25

These equations assume only that the flow is irrotational and incompressible. The

velocity potential function is defined such that the velocity components are Z *u*a ,*w*. s

ZEfrE *x*a ,EfrE *z *..

If it is further assumed that the bottom is locally horizontal, the bottom boundary

condition becomes just:

Z 24.

Nonlinear wave solutions of the form:

cosh *jk *Z h q *z *.

fZ xa , *z *, *t *. s *U*a xa qA j

sin *j*Z ka xa y v *t *. ,

Z 25.

cosh *jkh*

exactly satisfy the field ZEq. Z20.. and the reduced bottom boundary condition ZEq.

Z24... The current *U*a must be steady and depth-uniform, excluding any consideration of

velocity shear.

From the known velocity potential function, the velocity components are:

cosh *jk *Z h q *z *.

cos *j*Z ka xa y v *t *. ,

Z 26.

cosh *jkh*

sinh *jk *Z h q *z *.

sin *j*Z ka xa y v *t *. .

Z 27.

cosh *jkh*

Using also the irrotational Bernoulli equation, the dynamic pressure is:

Ef

1

Z ua ua q *w * 2 . ,

Z 28.

y

E*t*

2

in which:

Ef

sinh *jk *Z h q *z *.

s yjv *A * j

cos *j*Z ka xa y v *t *. ,

Z 29.

E*t*

cosh *jkh*

and the Bernoulli constant is ZSobey, 1992.:

2

1

1

z

/

.

Z 30.

cosh *jkh*

2

4

A prediction of the local kinematics is sought throughout the water column in the

immediate neighborhood of the PUV gauge at horizontal position *x*a . The pressure

sensor is at known elevation *z * P and the directional current meter at known elevation

o

times *t*i , and measured velocity component traces *u*abs s *u*a Z *t*i; *x*a , *z * UV . are available at

o

the same discrete times. The local water depth *h *and depth-uniform current *U*a are also

known. The current can be estimated from the UV traces as a time-average over

sufficient time to average over the local waves but not over local astronomical and storm

tide currents.

The unknowns in this local solution are radian frequency v , the wave number

components *k*a , the spatial phase *k*a xa , the Fourier coefficients *A * j and the local water

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