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

pixel array known in real-world coordinates, timestacks are analyzed in terms of

an estimated root-mean-square average wave number *k*rms( *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 *k*y( *f *) and *k*x ( *f *) by

linear combination of normalized quadspectra:

(2)

where

system [*xp, yp*] and [*xq, yq*]

Least-squares fit solutions of wave number coefficients αpq are obtained by

singular value decomposition of

( *x * p - *x*q ) n-*m *( *y * p - *y * q ) m

∑∑α

=1

(3)

(*n *- *m*)!*m*!

for *n *= 2, *m *= 0 and *n *= 2, *m *= 1, and

( *x * p - *x*q ) n-*m *( *y * p - *y * q ) m

∑∑α

=0

(4)

(*n *- *m*)!*m*!

for *n *and *m *= other and *i*n = -*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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