In this chapter, we summarize the basic principles of digital waveguide models. Such models are used for efficient synthesis of string and wind musical instruments (and tonal percussion, etc.), as well as for artificial reverberation. They can be further used in modal synthesis by efficiently implementing a quasi harmonic series of modes in a single ``filtered delay loop''.
We begin with the simplest case of the infinitely long ideal vibrating string, and the model is unified with that of acoustic tubes. The resulting computational model turns out to be a simple bidirectional delay line. Next we consider what happens when a finite length of ideal string (or acoustic tube) is rigidly terminated on both ends, obtaining a delay-line loop. The delay-line loop provides a basic digital-waveguide synthesis model for (highly idealized) stringed and wind musical instruments. Next we study the simplest possible excitation for a digital waveguide string model, which is to move one of its (otherwise rigid) terminations. Excitation from ``initial conditions'' is then discussed, including the ideal plucked and struck string. Next we introduce damping into the digital waveguide, which is necessary to model realistic losses during vibration. This much modeling yields musically useful results. Another linear phenomenon we need to model, especially for piano strings, is dispersion, so that is taken up next. Following that, we consider general excitation of a string or tube model at any point along its length. Methods for calibrating models from recorded data are outlined, followed by modeling of coupled waveguides, and simple memoryless nonlinearities are introduced and analyzed.