Passive String Terminations

In Chapter 4 we analyzed the effect of rigid string terminations on traveling waves. We found that waves derived by time-derivatives of displacement (displacement, velocity, acceleration, and so on) reflect with a sign inversion, while waves defined in terms of the first spatial derivative of displacement (force, slope) reflect with no sign inversion. In this appendix, we will look at the more realistic case of yielding terminations for strings. This analysis can be considered a special case of the loaded string junction analyzed in §H.10.

Yielding string terminations (at the bridge) have a large effect on the sound produced by acoustic stringed instruments. Rigid terminations can be considered a reasonable model for the solid-body electric guitar in which maximum sustain is desired for played notes. Acoustic guitars, on the other hand, must transduce sound energy from the strings into the body of the instrument, and from there to the surrounding air. All audible sound energy comes from the string vibrational energy, thereby reducing the sustain (decay time) of each played note. Furthermore, because the bridge vibrates more easily in one direction than another, a kind of ``chorus effect'' is created from the detuning of the horizontal and vertical planes of string vibration (as discussed further in §4.12.1). A perfectly rigid bridge, in contrast, cannot transmit any sound into the body of the instrument, thereby requiring some other transducer, such as the magnetic pickups used in electric guitars, to extract sound for output.^M.1

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 .

For passive systems (i.e., for all unamplified acoustic musical instruments), the driving-point impedance is positive real (see §M.4) [437,532], which means (1) is real when is real, and (2) the real part of is nonnegative when the real part of is nonnegative, i.e., re   re.
This seemingly simple property has deep 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