226
DOC.
8
ANALYSIS
OF A RESONATOR'S MOTION
relativity
is
not
necessary.9
It
suffices to
expand
the electric and
magnetic
forces at
the
origin
in
a
Fourier
series, dependent
on
time
alone,
as long as
one
can prove
that the
individual
components
of
force
appearing
in this
expression are
independent
of
each
other.
The
momentum
that the oscillator
experiences
in
the
x-direction
during
time
x
is
j
-
Ik/k. i
0 0
ae
-Is#
dz
f
dt
dt.
Integration
by
parts
yields
J
o
dldt-
dt
Id*
*
-
f
-JLfdt
The
first
summand
vanishes
if
t
is appropriately
chosen,
i.e.,
if
t
is
large enough.
If,
in
accordance
with Maxwell's
equation, one
also
puts
d*
ae
X
i a»
aez
y
-
_
c
dt dx
dz9
one
arrives at the
simple expression
(10)
Tfae
j
=
j _ifdt.
0
dx
Now
only
the
component
Ez
and its derivative
d&Jdx
appear
in
our
expression.
However,
their
independence
can
easily
be
proved.
For
if
we
just
consider
two
wave
trains
(with
identical
solid
angles)
approaching
each
other,
we
can
write
Ez =
Eksi
sin
2tui
'
_
ax + ßy
+
b
cos
n
2im
^
_
ou: + +
yz
+
,
.
2wi
r
ax +
ßy
+
yzL-
a
'sin it
+
- n
rj!
r
/
2
r
ax + ßy +
yz
+
b
cos
\tm
+
-
ft
rjr
and
9
Because
the
momenta
with variable
signs,
which
result from the
irregularities
of the radiation
process, can
be
determined for
a
resonator at rest.