D'Alembert Derived

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

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Traveling-Wave Solution

Converting String-State to Traveling-Waves

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

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

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 sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

The wave equation refers to the partial differential equation governing wave phenomena in elastic media such as air and isotropic solids. — Click for https://en.wikipedia.org/wiki/Wave_equation

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The Fourier theorems establish important, elementary time-frequency relationships. The theorems are largely the same whether stated for the case of the Fourier Transform (FT), Discrete Fourier Transform (DFT), Discrete Time Fourier Transform (DTFT), or Fourier Series (FS). — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Theorems.html

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Traveling-Wave Solution

Converting String-State to Traveling-Waves

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right)$

(C.14)

**Figure C.2:** An infinitely long string, ``plucked'' simultaneously at *three* points, labeled ``p'' in the figure, so as to produce an initial triangular displacement. The initial displacement is modeled as the sum of two identical triangular pulses which are exactly on top of each other at time 0 . At time shortly after time 0 , the traveling waves centers are separated by meters, and their sum gives the trapezoidal physical string displacement at time which is also shown. Note that only three short string segments are in motion at that time: the flat top segment which is heading to zero where it will halt forever, and two short pieces on the left and right which are the leading edges of the left- and right-going traveling waves. The string is not moving where the traveling waves overlap at the same slope. When the traveling waves fully separate, the string will be at rest everywhere but for two half-amplitude triangular pulses heading off to plus and minus infinity at speed .
$\includegraphics[width=\twidth]{eps/f_t_waves_no_term}$

D'Alembert Derived

``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