Show that, for the classical solutions of the wave equation, the following quantity is independent of $t$.

Question

For any classical solution of the equation $$ \frac{\partial^2 u}{\partial t^2} - \Delta u=0\quad (x,t)\in\mathbb{R}^{d+1}, $$ show that the following quantity is independent of $t$: $$ M_1(t)=\int\limits_{\mathbb{R}^d}\frac{\partial u}{\partial t}(\vec{x},t)\frac{\partial u}{\partial x_1}(\vec{x},t)\,\mathrm{d}\vec{x}. $$