R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
20
2. Global linear analysis
Linear analyses of direction-sensing field instruments have mostly followed the
floating buoy analysis of Longuet-Higgins et al. Z1963. Zsee also Horikawa, 1988.. This
is a frequency domain analysis, which uses the entire burst sample for each kinematic
quantity measured. A typical burst sample has a duration of about 20 min, such that it
would include upwards of a hundred waves. There are essentially equivalent implemen-
tations in the time domain Ze.g., Lee and Wang, 1984..
The fundamental basis of the methodology is the Gaussian random wave model, in
which the irregular water surface is represented as the superposition of very many linear
waves of different frequencies, directions and amplitudes. In the limit, summing over all
positive frequencies v and directions u , the water surface h is represented as:
1
p
`
hZ xa , t . s
H H F Z v ,u . exp yi Z k
xa y v t . d v du ,
Z 1.
a
2p yp 0
where FZ v ,u . is the complex Fourier transform of h in radian frequency-direction
space, defined by the inverse Fourier transform:
p
`
FZ v ,u ; xa , t . s
HypH0 hZ x
, t . exp i Z ka xa y v t . d v du ,
Z 2.
a
TM
and ka are the cartesian components of the vector wave number k. ka and v are related
through the linear dispersion relationship for waves on a steady current:
2
Z v y ka Ua . s gk tanh kh ,
Z 3.
in which Ua is the local depth-uniform and steady current.
It follows directly from linear wave theory that the Fourier transforms of the dynamic
pressure pd and the horizontal velocity components ua are:
Fp Z v ,u ; xa , z , t . s K p Z v ; z . FZ v ,u ; xa , t . ,
Z 4.
ka
FuaZ v ,u ; xa , z , t . s K u Z v ; z .
FZ v ,u ; xa , t . ,
Z 5.
k
respectively, where kark is the vector Zcos u , sin u . and:
cosh k Z h q z .
cosh k Z h q z .
KpZ v ; z. sr g
and K u Z v ; z . s Z v y ka Ua .
.
cosh kh
sinh kh
Z 6.
The K p and K u transfer functions are not dependent on the observations. In practice,
the Doppler adjustment for local current is mostly omitted, a common practice being to
remove the mean level Zi.e., the current. from the velocity traces prior to analysis.
Longuet-Higgins et al. Z1963. approximated the local directional variance spectrum
EZ v ,u ., strictly Ehh Z v ,u ., as the first five terms in a Fourier series in u :
2
1
E Z v ,u . s a0 q
Z a1 cos u q b1 sin u . q Z a2 cos 2u q b2 sin 2u . q . . .
Z 7.
3
6
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