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Displacement-Wave Simulation

As discussed in [121], displacement waves are often preferred over force or velocity waves for guitar-string simulations, because such strings often hit obstacles such as frets or the neck. To obtain displacement from velocity at a given $ x$, we may time-integrate velocity as above to produce displacement at any spatial sample along the string where a collision might be possible. However, all these integrators can be eliminated by simply going to a displacement-wave simulation, as has been done in nearly all papers to date on plucking models for digital waveguide strings.

To convert our force-wave simulation to a displacement-wave simulation, we may first convert force to velocity using the Ohm's law relations $ f^{{+}}_i=Rv^{+}_i$ and $ f^{{-}}_i=-Rv^{-}_i$ and then conceptually integrate all signals with respect to time (in advance of the simulation). $ R$ is the same on both sides of the finger-junction, which means we can convert from force to velocity by simply negating all left-going signals. (Conceptually, all signals are converted from force to velocity by the Ohm's law relations and then divided by $ R$, but the common scaling by $ 1/R$ can be omitted (or postponed) unless signal values are desired in particular physical units.) An all-velocity-wave simulation can be converted to displacement waves even more easily by simply changing $ v$ to $ y$ everywhere, because velocity and displacement waves scatter identically. In more general situations, we can go to the Laplace domain and replace each occurrence of $ V^{+}(s)$ by $ sY^{+}(s)$, each $ V^{-}(s)$ by $ sY^{-}(s)$, divide all signals by $ s$, push any leftover $ s$ around for maximum simplification, perhaps absorbing it into a nearby filter. In an all-velocity-wave simulation, each signal gets multiplied by $ s$ in this procedure, which means it cancels out of all definable transfer functions. All filters in the diagram (just $ \hat{\rho}_f(s)$ in this example) can be left alone because their inputs and outputs are still force-valued in principle. (We expressed each force wave in terms of velocity and wave impedance without changing the signal flow diagram, which remains a force-wave simulation until minus signs, scalings, and $ s$ operators are moved around and combined.) Of course, one can absorb scalings and sign reversals into the filter(s) to change the physical input/output units as desired. Since we routinely assume zero initial conditions in an impedance description, the integration constants obtained by time-integrating velocities to get displacements are all defined to be zero. Additional considerations regarding the choice of displacement waves over velocity (or force) waves are given in §E.3.3. In particular, their initial conditions can be very different, and traveling-wave components tend not to be as well behaved for displacement waves.

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