5'. Dispersion of Waves (30 pts) When a transverse wave travels down a real wire there are forces acting on each portion of the wire in addition to the force resulting from the tension in the wire. An equation that gives an improved description of a wave on a real wire is 32y T 32y @ = E (3) a?! (9) where T > 0 is the tension, ,u > 0 is the mass per unit length and o: > 0 is a constant. (a) Find the relationship between to, T, who: for y(a:,t) : Acos(wt hat) to be a solution where w and k are constants. (6 pts) (6) What is the lowest angular frequency that the wire can support according to the condition found in the previous question. (6 pts) (c) Obtain the phase velocity, vph and the group velocity v9 of the waves. Write your result in terms of w,THu, 16,05. (6' pts) (d) How much is the product of up}, and v9: vth9? (6 pts) (e) Is v9 greater or smaller than 1/T/tt (the wave speed when a : 0)? (3 pts) ( f ) Is the dispersion of the waves in the wire when a > 0 non-dispersive, normal or anomalous? What's the answer ifa = 0'? Justify your answer. (5' pts)