Passive String Terminations

When a traveling wave reflects from the bridge of a real stringed instrument, the bridge moves, transmitting sound energy into the instrument body. How far the bridge moves is determined by the driving-point impedance of the bridge, denoted . The driving point impedance is the ratio of Laplace transform of the force on the bridge to the velocity of motion that results . That is, .

For passive systems (i.e., for all unamplified acoustic musical instruments), the driving-point impedance is positive real (a property defined and discussed in §C.11.2). Being positive real has strong implications on the nature of . In particular, the phase of cannot exceed plus or minus degrees at any frequency, and in the lossless case, all poles and zeros must interlace along the axis. Another implication is that the reflectance of a passive bridge, as seen by traveling waves on the string, is a so-called Schur function (defined and discussed in §C.11); a Schur reflectance is a stable filter having gain not exceeding 1 at any frequency. In summary, a guitar bridge is passive if and only if its driving-point impedance is positive real and (equivalently) its reflectance is Schur. See §C.11 for a fuller discussion of this point.

At , the force on the bridge is given by (§C.7.2)

where is the string tension as in Chapter 6, and is the slope of the string at . In the Laplace frequency domain, we have

due to linearity, and the velocity of the string endpoint is therefore

---

Previous: Bridge Modeling
Next: A Terminating Resonator

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.