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.

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

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

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

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

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

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

2. Changing the inclination of the friction curve