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Accounting for Torsional Waves

Torsional waves can be modeled as an additional couple of waveguides whose speed is about $ 5.2$ times the transverse wave speed.

$\displaystyle v_{\hbox{\small h}} = v_{\hbox{\small in}}+ v_{\hbox{\small ib}} +{v}_{\hbox{\small
int}} + {v}_{\hbox{\small ibt}}

$\displaystyle v_{\hbox{\small on}} = v_{\hbox{\small ib}}+ \frac{f}{2 Z}

$\displaystyle v_{\hbox{\small ob}} = v_{\hbox{\small in}}+ \frac{f}{2 Z}

$\displaystyle v_{\hbox{\small ont}} = v_{\hbox{\small ibt}}+ \frac{f}{2 Z_{\hbox{\small t}}}

$\displaystyle v_{\hbox{\small obt}} = v_{\hbox{\small int}}+ \frac{f}{2 Z_{\hbox{\small t}}}

Structure of the basic model with filter in the nut side and torsional waves.

where $ f = $ applied force

$ Z=$ string transverse wave impedance

$ Z_t=$ string torsional wave impedance,

Torsional waves facilitate the establishment of Helmholtz motion because they are more damped than the transversal waves.
Their contribution at the bow point can be modeled in two ways:

  1. Changing the slope of the straight line:

    $\displaystyle Z_{\hbox{\small s}}=\frac{1}{\frac{1}{2Z}+\frac{1}{2Z_{\hbox{\small t}}}}

  2. Changing the inclination of the friction curve


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Download stiffbowed.pdf
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Download stiffbowed_4up.pdf

``Impact of String Stiffness on Virtual Bowed Strings'', by Stefania Serafin<>, (From CCRMA DSP Seminar Presentation, Music 423).
Copyright © 2014-03-24 by Stefania Serafin<>
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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