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 the Laplace-transform of the force on the bridge, , divided by 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