8.4. WAVE REFLECTION ANALYSIS
509
and
oo
A2, cos o'it + B2, sin (Tit
(8.38'
1=1
where the Fourier coefficients in Eqns. 8.37 and 8.38 are given by the expressions
-Ai, = a/, cos
+ aRi cos (8.39)
Bi, = aIt sin 4>y, - aR, sin
(8.40)
A2, = aj, cos(fctAl + /,) + aR, cos(fctA^ + $Rt)
(8.41)
B2i = aIt sin(^A^ + (8-42)
with
= (kiXi + €/,)
and
= (fc.-xi + efl>)
(8-43)
Equations 8.39 - 8.42 represent a system of four equations with four
unknowns (a/,, aR,,
and
for each frequency component. The
unknowns are expressed in terms of the Fourier coefficients (Xi,, Bi,, A2,,
and B2,) that can be determined by Fourier analysis of the two measured
sea surface elevation time series represented by Eqns. 8.37 - 8.38. Solving
Eqns. 8.39 - 8.42 results in the following set of equations:
aL ~
1rm [(^2, - Aït cos^A^- Bi, sinfctA/’)2 +
2| sin ki A£|
+ (B2, — Bi, cos kiA£ + Xi, sin kiA^)2]
(8.44)
aRi ~
a
[(A2, - Ai, cos^Al+Bb sinfc.Al)2 +
2|sinK,A£|
+ (B2, - Bi.cos^A^-Xi.sin^A^2]172
(8.45)
$/, = tan 1
~(A2, - >1i, cos fc,-Al - Bi, sin kjM) _ Xi + C/j (8.46)
(B2, - Bi, cos kiA£ + Ai, sin ktJ