Infinite Harmonic chain with heavy mass

Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to 1, except for one heavy particle which has mass M . We investigate the macroscopic behaviour of such a system when M is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle