We consider decentralized control of a platoon of N identical vehicles moving in a straight line following a single lead vehicle moving at constant velocity. The control objective is for each vehicle to maintain the velocity of the leader and inter-vehicular separation using only the local information from itself and its two nearest neighbors. Each vehicle is modeled as a double integrator. To aid the analysis, we derive a continuous partial differential equation (PDE) approximation of the discrete platoon dynamics. The PDE model is used to explain the progressive loss of closed-loop stability with increasing number of vehicles, and to devise ways to combat this loss of stability. If every vehicle uses the same controller, we show that the least stable closed-loop eigenvalue approaches zero as O(1/N2) in the limit of a large number (N) of vehicles. We then show how to ameliorate this loss of stability by small amounts of "mistuning", i.e., changing the controller gains from their nominal values. We prove that with arbitrary small amounts of mistuning, the asymptotic behavior of the least stable closed loop eigenvalue can be improved to O(1/N). These conclusions are validated for the discrete platoon via numerical calculations.