R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
22
A prediction of the local kinematics is sought throughout the water column in the
immediate neighborhood of the PUV gauge at horizontal position xa . The pressure
sensor is at known elevation z P and the directional current meter at known elevation
z UV . Measured dynamic pressure traces pd bs s pdZ ti; xa , z P . are available at discrete
o
times ti , as are measured velocity component traces uabs s ua Z ti; xa , z UV . at the same
o
discrete times. The local water depth h and depth-uniform current Ua 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.
ZEq. Z3.. plus the three observational equations ZEqs. Z11. and Z12... These are the
simultaneous implicit algebraic equations:
2
f1 Z v , ka ,h . s Z v y ka Ua . y gk tanh kh s 0
cosh kZ h q z P .
h y pd bs s 0
o
f2 Z v , ka ,h . s r g
cosh kh
Z 14.
k1
cosh kZ h q z UV .
h y u1bs s 0
o
f3 Z v , ka ,h . s U1 q
Z v y ka Ua .
k
sinh kh
k2
cosh kZ h q z UV .
h y uobs s 0,
f4 Z v , ka ,h . s U2 q
Z v y ka Ua .
2
k
sinh kh
In principle, a unique local solution at each time ti can be computed. An analytical
solution does not seem feasible. A direct numerical algorithm is the NewtonRaphson
method. Writing Eq. Z14. as fiZ x j . s 0 where i, j s 1,2,3,4 and vector x Zj n. as the
solution estimate at iteration n, the correction D x Zj n. is suggested by the local Taylor
series expansion:
Z n.
E fi
fi Z xZjn . . q
D x Zj n. q . . . s 0.
Z 15.
E xj
The Jacobian E firE x j is evaluated analytically and Eq. Z15. is solved by matrix
inversion.
The predictive potential of this algorithm was evaluated for a theoretical steady wave
train of period 10 s and height 10 m, directed at qpr10 to the x-axis in 20 m of water
and an opposing current of y1 mrs. The PUV gauge is located at z P s z UV s y10 m.
The initial theoretical PUV trace was computed from Airy theory at a sampling time
interval of 0.5 s. With this trace as the measured PUV record, the local linear algorithm
predicted the local wave frequency, the local wave number components and the local
water surface elevation. Except in the immediate neighborhood of zero-crossings, where
profile curvature is minimal and Eq. Z14. is ill-conditioned, there was consistent
agreement with Airy theory to four significant figures.
A second theoretical PUV trace Zrecord `Twenty'; see Table 1. was computed from
near-exact Zglobal. Fourier wave theory ZSobey, 1989., also at a sampling time interval
of 0.5 s. With this trace as the measured PUV record, the local linear algorithm was
significantly less successful.
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