Ocean Wave Spectra and Shtp Appltcattons 
of measuring the directions of the ocean waves and generating direc- 
tional waves in a seakeeping basin. In this section the result of pre- 
liminary work on basin-generated directional wave spectra is pre- 
sented. 
The elevation of the sea surface n is considered to bea 
stationary and homogeneous random process in time t and space 
X, respectively. In the linear theory the spectral representation of 
n is given by 
CAS ee ref f eR. x -ot) ae Z) (2.21) 
“ao 
where R(x, »x.) is the position vector, with x, and x, the two sur- 
face coordinates ; k = (k cos@, k sin 0) = (ky, ns ) is the wave number 
vector, with k = (Ic, + ee V2 and @ = me k,/k, the wave num- 
ber and wave direction, respectively ;k, andk, are respectively 
the wave-number components in the x, and x, directions ; and 
d&(k) is the random variable. oe be Ble theory the wave 
frequency, w , is given byw? = = g(k/ + Ke )V@ in deep water. 
In applications, one assumes that di(K ) satisfies the fol- 
lowing expected value condition 
aon [dé (Rk) dé(k') ] = (2.2) 
z 
where a bar denotes the complex conjugate. 
In the above expression S(k) is the directional wave spec- 
trum. S/( Kk ) dk can be interpreted as the mean-square value of 7 
arising from wave elements which lie in the infinitesimal range of 
wave number components (k, »k, + dk, ) and (k,,k, + dk, ). Knowing 
the directional wave spectrum, S( K ) , of the sea determines the 
composition of waves in all directions, 
In applications it is usually necessary to parameterize S( K ) 3 
A general representation of S( K) is given by its Fourier series. 
By decomposing the directional wave spectrum S( k; w a) ata given 
frequency w, intoa Fourier series with respect to gi ncchos ae 
Longuet-Higgins 8 was able to relate the Fourier coefficients to the 
Weer 
