Lattice vibrations.

I am thinking that we could study lattice vibrations this week. These are the vibrations of the nuclei. These vibrations involve small deviations away from the equilibrium positions of the nuclei in the lattice.

When sound travels through a solid, it travels via lattice vibrations. Lattice vibrations involve small deviations from equilibrium which occur in a collective, wave-like manner. Some aspects of lattice vibration mathematics are similar to the mathematical formalism of Bloch states, but they also differ in critical and interesting ways!...

Here is a warm-up problem, a pre-lecture problem, to do if you have time:
1. Consider two equal masses, of mass m, connected by a spring of spring constant k.
mass==k==mass      <--(this is a "picture", not an equation.)
a) Find the normal modes of this two mass system.
b) Express the equations of motion for this system using a 2x2 matrix.

Here is another warm-up problem. Basically the same thing but with 3 masses.
2. Consider three equal masses in a line, connected by a 2 springs of spring constant k.
mass==k==mass==k==mass       <--(this is a "picture", not an equation.)
a) Express the equations of motion for this system using a 3x3 matrix.
b) Find the normal modes of this three mass system.