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I am solving a initial value problem for the wave equation $$ u_{tt}=u_{xx}\ \ in \ \ \mathbb{R}\times (0,\infty), \ \ \ u=g, \ u_{t}=h \ \ on \ \ \mathbb{R}\times \{0\} $$ for some com[actly suppoerted functions $f,g\in C_{c}(\mathbb{R})$. Let $u$ be a solution to the wave equation above. The kinetic energy $k(t)$ and the potential energy $p(t)$ of $u$ are defined respectively $$ k(t)=\int_{\mathbb{R}}u_t^2(x,t)dx, \ \ \ \ p(t)=\int_{\mathbb{R}}u_x^2(x,t)dx. $$ I have trouble in showing that $k(t)=p(t)$ for sufficiently large $t$. I would appreciate it if someone could help me proving this equality for large time $t$.

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Here are some useful hints, given that I think this is a homework question you're asking about. Use d'Alembert's formula for the solution for $u(x,t)$, which gives $$ u(x,t) = \frac{1}{2}\left[g(x+t) + g(x-t)\right] + \frac{1}{2} \int_{x-t}^{x+t} h(y) \, dy$$ Now calculate $u_t$ and $u_x$ from this, plug those into your expressions for $k(t)$ and $p(t)$, and try calculating $k(t) - p(t)$ for large time $t$. Remember that you're given that $g,h$ are compactly supported functions, so somewhere down the line you're going to be using this information.

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  • $\begingroup$ Thank you for kind guidance, Christopher. I got the correct conclusion. This problem was a quite surprise to me, as the kinetic and potential energy coincide alter a long time. $\endgroup$ – Pooya Oct 8 '12 at 19:05

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