A given crystalline material has 3 atoms per lattice point. (i) How many acoustic phonon modes are there? (ii) How many of these are transverse, and how many longitudinal?

b) Given that the attenuation constant for TEM waves propagating on an air-filled cable is: α R 4πZ a b wp 20 +2) where R and Z₁ = 60 In In (1) Where a and b are the inner and outer radii of the cable respectively. Show that for minimum losses the ratio of the inner and outer radii is b/a = 3.6. Evaluate the corresponding characteristic impedance. HINT: Partially differentiate with respect to a holding b constant