3. A carbon nanotube may be visualized as a nearly one-dimensional material. A carbon nanotube of length L supports standing waves. The dispersion relation for these waves takes the form: 2 co (n; where o is the angular frequency, v is the wave velocity and nx is a positive integer. (a) Derive an expression for the number of standing waves, N(@), with frequency less that some value o. (Justify and explain each step in the derivation). (b) Derive an expression for the density of vibrational states, g(@), in that one- dimensional system. Sketch g(@) versus @ over the appropriate range of frequencies