Effects of Stiffness

• Phase velocity increases with frequency
  $$c(\omega) \mathrel{\stackrel{\Delta}{=}}c_0\left(1 + \frac{\kappa\omega^2}{2Kc_0^2}\right)$$
  where $c_0= \sqrt{K/\epsilon } =$ zero-stiffness phase velocity

• Note ideal-string (LF) and ideal-bar (HF) limits
• Traveling-wave components see a frequency-dependent sound speed
• High-frequency components ``run out ahead'' of low-frequency components (``HF precursors'')
• Traveling waves ``disperse'' as they travel
  (``dispersive transmission line'')
• String overtones are ``stretched'' and ``inharmonic''
• Higher overtones are progressively sharper
  ( $\hbox{Period}(\omega)$ = 2 $\times$ Length / $c(\omega)$)
• Piano strings are audibly stiff

Reference: L. Cremer: Physics of the Violin