Localized Velocity Excitations

Initial velocity excitations are straightforward in the DW paradigm, but can be less intuitive in the FDTD domain. It is well known that velocity in a displacement-wave DW simulation is determined by the difference of the right- and left-going waves [437]. Specifically, initial velocity waves can be computed from from initial displacement waves by spatially differentiating to obtain traveling slope waves , multiplying by minus the tension to obtain force waves, and finally dividing by the wave impedance to obtain velocity waves:

where denotes sound speed. The initial string velocity at each point is then . (A more direct derivation can be based on differentiating Eq.(E.4) with respect to and solving for velocity traveling-wave components, considering left- and right-going cases separately at first, and arguing the general case by superposition.)

We can see from Eq.(E.11) that such asymmetry can be caused by unequal weighting of and . For example, the initialization

corresponds to an impulse velocity excitation at position . In this case, both interleaved grids are excited.

---

Previous: Localized Displacement Excitations
Next: More General Velocity Excitations

Order a Hardcopy of Physical Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.