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Digital Waveguide Equivalent Circuits

Figure 6.5: Digital waveguide ``equivalent circuits'' for the rigidly terminated ideal string with a moving termination. a) Velocity-wave simulation. b) Force-wave simulation.

Two digital waveguide ``equivalent circuits'' are shown in Fig.6.5. In the velocity-wave case of Fig.6.5a, the termination motion appears as an additive injection of a constant velocity $ v_0$ at the far left of the digital waveguide. At time 0 , this initiates a velocity step from 0 to $ v_0$ traveling to the right. When the traveling step-wave reaches the right termination, it reflects with a sign inversion, thus sending back a ``canceling wave'' to the left. Behind the canceling wave, the velocity is zero, and the string is not moving. When the canceling step-wave reaches the left termination, it is inverted again and added to the externally injected dc signal, thereby sending an amplitude $ 2v_0$ positive step-wave to the right, overwriting the amplitude $ v_0$ signal in the upper rail. This can be added to the amplitude $ -v_0$ signal in the lower rail to produce a net traveling velocity step of amplitude $ v_0$ traveling to the right. This process repeats forever, resulting in traveling wave components which grow without bound, but whose sum is always either 0 or $ v_0$ . Thus, at all times the string can be divided into two segments, where the segment to the left is moving upward with speed $ v_0$ , and the segment to the right is motionless.

At this point, it is a good exercise to try to mentally picture the string shape during this process: Initially, since both the left end support and the right-going velocity step are moving with constant velocity $ c$ , it is clear that the string shape is piece-wise linear, with a negative-slope segment on the left adjoined to a zero-slope segment on the right. When the velocity step reaches the right termination and reflects to produce a canceling wave, everything to the left of this wave remains a straight line which continues to move upward at speed $ v_0$ , while all points to the right of the canceling wave's leading edge are not moving. What is the shape of this part of the string? (The answer is given in the next paragraph, but try to ``see'' it first.)

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University