3a.3. Consider a longitudinal wave us propagating in a linear monoatomic chain of mass M and with lattice spacing a and the nearest neighbor interaction C. us = u cos(wt - ska) a) Show that the total energy (summed for all particles) is given by, 2 Erotal = K + U = [/2M (dus) = [ { M (+) + C(U - 1+) us+132 S b) Show that the dispersion relation is given by dispersion relation 4C = M Ka |sin (1) @= c) time-averaged total energy per atom is 1 1 Eaug = Mau+ C(1-cosKa)u = Mwu Savg = =