Wave energy ($E_{\mathrm{lin}}=\frac{1}{2}\int (|u_t{|}^2+|\nabla u{|}^2)dx$)

For a real scalar wave equation, the kinetic and gradient energy is $E_{\mathrm{lin}}(t)=\frac{1}{2}\int (|u_t{|}^2+|\nabla u{|}^2)dx$. A potential term is added for a semilinear wave equation $u_{tt}-\Delta u+V^{\prime }(u)=0$, giving the conserved wave energy $E=E_{\mathrm{lin}}+\int V(u)dx$.