= Torsional waves in a solid circular rod
{title2=$\phi_{tt}=(G/\rho)\phi_{xx}$}
For small torsional displacement $\phi(x,t)$, the transmitted <torque> is $C=GJ_p\phi_x$. A slice of length $dx$ receives net torque $GJ_p\phi_{xx}dx$ and has axial <moment of inertia> $\rho J_pdx$. Its angular equation therefore gives the displayed <wave equation>, with wave speed $\sqrt{G/\rho}$. The same geometric factor cancels from inertia and stiffness, making this ideal torsional speed independent of rod radius. This long-wave circular-rod model neglects additional elastic modes and dissipation.
Back to article page
