The chain has particles of unit mass with displacement for
. The ends of the chain are attached to a rigid
wall. The equation of motion then reads

(1) 
for
and at the ends we have
This system of equations can be rewritten in matrix form as

(4) 
where is an tridiagonal matrix with 2's on the diagonal
and 's on the super and subdiagonals.
The matrix equation is solved by first diagonalizing . The
eigenvectors and eigenvalues (diagonal values) satisfy

(5) 
for
. These eigenvectors define the normal modes of
vibration. The exact analytic solution gives the eigenvalues

(6) 
with eigenvectors proportional to

(7) 
You should normalize each eigenvector so the sum of the squares of the
components is 1. For a given normal mode we can plot the
components of the eigenvector
vs . For the
lowest mode , there are no nodes, for , there is one
node, etc. You are asked to plot these.
The general solution to the linear system can then be written as a
linear combination of the eigenvectors

(8) 
where the coefficient functions for the linear system,
only are just

(9) 
The initial values and can be obtained from the
initial displacement vector and initial velocity from
where we have assumed that the eigenvectors are normalized to 1 and
are real.
The energy of the system is easily found to be

(12) 
and using the orthonormal properties of the eigenvectors, we can
easily show that

(13) 
where

(14) 
The normal mode velocities and amplitudes can be derived from the
diplacement velocities and amplitudes using