 Energy flux of a torsional wave

The local elastic strain energy and rotational kinetic energy per length are $u_s=G{J}_p{\theta }_x^2/2$ and $u_k=\rho J_p{\theta }_t^2/2$. Multiplying the torsional wave equation in a circular shaft by $\rho J_p{\theta }_t$ and integrating the product derivative gives ${\partial }_t(u_s+u_k)+{\partial }_x\mathcal{P}=0$. Thus the energy flux is torque times angular speed, with the sign set by the positive direction. The inequality $|\mathcal{P}|=2\sqrt{G/\rho }\sqrt{u_s{u}_k}\le \sqrt{G/\rho }(u_s+u_k)$ follows from a square. A one-way travelling wave has $u_s=u_k$; a steadily rotating twisted shaft need not. Stored energy divided by transit time is not a separate bound for each energy component in a general superposition.