Two-Port Series Adaptor for Force Waves

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

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The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

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In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

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A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

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Two-Port Series Adaptor for Force Waves

**Figure F.7:** a) Two-port description of the adaptor implementing a series connection between reference impedances and . b) Corresponding series force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form.
$\includegraphics[width=\twidth]{eps/lAdaptorSeries}$

As discussed in §7.2, a series connection is characterized by a common velocity and forces which sum to zero at the junction:

$\begin{eqnarray*} && f_1(n) + f_2(n) = 0\\ && v_1(n) = v_2(n) \isdef v_J(n) \end{eqnarray*}$

The derivation can proceed exactly as for the parallel junction in §F.2.1, but with force and velocity interchanged, i.e., $f\leftrightarrow v$ , and with impedance and admittance interchanged, i.e., $R\leftrightarrow \Gamma$ . In this way, we may take the dual of Eq.(F.11) to get

$\begin{eqnarray*} v^{-}_1 &=& -\rho v^{+}_1 + (1+\rho) v^{+}_2\\ v^{-}_2 &=& (1-\rho)v^{+}_1 + \rho v^{+}_2 \end{eqnarray*}$

diagrammed in Fig.F.8. Converting back to force wave variables via $f^{{+}}_i=R_iv^{+}_i$ and $f^{{-}}_i=-R_iv^{-}_i$ , and noting that $(1+\rho)R_1/R_2 = 1-\rho$ , we obtain, finally,

$\begin{eqnarray*} f^{{-}}_1 &=& \rho f^{{+}}_1 - (1-\rho) f^{{+}}_2\\ f^{{-}}_2 &=& -(1+\rho)f^{{+}}_1 - \rho f^{{+}}_2 \end{eqnarray*}$

$\begin{eqnarray*} f^{{-}}_1 &=& -f^{{+}}_2 + \rho(f^{{+}}_1 + f^{{+}}_2)\\ f^{{-}}_2 &=& -f^{{+}}_1 - \rho(f^{{+}}_1 + f^{{+}}_2). \end{eqnarray*}$

**Figure F.8:** Series velocity scattering junction in Kelly-Lochbaum form.
$\includegraphics[scale=0.9]{eps/lscat_vel_series_renum}$

Two-Port Series Adaptor for Force Waves

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University