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Moving Rigid Termination

It is instructive to study the ``waveguide equivalent circuit'' of the simple case of a rigidly terminated ideal string with its left endpoint being moved by an external force, as shown in Fig.6.4. This case is relevant to bowed strings9.6) since, during time intervals in which the bow and string are stuck together, the bow provides a termination that divides the string into two largely isolated segments. The bow can therefore be regarded as a moving termination during ``sticking''.

Figure 6.4: Moving rigid termination for an ideal string at time $ t_0$ .

Referring to Fig.6.4, the left termination of the rigidly terminated ideal string is set in motion at time $ t=0$ with a constant velocity $ v_0$ . From Eq.(6.5), the wave impedance of the ideal string is $ R=\sqrt{K\epsilon }$ , where $ K$ is tension and $ \epsilon $ is mass density. Therefore, the upward force applied by the moving termination is initially $ f_0=Rv_0$ . At time $ t_0<L/c$ , the traveling disturbance reaches a distance $ c t_0$ from $ x=0$ along the string. Note that the string slope at the moving termination is given by $ -v_0 t_0/(c t_0) = -v_0/c = -(f_0/R)/c = -f_0/K$ , which derives the fact that force waves are minus tension times slope waves. (See §C.7.2 for a fuller discussion.)

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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