Monatomic linear lattice consider a longitudinal wave us = u cos (wt ? sKa) which propagates in a monatomic linear lattice of atoms of mass M, spacing a, and nearest-neighbor interaction C.

(a) Show that the total energy of the wave is where s runs over all atoms.

(b) By substitution of u_s, in this expression, show that the time-average total energy per atom is where in the last step we have used the dispersion relation (9) for this problem.

Part (a) E = } M 2 (du/dt)2 + } c2u, - 11+1) Part (b)