1. We saw that any translation-invariant medium admits wave solutions of the form was, t) = e'fkm'fwt, for appropriate k and w. Waves in many media (including transverse oscillations of a continuous string) satisfy the wave equation, in which X is a constant: (a) Show that wave solutions to this equation satisfy a linear diSpersion relation w(k), and nd the phase velocity and group velocity jwk as a function of the constant X. (b) You should nd in (a) that waves of any frequency move with the same velocity c. This suggests that pulses could move with that velocity without distorting. Show that the following is indeed a solution to the wave equation: Marat) = at - ct) + 9(56 + ct)- where f, g are arbitrary functions of one variable. If f (u) = (\"2 and 9(a) = e'\"2 are both Gaussian-shaped bumps, deseribe qualitatively what this solution represents (consider for examples negative times, t = 0, and late positive times). (c) Another wave-describing equation is Schrodinger's equation for the \"wavefunction\" 1/283, t) of a free particle of mass m: at 2m 33:2 ' While the physical interpretation of 1,!) is not too straightforward, its wave properties are. 2 2 1.5%: a an Compute (as in (a)) the dispersion relation w(k) for the free particle, and nd its phase velocity and group velocity. Which of these results (if any) looks more familiar to you, when re-expressed in terms of energy E = nd and momentum p = k