The goal of this work is to design and analyze quadratic finite elements for problems of time-harmonic acoustics, and to compare the computational efficiency of quadratic elements to that of lower-order elements. Non-reflecting boundary conditions yield an equivalent problem in a bounded region which is suitable for domain-based computation of solutions to exterior problems. Galerkin/least-squares technology is utilized to develop robust methods in which stability properties are enhanced while maintaining higher-order accuracy. The design of Galerkin/least-squares methods depends on the order of interpolation employed, and in this case quadratic elements are designed to yield dispersion-free solutions to model problems. The accuracy of Galerkin/least-squares and traditional Galerkin elements is compared, as well as the accuracy of quadratic versus standard linear interpolation, incorporating the effects of representing the radiation condition in exterior problems. The efficiency of the various methods is measured in terms of the cost of computation, rather than resolution requirements. In this manner, clear guidelines for selecting the order of interpolation are derived. Numerical testing validates the superior performance of the proposed methods. This work is a first step to gaining a thorough analytical understanding of the performance of p refinement as a basis for the development of h-p finite element methods for large-scale computation of solutions to acoustic problems.