504 CHAPTER 8. LABORATORY MEASUREMENT AND ANALYSIS
Eqn. 8.16 can be written as
rj(t) = aj A cos( (8.19)
The coefficient, A, is found by squaring Eqns. 8.17 and 8.18 and summing
to yield
A = ^/1 4- 7 (8.20)
and the phase, fi, is given by the ratio of Eqns. 8.18 and 8.17, i.e.,
sin(fcz + c) — Kr sin(kx 4- e + 0)
tan p —
7
~
~7~,
TT"
cos(fcx + 6) 4- 1\R COS{kx + € + 0)
(8.21)
Substituting for the coefficient, A, in Eqn. 8.19 gives an alternate expression
for the superimposed incident and reflected waves, i.e.,
7/(t) = a/ yjl + Kr + 2Kr cos[2(fcx + c) + 0] cos(o4 — /?)
(8.22)
The maximum wave height of the envelope occurs at the antinode of
the standing wave system when
cos[2(fcz 4- e) 4- 0] = 1
and its magnitude is given as twice the amplitude, or
Hmax —
1 4~ 271// 4- Kr
or . Hrnax = HJ(l + KR) (8.23)
where the incident wave height is defined as 77/ = 2a/. Similarly, the
minimum wave height of the envelope occurs at the node of the standing
wave system when
cos[2(fcx 4- e) 4- 0] = — 1
and its magnitude is
Hmin — 2a/^1 — “ 2Kr 4- 71'^
or
Hmin — Hj(I — 71//) (8.24)
Equations 8.23 and 8.24 can be solved for the incident wave height, i.e.,
Hj =
Hmax T Hmin
(8.25)
2