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Digital Waveguide Bowed-String

A more detailed diagram of the digital waveguide implementation of the bowed-string instrument model is shown in Fig.9.52. The right delay-line pair carries left-going and right-going velocity waves samples $ v_{s,r}^{+}$ and $ v_{s,r}^{-}$ , respectively, which sample the traveling-wave components within the string to the right of the bow, and similarly for the section of string to the left of the bow. The `$ +$ ' superscript refers to waves traveling into the bow.

Figure 9.52: Waveguide model for a bowed string instrument, such as a violin.

String velocity at any point is obtained by adding a left-going velocity sample to the right-going velocity sample immediately opposite in the other delay line, as indicated in Fig.9.52 at the bowing point. The reflection filter at the right implements the losses at the bridge, bow, nut or finger-terminations (when stopped), and the round-trip attenuation/dispersion from traveling back and forth on the string. To a very good degree of approximation, the nut reflects incoming velocity waves (with a sign inversion) at all audio wavelengths. The bridge behaves similarly to a first order, but there are additional (complex) losses due to the finite bridge driving-point impedance (necessary for transducing sound from the string into the resonating body). According to [95, page 27], the bridge of a violin can be modeled up to about $ 5$ kHz, for purposes of computing string loss, as a single spring in parallel with a frequency-independent resistance (``dashpot''). Bridge-filter modeling is discussed further in §9.2.1.

Figure 9.52 is drawn for the case of the lowest note. For higher notes the delay lines between the bow and nut are shortened according to the distance between the bow and the finger termination. The bow-string interface is controlled by differential velocity $ v_{\Delta}^{+}$ which is defined as the bow velocity minus the total incoming string velocity. Other controls include bow force and angle which are changed by modifying the reflection-coefficient $ \rho(v_{\Delta}^{+})$ . Bow position is changed by taking samples from one delay-line pair and appending them to the other delay-line pair. Delay-line interpolation can be used to provide continuous change of bow position [269].

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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