8,4. WAVE REFLECTION ANALYSIS
517
Co-Located Velocities
Hughes (1993) also presented a frequency domain analysis procedure to
resolve incident and reflected irregular wave spectra using synoptic time
series of horizontal and vertical velocity collected at the same point in the
water column. Just as in the case of a vertical sensor array, this method
uses linear wave theory, and it assumes normal incidence and reflection of
nonbreaking waves. Details of this method are presented below.
The time series of horizontal velocity is represented by Eqn. 8.30, and
a time lag, r, is added to Eqn. 8.31 to correct for systems in which the
velocity channels are sequentially or alternately sampled. This gives
oo
w(x,z,t) = ^2 {aLyi sinful, - ai(t + r)] -
Y,-sin[$Bi +
+ r)]}
»=1
(8.72)
A positive value of r means that the vertical velocity sample was collected
“t seconds” after the horizontal velocity sample.
Equations 8.30 and 8.72 are expanded using trigonometric identities and
rearranged into the forms
OO
u(x, z, t) = ^2 [C« cos(o-,t) + Di sin(cr2/)]
(8.73)
i—1
and
oo
w(z, z, f) =
[G» cos(crlf) + Hi sin(orI/)]
(8.74)
t=i
where the Fourier coefficients in Eqns. 8.73 and 8.74 are given by the expressions
— = aji cos(<>/1) - ajit cos($a,)
(8.75)
Zi
— aix sin( Zi
(8.76)
—L = aji sin($z,) -
sin( (8.77)
— — —dJi cos($jt) — a/l, COS( yi
(8.78)
and Ei and Fi are defined for convenience as
Ei = Gi cos( (8.79)