R.J. Sobey, S.A. Hughes r Coastal Engineering 36 (1999) 1736
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 ua ,w. s
ZEfrE xa ,EfrE z ..
If it is further assumed that the bottom is locally horizontal, the bottom boundary
condition becomes just:
w s 0 at z s yh.
Z 24.
Nonlinear wave solutions of the form:
cosh jk Z h q z .
fZ xa , z , t . s Ua xa qA j
sin jZ ka xa y v t . ,
Z 25.
cosh jkh
j
exactly satisfy the field ZEq. Z20.. and the reduced bottom boundary condition ZEq.
Z24... The current Ua 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 jZ ka xa y v t . ,
uaZ xa , z , t . s Ua qjka A j
Z 26.
cosh jkh
j
sinh jk Z h q z .
sin jZ ka xa y v t . .
wZ xa , z , t . sjkA j
Z 27.
cosh jkh
j
Using also the irrotational Bernoulli equation, the dynamic pressure is:
Ef
1
Z ua ua q w 2 . ,
pd Z xa , z , t . s B y
Z 28.
y
Et
2
in which:
Ef
sinh jk Z h q z .
s yjv A j
cos jZ ka xa y v t . ,
Z 29.
Et
cosh jkh
j
and the Bernoulli constant is ZSobey, 1992.:
2
jkA j
1
1
z
/
Bs
Ua Ua q
.
Z 30.
cosh jkh
2
4
j
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 , and measured velocity component traces uabs s ua Z ti; xa , z UV . are available at
o
the same 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.
The unknowns in this local solution are radian frequency v , the wave number
components ka , the spatial phase ka xa , the Fourier coefficients A j and the local water
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