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Reflectance and Transmittance of a Yielding String Termination

Figure C.28: Ideal vibrating string terminated on the right by a passive impedance $ R_b(s)$.

Consider the special case of a reflection and transmission at a yielding termination, or ``bridge'', of an ideal vibrating string on its right end, as shown in Fig.C.28. Denote the incident and reflected velocity waves by $ v^{+}(t)$ and $ v^{-}(t)$, respectively, and similarly denote the force-wave components by $ f^{{+}}(t)$ and $ f^{{-}}(t)$. Finally, denote the velocity of the termination itself by $ v_b(t)=v^{+}(t)+v^{-}(t)$, and its force-wave reflectance by

$\displaystyle \hat{\rho}_f(s) \isdefs \frac{F^{-}(s)}{F^{+}(s)} \eqsp -\frac{V^{-}(s)}{V^{+}(s)}
\eqsp \frac{R_b(s)-R_0}{R_b(s)+R_0},

where $ R_0$ denotes the string wave impedance. The bridge velocity is given by

$\displaystyle V_b(s) \eqsp V^{+}(s) + V^{-}(s),

so that the bridge velocity transmittance is given by

$\displaystyle \hat{\tau}_v(s) \isdefs \frac{V_b(s)}{V^{+}(s)}
\eqsp \frac{V^{+}(s)+V^{-}(s)}{V^{+}(s)}
\eqsp 1+\hat{\rho}_v(s)
\eqsp 1-\hat{\rho}_f(s),

and the bridge force transmittance is given by

$\displaystyle \hat{\tau}_f(s) \isdefs \frac{F_b(s)}{F^{+}(s)}
\eqsp \frac{R_0[V^{+}(s)-V^{-}(s)]}{R_0V^{+}(s)}
\eqsp 1+\hat{\rho}_f(s)

where the bridge force is defined as ``up'' so that it is given for small displacements by the string tension times minus the string slope at the bridge. (Recall from §C.7.2 that force waves on the string are defined by $ f = -Ky'$ where $ K$ and $ y'$ denote the string tension and slope, respectively.
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