Suppose that all the vibrational branches of a one-dimensional crystal consisting of two different masses with equal spacing, a=3.0 can be represented by a single optical mode branch and a single acoustical mode branch. If the frequency of the long-wavelength optical waves is 2x1013 rad/s and the maximum frequency of acoustical waves is 1013 rad/s, what is the group velocity of long wavelength acoustical waves?

Suppose that all the vibrational branches of a one-dimensional crystal consisting of two different masses with equal spacing, a=3.0 can be represented by a single optical mode branch and a single acoustical mode branch. If the frequency of the long-wavelength optical waves is 2x1013 rad/s and the maximum frequency of acoustical waves is 1013 rad/s, what is the group velocity of long wavelength acoustical waves?