Formulas for the Schelleng diagram

$$f_{\hbox{\small max}}=\frac{2Z v_{\hbox{\small b}}}{(\mu_{\hbox{\small s}}-\mu_{\hbox{\small d}})\beta}$$

$$f_{\hbox{\small min}}=\frac{Z^{2}}{2r_{\hbox{\small 1}}} \cdot \frac{v_{\hbox{\small b}}}{(\mu_{\hbox{\small s}}-\mu_{\hbox{\small d}})\beta^{2}}$$

where $\mu_{\hbox{\small s}}$ =coefficient of static friction
$\mu_{\hbox{\small d}}$ =coefficient of dynamic friction
$v_{\hbox{\small b}}$ =bow velocity
$\beta$ =bow position
$Z$ =characteristic impedance of the string,
$Z=\sqrt{T \rho}$
$r_{\hbox{\small 1}}$ =term that represents losses of the string