``Traveling Waves'' in Lumped Systems

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Physical Audio Signal Processing

Digital Waveguide Theory

Power-Normalized Waveguide Filters

Reflectance of an Impedance

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Traveling waves propagate (i.e. travel) through space. — Click for http://ccrma.stanford.edu/realsimple/travelingwaves

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

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

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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

Wave Digital Filters (WDF) digitize analog circuits element-by-element using the bilinear transform and virtual traveling waves. — Click for https://ccrma.stanford.edu/~jos/pasp/Wave_Digital_Filters.html

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Physical Audio Signal Processing

Digital Waveguide Theory

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Reflectance of an Impedance

``Traveling Waves'' in Lumped Systems

One of the topics in classical network theory is the reflection and transmission, or scattering formulation for lumped networks [35]. Lumped scattering theory also serves as the starting point for deriving wave digital filters (the subject of Appendix F). In this formulation, forces (voltages) and velocities (currents) are replaced by so-called wave variables

$\begin{eqnarray*} f^{{+}}(t) &\isdef & \frac{f(t) + R_0v(t)}{2} \\ f^{{-}}(t) &\isdef & \frac{f(t) - R_0v(t)}{2} \end{eqnarray*}$

where is an arbitrary reference impedance. Since the above wave variables have dimensions of force, they are specifically force waves. The corresponding velocity waves are

$\begin{eqnarray*} v^{+}(t) &\isdef & \frac{1}{2}[v(t) + f(t)/R_0], \\ v^{-}(t) &\isdef & \frac{1}{2}[v(t) - f(t)/R_0]. \end{eqnarray*}$

Dropping the time argument, since it is always `(t)', we see that

$\displaystyle f^{{+}}$	$\displaystyle =$	$\displaystyle R_0v^{+}$
$\displaystyle f^{{-}}$	$\displaystyle =$	$\displaystyle -R_0v^{-}\protect$	(C.73)

and

$\displaystyle f$	$\displaystyle =$	$\displaystyle f^{{+}}+ f^{{-}}$
$\displaystyle v$	$\displaystyle =$	$\displaystyle v^{+}+ v^{-}\protect$	(C.74)

These are the basic relations for traveling waves in an ideal medium such as an ideal vibrating string or acoustic tube. Using voltage and current gives elementary transmission line theory.

Subsections

Reflectance of an Impedance

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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

``Traveling Waves'' in Lumped Systems

``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