
229Harmonic Oscillator
© 2010 Taylor & Francis Group, LLC
Energies are time-dependent, and our analysis is simplied if we focus on time-average values.
Now,
1
2
1
2
1
2
2
0
2
2
0
2
ωδ
ωδ
πω π
/
d
/
dsin( )cos ()
/
tt
−=
//ω
.
Thus, for an average over one period of oscillation, we have
T
mB
V
kB
22 2
ω
,
and
E
Bm k
mB
=
+
=
−
22
22
0
2
()ω
.
Let us now nd the frequency ω
k
at which 〈T〉 is a maximum:
d
d
T
k
ω
ωω=
= 0 (7.80)
from which we obtain
ω
k
= ω
0
. (7.81)
Thus, the kinetic energy resonance occurs at the natural frequency ω
0
of the system for an undamped
oscillator. Because the potential energy is proportional to the square of the amplitude, we expect
that the potential ...