Torsional wave equation in a circular shaft (${\theta }_{tt}=(G/\rho ){\theta }_{xx}$)

For a slender homogeneous circular shaft in small-amplitude linear elasticity, torque is $G{J}_p{\theta }_x$ and axial angular inertia per length is $\rho J_p$. Their common geometric factor cancels in the angular equation of motion, leaving the displayed wave equation. Here $J_p=\int r^{2}dA$ is the polar second moment of area, rather than the mass moment of inertia itself.