#### 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

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