Torsional wave equation in a circular shaft
= Torsional wave equation in a circular shaft
{title2=$\theta_{tt}=(G/\rho)\theta_{xx}$}
For a slender homogeneous circular shaft in small-amplitude <linear elasticity>, torque is $GJ_p\theta_x$ 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 $J_p=\int r^2dA$ is the polar second moment of area, rather than the mass <moment of inertia> itself.