R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
26
surface elevations hn s hZ tn; xa . in the neighborhood of time ti. 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 EhrEt and EhrE xa . The
temporal gradient EhrEt 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
fK
sfK q
g Dt
Z 31.
1 D Ef
ub Dub
w Dw
z/
at z s h .
swq
q
q
Et
g Dt
g
Dt
g Dt
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 EhrEt 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 ti. 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
z UV :
Ef
1 D
ub Dub
w Dw
z/
f K Z v , ka , ka xa , A j ,h . s w q
s 0 at z s h
q
q
g Dt E t
g Dt
g Dt
Ef
1 2
q Z ub q w 2 . q gh s 0 at z s h
f D Z v , ka , ka xa , A j ,h . s
Et
2
f p Z v , ka , ka xa , A j ,h . s pd y pd bs s 0 at z s z P
o
fuaZ v , ka , ka xa , A j ,h . s ua y uabs s 0 at z s z UV .
o
Z 32.
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