In this section, we interpret the sampled d'Alembert traveling-wave solution of the ideal wave equation as a digital filtering framework. This is an example of what are generally known as digital waveguide models [434,435,437,441,445].
The term in Eq.(C.16) can be thought of as the output of an -sample delay line whose input is . In general, subtracting a positive number from a time argument corresponds to delaying the waveform by samples. Since is the right-going component, we draw its delay line with input on the left and its output on the right. This can be seen as the upper ``rail'' in Fig.C.3
Similarly, the term can be thought of as the input to an -sample delay line whose output is . Adding to the time argument produces an -sample waveform advance. Since is the left-going component, it makes sense to draw the delay line with its input on the right and its output on the left. This can be seen as the lower ``rail'' in Fig.C.3.
Note that the position along the string, meters, is laid out from left to right in the diagram, giving a physical interpretation to the horizontal direction in the diagram. Finally, the left- and right-going traveling waves must be summed to produce a physical output according to the formula
Any ideal, one-dimensional waveguide can be simulated in this way. It is important to note that the simulation is exact at the sampling instants, to within the numerical precision of the samples themselves. To avoid aliasing associated with sampling, we require all waveshapes traveling along the string to be initially bandlimited to less than half the sampling frequency. In other words, the highest frequencies present in the signals and may not exceed half the temporal sampling frequency ; equivalently, the highest spatial frequencies in the shapes and may not exceed half the spatial sampling frequency .
A C program implementing a plucked/struck string model in the form of Fig.C.3 is available at http://ccrma.stanford.edu/~jos/pmudw/.