#### 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)

*slope*of the string at . In the Laplace frequency domain, we have

**Next Section:**

A Terminating Resonator

**Previous Section:**

Software for Cubic Nonlinear Distortion