8.4. WAVE REFLECTION ANALYSIS
515
0 = 52
COS[$A “
+ r)] + aRx cos[$Bi + (Tt(/ + r)]} (8.59)
1=1
with a positive value of r meaning that the sea surface elevation sample
was collected “r seconds” after the horizontal velocity sample.
After expansion and rearranging, Eqns. 8.30 and 8.59 become
OO
u(æ, M) = 52
cos(a«0 + Di sin((M)]
(8.60)
»=i
and
OO
7/(z, * ) = 52
cos(a«0 + Qi sin(o-it)]
(8.61)
where the Fourier coefficients in Eqns. 8.60 and 8.61 are given by the expressions
Q.
cos(#A) - aRx cos(flt)
(8.62)
y- = aJx sin($/J + aRx sin($fit)
(8.63)
Ai = aIx cos($yt) + aRx cos($^i)
(8.64)
Bi = ajx sin($/t) - aRx sin($K,)
(8.65)
and Ai and Bi are defined for convenience as
Ai = Pi cos(cr,r) - Qi sin( (8.66)
and
Bi — Qi cos(atr) + P, sin(aJr)
(8.67)
As before, Zt is given by Eqn. 8.33.
Equations 8.62 - 8.65 are solved for the four unknowns (a/t,
and in terms of the Fourier coefficients (C,, Di, Pi, and Qi) obtained via fastFourier transform from the two times series represented by Eqns. 8.60 -
8.61. The solution is given as
“'■=î\/^+ï) +(Bi+z)
(8- 68)
Z y \
J
\
/