In Chapter 7, a general model of linear string motion was presented. This model, which holds under low-amplitude vibration conditions, is sufficient in many cases of musical interest, but not all. If vibration amplitude becomes large (under, say, high amplitude striking or plucking conditions), various nonlinear effects begin to appear, and can become perceptually dominant. The effect which is perhaps the most familiar to the reader will be that of the pitch glide, which is common across many instruments, and not merely strings (it occurs in struck percussion instruments as well--see §11.1.4). Under a high amplitude strike or pluck, there is often a downward change in the pitch of the string, due to increased tension in string, (or, equivalently, to the increased length of the deformed string). Such an effect, often called tension modulation[220,232,74], cannot be captured by a linear model, for which modal frequencies are, by linear system theory, fixed. Other more subtle phenomena also play an important perceptual role. The generation of audible so-called phantom partials [52,12] in piano strings under high striking velocities is due to coupling between longitudinal and transverse vibration, and beating can result from the instability of motion of a nonlinear string in a single polarization, which is a purely three-dimensional effect.
A useful starting point, and perhaps the simplest nonlinear partial differential equation in musical acoustics, is the Kirchhoff-Carrier model of string vibration, introduced in §8.1, which models the effect of tension modulation, and thus gives rise to pitch glide phenomena. Finite difference schemes and modal methods are discussed. In order to deal with more realistic cases of string vibration, more general models in both one and two polarizations are introduced in §8.2 and §8.3, respectively. Finite difference schemes are developed, and various effects of musical interest such as phantom partials and whirling are examined.
References for this chapter include: [31,139,244,66,67,233,232,220,3,185,149,99,186,132,156,113,140,12,11,129,74,42,4,52,13,158]