Brief Discussion of Torsional Vibrations

Abstract

As shown, for example, in [8], from equations of motion of the following type, describing the propagation of two wave modes in a solid:

$$ \rho \frac{{{\partial ^{2}}u}}{{\partial {t^{2}}}} - \left( {\lambda + \mu } \right)\frac{{\partial \Delta }}{{\partial x}} + \mu {\nabla ^{2}}u $$

((1))

where u is the displacement, ρ is the density, λ and μ are the Lamé constants, Δ is the volume expansion, and ${\nabla ^{2}} = \left( {\frac{{{\partial ^{2}}}}{{\partial {x^{2}}}} + \frac{{{\partial ^{2}}}}{{\partial {y^{2}}}} + \frac{{{\partial ^{2}}}}{{\partial {z^{2}}}}} \right)$, one obtains the following wave equation after differentiation with respect to the coordinates and separate addition of the left- and right-hand sides:

$$ \rho \frac{{{\partial ^{2}}\Delta }}{{\partial {t^{2}}}} = \left( {\lambda + 2\mu } \right){\nabla ^{2}}\Delta $$

((2))

.