11:1% Velocity averaging for specific heat. Derive an averaging rule analogous to eq. (11.16) to combine v , and v , into a single value of v, ., that gives the correct low-temperature specific heat for the combined system of longitudinal plus transverse acoustic phonons. Evaluate the numerical value of your v, ., for v,y =5200m's and v, =8840m/s (approximate average values for S1), and compare to the value of v, found from eq. (11.16) as well as to the simple weighted average v, =