Vibrational modes of a discrete particle string

[aaaa202]

I just want to make sure I understand the normal modes of a vibrating system of particles with discrete spacing. I have tried to drawn what I understand as the lowest and highest frequency mode of the standing waves. Is the drawing correct? Edit: actually I have drawn the maximum frequency where no particles are permitted to move..
The system is 4 particles and for standing waves their vibrations are described by:
y(s) = sin(sKa) with a time dependence and where s refers to particle number s and a is the particle spacing.
Now with y(0)=y(4) you have the modes:
k=\pi/4a, 2\pi/4a, k=3\pi/4a
The last one corresponds to no movement of any particle. Is this correctly understood? What happens if we take for instance k=5\pi/4a. Does this mode simply reproduce the motion of k=\pi/4a?

 

[BOYLANATOR]

I'm not sure that I understand your question exactly. If you are just trying to understand how standing waves work then it is easier to think of just 2 points which are the ends of a "string". The string must always be anchored to these two points, so the minimum energy mode is a half wave. There is no maximum frequency in the simplest models. The more vigorously you shake the string, the greater the frequency of the standing waves.

Have a look at this picture from wikipedia
http://en.wikipedia.org/wiki/File:Harmonic_partials_on_strings.svg

 

[aaaa202]

It is standing waves but not on a string, rather for a system of particles with a discrete spacing. Like when you examine vibrations of atomic planes.