In this section we will show that the digital waveguide simulation technique is equivalent to the recursion produced by the finite difference approximation (FDA) applied to the wave equation [445, pp. 430-431]. A more detailed derivation, with examples and exploration of implications, appears in Appendix E. Recall from (C.6) that the time update recursion for the ideal string digitized via the FDA is given by
The last identity above can be rewritten as
This results extends readily to the digital waveguide mesh (§C.14), which is essentially a lattice-work of digital waveguides for simulating membranes and volumes. The equivalence is important in higher dimensions because the finite-difference model requires less computations per node than the digital waveguide approach.
Even in one dimension, the digital waveguide and finite-difference methods have unique advantages in particular situations, and as a result they are often combined together to form a hybrid traveling-wave/physical-variable simulation [354,355,223,124,123,225,265,224]. In this hybrid simulations, the traveling-wave variables are called ``W variables'' (where `W' stands for ``Wave''), while the physical variables are caled ``K variables'' (where `K' stands for ``Kirchoff''). Each K variable, such as displacement on a vibrating string, can be regarded as the sum of two traveling-wave components, or W variables:
Conversion between K variables and W variables can be non-trivial due to the non-local dependence of one set of state variables on the other, in general. A detailed examination of this issue is given in Appendix E.