# Equipartition of energy in the wave equation

(Equipartition of energy). Let $u \in C^2(\R \times [0,\infty))$ solve the initial-value problem for the wave equation in one dimension: $$\begin{cases} u_{tt}-u_{xx}=0 & \text{ in } \R \times (0, \infty) \\ u=g, u_t=h & \text{ on } \R \times \{t=0\}. \end{cases}$$ Suppose $g, h$ have compact support. The kinetic energy is $k(t) := \frac{1}{2} \int_{-\infty}^{\infty} u_t^2 (x,t) \, dx$ and the potential energy is $p(t) := \frac{1}{2} \int_{-\infty}^{\infty} u_x^2 (x,t) \, dx$. Prove that

(i) $k(t)+p(t)$ is constant in $t$,
(ii) $k(t)=p(t)$ for all  $t$ large enugh.