26

surface elevations hn s hZ *t*n; *x*a . in the neighborhood of time *t*i. With *J *Fourier

coefficients, *N *local water surface elevations, and treating the spatial phase as a single

unknown, there are 4 q *J *q *N *unknowns.

Active equations are provided by the PUV observations and by the kinematic and

dynamic free surface boundary conditions. Note that the primitive form of the kinematic

free surface boundary condition, Eq. Z21., requires estimates of EhrE*t *and EhrE *x*a . The

temporal gradient EhrE*t *might be estimated from interpolation among the local hn

which is part of the solution, but there is no spatial information from the PUV traces.

Sobey Z1992. estimated these spatial gradients by imposing a locally steady approxima-

tion on the waves.

This additional approximation can be avoided. The problem formulation is not

compromised by redefinition ZLonguet-Higgins, 1962. of the kinematic free surface

boundary condition as:

1 *Df * D

X

s*f*K q

Z 31.

1 *D * Ef

z/

at *z *s h .

s*w*q

q

q

E*t*

This modified form of the kinematic free surface boundary condition excludes both

temporal and spatial gradients of h.

As an initial evaluation of this form of the kinematic free surface boundary condition,

it was substituted in the Sobey Z1992. code for local irregular wave kinematics from a

single water surface trace Zlocal Fourier irregular; LFI-E.. For theoretical traces where

the locally steady approximation was exact, the results were visually identical. For

measured traces, the results were similar. Without the need to estimate EhrE*t *from

measured water surface elevations, the local numerical solutions appeared to be much

more robust. The Eq. Z31. form of the kinematic free surface boundary condition was

adopted in the present PUV analysis.

At any time *t *where there is an assigned water surface node, active theoretical

equations are provided by both free surface boundary conditions. In addition, there are

PUV observational equations at each discrete measurement time *t*i. Each of these

equations are implicit and algebraic. Available equations are the modified KFSBC at h,

the DFSBC at h, a pressure equation at *z * P and two horizontal velocity equations at

Ef

1* D*

z/

s 0 at *z *s h

q

q

Ef

1 2

q Z ub q *w * 2 . q *g*h s 0 at *z *s h

E*t*

2

o

o

Z 32.

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