A new algorithm for computing one-dimensional acoustic waves in a pipe is demonstrated by solving the acoustic equations as an initial-boundary-value problem. Conventional dissipation-free second-order finite difference methods suffer severe phase distortion for grids with less that about ten mesh points per wavelength. Using the signal generated by a piston in a duct as an example, transient acoustic computations are presented using a new compact three-point algorithm which allows about 60% fewer mesh points per wave length. Both pulse and harmonic excitation are considered. Coupling of the acoustic signal with the pipe resonant modes are shown to generate a complex transient wave with rich harmonic content.