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D'Alembert Derived

Setting $ s\isdeftext j \omega $ , and extending the summation to an integral, we have, by Fourier's theorem,

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right)$ (C.14)

for arbitrary continuous functions $ y_r(\cdot)$ and $ y_l(\cdot)$ . This is again the traveling-wave solution of the wave equation attributed to d'Alembert, but now derived from the eigen-property of sinusoids and Fourier theory rather than ``guessed''.

An example of the appearance of the traveling wave components shortly after plucking an infinitely long string at three points is shown in Fig.C.2.

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 $ t_0$ shortly after time 0 , the traveling waves centers are separated by $ 2ct_0$ meters, and their sum gives the trapezoidal physical string displacement at time $ t_0$ 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 $ c$ .

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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