We will now derive a finite-difference model in terms of string displacement samples which correspond to the lossy digital waveguide model of Fig.C.5. This derivation generalizes the lossless case considered in §C.4.3.
Figure C.7 depicts a digital waveguide section once again in ``physical canonical form,'' as shown earlier in Fig.C.5, and introduces a doubly indexed notation for greater clarity in the derivation below [445,223,124,123].
Referring to Fig.C.7, we have the following time-update relations:
Adding these equations gives
with , and . In , it was shown by von Neumann analysis (§D.4) that these parameter choices give rise to a stable finite-difference scheme (§D.2.3), provided . In the present context, we expect stability to follow naturally from starting with a passive digital waveguide model.