Timestacks are analyzed to provide estimates of vector-mean wave direction
θm( f ). Directions are derived from the phase difference between the signals
measured at one location relative to the signal at another for all array elements.
Figure 35 shows a visual example of the phase structure observed by the
alongshore elements of a pixel array during a random wave test for a duration of
10 sec. Focusing on the record between 3 and 4 sec, the slight temporal offset of
the low intensity between sampling locations is indicative of a wave propagating
obliquely through the array. With the separation distance between elements of a
an estimated root-mean-square average wave number krms( f ) following Herbers,
Elgar, and Guza (1995). This method is based on an expansion of the theoretical
cross-spectrum of gravity waves for small sensor separations relative to
wavelength. The computed wave number moments are extended to calculations
of alongshore and cross-shore components of wave number ky( f ) and kx ( f ) by
linear combination of normalized quadspectra:
Q pq ( f )
N
N
k x = ∑∑ α pq
(2)
Ppp ( f ) Pqq ( f )
p=1 q=1
where
Qpq = quad-spectrum of pixel sensor pair locations in 3-D coordinate
system [xp, yp] and [xq, yq]
N = number of array elements
Ppp and Pqq = autospectra for pixels sensors located at [xp, yp] and [xq, yq]
Least-squares fit solutions of wave number coefficients αpq are obtained by
singular value decomposition of
( x p - xq ) n-m ( y p - y q ) m
N
N
∑∑α
=1
n
i
(3)
pq
(n - m)!m!
p=1 q=1
for n = 2, m = 0 and n = 2, m = 1, and
( x p - xq ) n-m ( y p - y q ) m
N
N
∑∑α
=0
in
(4)
pq
(n - m)!m!
p=1 q=1
for n and m = other and in = -i . The number of terms in the expansion was
constrained to 8, and the truncation value for the smallest eigenvalue (relative to
the largest eigenvalue) was 10-4.
48
Chapter 5 Video-Based Wave Direction Measurement
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