In real string instruments, wave motion is possible in at least four dimensions corresponding to horizontal and vertical transverse motion, longitudinal motion, and especially for bowed strings, torsional waves.
If the string terminations were perfectly rigid, the horizontal and vertical transverse polarizations would be largely independent and we could model them with two identical, uncoupled, filtered delay loops, as shown in Fig. 10.
Figure 10:
A digital waveguide simulation of a rigidly terminated string vibrating in two uncoupled planes of vibration.
For terminations that are yielding to some extent, there will be coupling between various planes of vibration. We can simulate linear coupling with digital filters that connect the two planes of vibration together (as shown in Fig. 11 below).
Figure 11:
A digital waveguide string simulation with coupled horizontal and vertical planes of vibration.
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