8.4. WAVE REFLECTION ANALYSIS
513
7/(x,Z) =
+
where the incident wave train is given by
(8.50)
oo
0 = $2 ab c°s($A - a{t)
(8.51)
:=i
and the reflected wave train is
OO
t]r(x, t) = 52 Ofii cos($fi, 4- ait)
(8.52)
ï=i
with incident and reflected phase angles given by Eqn. 8.32.
The measured horizontal velocity aligned with the direction of wave
travel can be represented by Eqn. 8.30. However, if the analysis is limited
to just long-wave frequencies, the velocity transfer function (ZJ given by
Eqn. 8.33 simplifies to a constant, i.e.,
(8.53)
and the horizontal velocity time series becomes
I— f OO
00
y 152 aL cos($a -
- 52aR' cos($ * . +
(8.54)
or in terms of the definitions given by Eqns. 8.51 and 8.52
(8.55)
Equations 8.50 and 8.55 are two equations with two unknowns, rj/(æ,0
and 7]R(x,t). Solving the equations yields
*?/(<) = | 7?(/) +
(8.56)
and
VR(t) = | î/(0~
(8.57)
(Note that the z-dependence has been dropped because both measurements
are made at the same z-position.)
513
7/(x,Z) =
+
where the incident wave train is given by
(8.50)
oo
0 = $2 ab c°s($A - a{t)
(8.51)
:=i
and the reflected wave train is
OO
t]r(x, t) = 52 Ofii cos($fi, 4- ait)
(8.52)
ï=i
with incident and reflected phase angles given by Eqn. 8.32.
The measured horizontal velocity aligned with the direction of wave
travel can be represented by Eqn. 8.30. However, if the analysis is limited
to just long-wave frequencies, the velocity transfer function (ZJ given by
Eqn. 8.33 simplifies to a constant, i.e.,
(8.53)
and the horizontal velocity time series becomes
I— f OO
00
y 152 aL cos($a -
- 52aR' cos($ * . +
(8.54)
or in terms of the definitions given by Eqns. 8.51 and 8.52
(8.55)
Equations 8.50 and 8.55 are two equations with two unknowns, rj/(æ,0
and 7]R(x,t). Solving the equations yields
*?/(<) = | 7?(/) +
(8.56)
and
VR(t) = | î/(0~
(8.57)
(Note that the z-dependence has been dropped because both measurements
are made at the same z-position.)