The eigenvalues and the corresponding eigenvectors are given by
, 
, 
, and
From the eigenvector 
it is clear that the first mode corresponds
to a vibrational pattern in which the atom in the middle is stationary while
the atoms at the ends vibrate with the same amplitude but 
out of phase
with each other.
The second mode corresponds to a situation in which the two atoms at the ends
vibrate in phase with each other but 
out of phase with the atom in the
middle.
In the case of 
the mass ratio 
is greater than
and so the amplitude of vibration is larger for the middle atom than for
the atoms at the two ends.
Because 
,
the eigenvalue is related to the square of the frequency of vibration.
And since a cannot be negative,
therefore 
and
mode 2 must always have a higher frequency of vibration than mode 1.
This can be seen in the animation.
For mode 2, as a is increased from 
,
the amplitude of vibration for the middle atom
becomes larger than those of the end atoms, and the frequency is increased.
As a is decreased from 
towards 0,
the amplitude of vibration for the end atoms becomes larger than that of the
middle atom, and the frequency decreases and approaches that of mode 1.
In fact in the limit that a goes to 0,
the two modes become exactly the same mode.
This can be seen easily in the animation if the amplitudes of vibration
for the end atoms are kept the same for both modes while a is gradually
decreased to 0.
The general vibration motion of the molecule in the center-of-mass system
is a linear combination of mode 1 and 2.
Since the ratio of the vibration frequencies of the modes
is in general not a rational number,
the general motion is aperiodic.
Periodic motion occurs only when
for 
.
One should set 
and 4 to see that the vibration frequencies for
mode 1 and 2 are indeed commensurate at these mass ratios..