Torsional waves in a solid circular rod (${\varphi }_{tt}=(G/\rho ){\varphi }_{xx}$)

For small torsional displacement $\varphi (x,t)$, the transmitted torque is $C=G{J}_p{\varphi }_x$. A slice of length $dx$ receives net torque $G{J}_p{\varphi }_{xx}dx$ and has axial moment of inertia $\rho J_{p}dx$. 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.