Force Waves

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Physical Audio Signal Processing

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The wave equation can be transformed into many equivalent forms involving different variables, such as displacement, velocity, acceleration, slope, and force. For a given problem, using one set of variables may be more convenient than another. — Click for http://ccrma.stanford.edu/~jos/VariableChoice/Overview_Wave_Variable_Choices.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

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Physical Audio Signal Processing

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Force Waves

**Figure C.14:** Transverse force propagation in the ideal string.
$\includegraphics[width=\twidth]{eps/fStringForce}$

Referring to Fig.C.14, at an arbitrary point along the string, the vertical force applied at time to the portion of string to the left of position by the portion of string to the right of position is given by

$\displaystyle f_{rl}(t,x) = K\sin(\theta) \approx K\tan(\theta) = Ky'(t,x)$

(C.41)

assuming $\left\vert y'(t,x)\right\vert \ll 1$ , as is assumed in the derivation of the wave equation. Similarly, the force applied by the portion to the left of position

to the portion to the right is given by

$\displaystyle f_{lr}(t,x) = - K\sin(\theta) \approx - Ky'(t,x).$

(C.42)

These forces must cancel since a nonzero net force on a massless point would produce infinite acceleration. I.e., we must have $f_{rl} + f_{lr} \equiv 0$ at all times

and positions

Vertical force waves propagate along the string like any other transverse wave variable (since they are just slope waves multiplied by tension ). We may choose either $f_{rl}$ or $f_{lr}$ as the string force wave variable, one being the negative of the other. It turns out that to make the description for vibrating strings look the same as that for air columns, we have to pick $f_{lr}$ , the one that acts to the right. This makes sense intuitively when one considers longitudinal pressure waves in an acoustic tube: a compression wave traveling to the right in the tube pushes the air in front of it and thus acts to the right. We therefore define the force wave variable to be

$\displaystyle f(t,x) \isdefs f_{lr}(t,x) = -Ky'(t,x).$

(C.43)

Note that a negative slope pulls up on the segment to the right. At this point, we have not yet considered a traveling-wave decomposition.

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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

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