Localized Velocity Excitations

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Physical Audio Signal Processing

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The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

Click for http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Click for http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe.html#c3

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

Click for https://ccrma.stanford.edu/~jos/pasp/Force_Waves.html

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 displacement of an object describes how far away and in what direction it has been moved, or displaced, from its rest position. Displacement is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Plucked_String_Displacement.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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Physical Audio Signal Processing

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Localized Velocity Excitations

Initial velocity excitations are straightforward in the DW paradigm, but can be less intuitive in the FDTD domain. It is well known that velocity in a displacement-wave DW simulation is determined by the difference of the right- and left-going waves [441]. Specifically, initial velocity waves $v^{\pm}$ can be computed from from initial displacement waves $y^\pm$ by spatially differentiating $y^\pm$ to obtain traveling slope waves $y'^\pm$ , multiplying by minus the tension to obtain force waves, and finally dividing by the wave impedance $R=\sqrt{K\epsilon }$ to obtain velocity waves:

$\displaystyle v^{+}$	$\displaystyle =$	$\displaystyle -cy'^{+}= \frac{f^{{+}}}{R}$
$\displaystyle v^{-}$	$\displaystyle =$	$\displaystyle \;cy'^{-}= -\frac{f^{{-}}}{R}, \protect$	(E.13)

where $c=\sqrt{K/\epsilon }$ denotes sound speed. The initial string velocity at each point is then $v(nT,mX)=v^{+}(n-m)+v^{-}(n+m)$ . (A more direct derivation can be based on differentiating Eq.(E.4) with respect to

and solving for velocity traveling-wave components, considering left- and right-going cases separately at first, and arguing the general case by superposition.)

We can see from Eq.(E.11) that such asymmetry can be caused by unequal weighting of $y_{n,m}$ and $y_{n,m\pm1}$ . For example, the initialization

$\begin{eqnarray*} y_{n-1,m+1} &=& +1\\ y_{n-1,m} &=& -1 \end{eqnarray*}$

corresponds to an impulse velocity excitation at position . In this case, both interleaved grids are excited.

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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

Localized Velocity Excitations

``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