We present a formal solution to the initial value problem for small perturbations of a straight vortex tube with constant vorticity, and show that any initial perturbation to such a tube evolves exclusively as a collection of Kelvin vortex waves. We then study in detail the evolution of the following particular initial states of the vortex tube: (i) an axisymmetric pinch in the radius of the tube, (ii) a deflection in the location of the tube, and (iii) a flattening of the tube's cross-secton. All of these initial states are localized in the direction along the tube by weighting them with a Gaussian function. In each case, the initial perturbation is decomposed into packets of Kelvin vortex waves which then propagate outward along the vortex tube. We discuss the physical mechanisms responsible for the propagation of the wave packets, and study the consequences of wave dispersion for the solution.