Terminated String Impedance

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Physical Audio Signal Processing

Moving Rigid Termination

Animation of Moving String Termination

The Ideal Plucked String

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

If the mass of a harmonic oscillator is excited, the system responds by vibrating back and forth. Whenever the mass is at its rest position, the system contains only kinetic energy. Conversely, whenever the mass velocity is zero, the system contains only potential energy. — Click for http://ccrma.stanford.edu/realsimple/lab_inst/Background.html

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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

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

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

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Physical Audio Signal Processing

Moving Rigid Termination

Animation of Moving String Termination

The Ideal Plucked String

Terminated String Impedance

Note that the impedance of the terminated string, seen from one of its endpoints, is not the same thing as the wave impedance $R=\sqrt{K\epsilon }$ of the string itself. If the string is infinitely long, they are the same. However, when there are reflections, they must be included in the impedance calculation, giving it an imaginary part. We may say that the impedance has a ``reactive'' component. The driving-point impedance of a rigidly terminated string is ``purely reactive,'' and may be called a reactance (§7.1). If denotes the force at the driving-point of the string and denotes its velocity, then the driving-point impedance is given by (§7.1)

$\displaystyle R(j\omega) \isdefs \left.\frac{F(s)}{V(s)}\right\vert _{s=j\omega},$

where

and

denote the Laplace transforms of

and

. In the case of a rigidly terminated string above, as well as in any system in which all energy delivered to the system is ultimately reflected back to the input, the impedance is purely imaginary at every frequency (a ``pure reactance''), as is easy to show:

$\displaystyle R(s) \isdefs \frac{F(s)}{V(s)} \eqsp \frac{F^{+}+F^{-}}{V^{+}+V^{-}} \eqsp \frac{F^{+}+e^{-s2L/c}F^{+}}{V^{+}-e^{-s2L/c}V^{+}} \eqsp R\frac{1+e^{-s2L/c}}{1-e^{-s2L/c}}$

where

denotes the string length. Let

denote the period of string vibration. Then on the frequency axis $s=j\omega$ we have

$\displaystyle R(j\omega) \eqsp R\frac{1+e^{-j\omega P}}{1-e^{-j\omega P}} \eqsp R\frac{2\cos(\omega P/2)}{2j\sin(\omega P/2)} \eqsp -jR\,\cot(\omega P/2).$

Thus, the driving-point impedance of a rigidly terminated string is purely reactive (imaginary), with alternating poles and zeros along the $j\omega$ axis. Impedance will be discussed further in §7.1 below.

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``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

Terminated String Impedance

``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