The wave digital hammer: A computationally efficient traveling wave model of the piano hammer and the felt mallet

Recent work has led to traveling wave string and membrane models using the digital waveguide and the 2-D digital waveguide mesh. This paper introduces a new development for these musical instrument models which extends their usefulness: a traveling wave model for the piano hammer, or felt mallet. When a mallet strikes an ideal membrane or string, it sinks down into it, feeling a pure resistive impedance. In the membrane case, the depression induces a circular traveling wave outward. If the membrane is bounded, reflected waves return to the strike point to throw the mallet away from the membrane. This complex mallet–membrane interaction can have very different and difficult to predict acoustical effects, particularly when a second or third strike occurs while the membrane is still in motion, as in a drum roll. The piano hammer, or felt mallet, is viewed as a nonlinear mass/spring (inductor/capacitor) system, the nonlinear spring representing the felt portion. By decomposing the system into appropriate traveling waves, a unit of delay is extracted in the discrete time version, greatly simplifying the implementation. This wave digital hammer can be attached to any waveguide string or membrane model at a time-varying lossless scattering junction.