## Linear Problem

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
 (2) (3)

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
 (10) (11)

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
 (15)