String Models

As mentioned in §A.5.2, the first known computational models of vibrating strings were based on finite difference modeling by Ruiz and Hiller [195]. In the field of musical acoustics, software simulations of bowed strings were being carried out by Michael E. McIntyre and James Woodhouse as early as the mid 1970s [310]. In these simulations, the string is represented by the ``Green's function'' (impulse response) seen by a traveling wave traversing the string once in both directions (i.e., one round trip). This model was evidently based on investigations in the 1960s by John Schelleng, Lothar Cremer, and Cremer's co-worker Hans Lazarus into the behavior of the so-called ``corner rounding function'' (round-trip impulse response) in the context of bowed-string dynamics [412,413,95].^A.19 The bow-string junction was based on theory worked out by Friedlander [151] and Keller [246], both published in 1953. These mathematical models of bowed-string dynamics were in turn preceded by influential investigations by Raman, published in 1918 [368], and Hermann von Helmholtz, published in 1863 [542]. An important enabling scientific measurement was that of the friction curve describing the bow-string contact, and many models, such as in [310,311] used a hyperbolic friction curve, approximating measurements by Lazarus [95]. Interestingly, while researchers in musical acoustics would often use their models to produce example waveforms illustrating the successful modeling of various visible physical effects, they apparently never listened to them as sound.

Robert T. Schumacher, who had been interested primarily in woodwind musical acoustics, collaborated with McIntyre and Woodhouse, and the result was a generalization of the Friedlander-Keller bowed-string model to wind instruments [311].

Incidentally, electrical equivalent circuits for bowed-string instruments appeared as early as 1952 [335, p. 122], with perhaps the most cited model being Schelleng's published in 1963 [412].