For a circular cross-section there are simple torsional waves with speed given again by c t , independent of the radius R . Flexion waves . These are waves in a thin beam that vibrates from side to side, when is larger than the lateral dimensions. Hence, the dispersion relation is where as usual . The phase velocity is and the group velocity is twice this.
Long waves move very slowly.
Waves in a string under tension. Note that is the mass per unit length of the string. The dispersion relation is At small k , large , the T term is dominant, and the contrary is true at small . That means that is imaginary and the amplitude grows exponentially, instead of oscillating.