100
10 Measuring-Rods and Clocks in Motion
-
6
x
q
a
2
λ o
- v
p p p p p p p p p
t 1
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p
t 2
t 3
Fig. 10.4 The moving breather solution (117) q III (x, t) =
2a
π arctan
sin
2π(t−vx/c 2
o )
To γ
cosh
π(x−vt)
Lo γ
√
2
moving with v =
0, 8 c o , γ = 0, 6 for the same points of times as in Fig. 9.3, thus t 1 = 0, t 2 =
πλo
√
2
2co , t 3 =
3
√
2πλo
2co .
Here, according to (102) T o = 2π
√
2 λ o /c o and L o = π λ o according to (99). The oscillating lines
in Fig. 9.3 undergo a change in their period of oscillation by the motion with the uniform velocity
v which we calculate in (120). Here a remarkable wavelike displacement of the oscillations can be
observed
amplitude of oscillation x = b,
q
III
ob (b, t) =
2a
π
arctan
sin
2πt
T o
1
−→ T o .
(118)
If we insert into (117) (into the equation for the moving clock U v ) the position of the
maximum oscillation after time t has passed, in other words x = vt we receive
q
III
(vt, t) =
2a
π
arctan
sin
2π(1 − v
2
/c
2
o ) t
γ T o
1
=
2a
π
arctan
sin
2π γ
2 t
γ T o
1
,
hence
q
III
(vt, t) =
2a
π
arctan
sin
2πt
T o /γ
1
.
(119)
From this the period of oscillation T
of the moving clock U v can be determined as
T
=
T o
1 − v 2 /c 2
o
.
(120)