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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 y*i *Z k

Z 1.

a

2p yp 0

where *F*Z v ,u . is the complex Fourier transform of h in radian frequency-direction

space, defined by the inverse Fourier transform:

p

`

HypH0 hZ x

, *t *. exp *i *Z ka xa y v *t *. d v du ,

Z 2.

a

TM

and *k*a are the cartesian components of the vector wave number *k*. *k*a and v are related

through the linear dispersion relationship for waves on a steady current:

2

Z v y *k*a Ua . s *gk *tanh *kh *,

Z 3.

in which *U*a is the local depth-uniform and steady current.

It follows directly from linear wave theory that the Fourier transforms of the dynamic

pressure *p*d and the horizontal velocity components *u*a are:

Z 4.

Z 5.

respectively, where *k*ar*k *is the vector Zcos u , sin u . and:

cosh *k *Z h q *z *.

cosh *k *Z h q *z *.

and *K * u Z v ; *z *. s Z v y *k*a 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

2

1

Z a1 cos u q *b*1 sin u . q Z a2 cos 2u q *b*2 sin 2u . q . . .

Z 7.

3

6

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