

 the ocean surface which must be measured." 



As in the case of the one-dimensional or scalar spec- 

 trum, it is desired to find an analytical representation 

 that will best define the directional ([ualities of waves. 

 The most profitable way, at this time, appears to be the 



brute-force method; that is, the directional spectrum is 

 measured at sea enough times to provide an empirical 

 evaluation of the general angular spreading characteris- 

 tics of ocean waves and this information is used to deter- 

 mine an empirical operator which may then be applied 

 to one of the spectrum formulations in Section 6. A 

 substantial amount of research has been done on the 

 method of collecting and reducing wave data to direc- 

 tional spectra and some of this work will be reported 

 here. 



Pierson (1952) was the first to try to show theoretically 

 how wa\'e records cr)uld be used to obtain information on 

 the directional properties of waves. As a simple case, 

 he suggested that a plane carrying a radar altimeter could 

 make a \'ariet.y of passes over the sea surface such that 

 the record showing the shortest waves would be indica- 

 tive of the mean wave travel. Flying very slowly and ob- 

 serving the waves visually would resolve the problem, 

 through the Doppler effect, of estimating wrong by 

 180 deg. Cartwright (1956) u,sed this principle on a 

 ship to determine mean wave direction; more will be 

 said on this later. Pier.son extended his idea of an air- 

 plane collecting wa\-e data, by flying rapidly over the 

 surface (/ ~' 0), to give a spatial definition of the sea 

 surface. He derived an elaborate transformation and 

 in\'erse transformation between the freciuency and wave- 

 number domains to show that it was possible to compute 

 the directional spectrum if one could evaluate 77 linear 

 inhomogeneous simultaneous equations with 77 un- 

 knowns. It is belie\'ed that no attempt has been made 

 to apply any data to the matrix of this set of ecjuations. 



Pierson (1952) further suggested (after a conference 

 with Tukey) that aerial-stereophotography would yield a 

 grid of points that might lend itself to determination of 

 the directional spectrum. The possible methodology 

 was not discussed, l)ut it did materialize a few years later 

 in the work of Longuet-Higgins (1957) and in Project 

 SWOP (1957). Both of the.se treatments will be dis- 

 cussed. 



One final method suggested by Pierson invol\-ed the re- 

 cording of waves from distant storms at shore stations 

 spaced several hundred miles apart. With knowledge of 

 the dispersi\'e properties of waves and the geometry of 

 the originating storm, one conceivably could extrapolate 

 back from the measured swell spectra to the two-dimen- 

 sional spectrum at the edge of the wave-generating area. 

 An investigation not imlike the one suggested by Pierson 

 here is being undertaken at the Scripps Institution of 

 Oceanography where attention is being paid to the fric- 

 tional dissipative proce.s.ses in wave dispersion with dis- 

 tance, which was considered by Pierson to be negligible. 

 This work will also be discussed later in this section. 



The embryo ideas set forth by Pierson ha\-e since been 

 prosecuted by other investigators (independently), but 

 aside from these, no novel contributions have been made 

 in measuring the directional spectrum except by N. F. 

 Barber (1957) who propo,sed no less than eleven methods 

 for such measurement. These methods, through the 

 mechanics of measurement, lend themselves best to ir- 



