= Energy flux of a torsional wave
{title2=$\mathcal P=-GJ_p\theta_x\theta_t$}
The local <elastic strain energy> and <rotational kinetic energy> per length are $u_s=GJ_p\theta_x^2/2$ 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 $|\mathcal P|=2\sqrt{G/\rho}\sqrt{u_su_k}\le\sqrt{G/\rho}(u_s+u_k)$ 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.
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