Equivalence of Digital Waveguide and Finite Difference Schemes

It was shown in §C.4.3 that the digital waveguide (DW) model for the ideal vibrating string performs the same ``state transitions'' as the more standard finite-difference time-domain (FDTD) scheme (also known as the ``leapfrog'' recursion). This appendix, initially published in [447], further establishes that the solution spaces of the two schemes are isomorphic. That is, a linear, one-to-one transformation is derived which converts any point in the state-space of one scheme to a unique point in the other scheme. Since boundary conditions and initial values are more intuitively transparent in the DW formulation, the simple means of converting back and forth can be useful in initializing and constructing boundaries for FDTD simulations, as we will see.