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**Final Exam / problem 4**

« **on:**December 20, 2012, 01:33:07 PM »

Let $u$ solve the initial value problem for the wave equation in one dimension

\begin{equation*}

\left\{\begin{aligned}

& u_{tt}- u_{xx}= 0 ,\qquad&& ~{\mbox{in}} ~\mathbb{R} \times (0,\infty),\\[3pt]

&u (0,x) = f(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} ,\\[3pt]

&u_t(0,x)= g(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} .

\end{aligned}\right.

\end{equation*}

Suppose $f(x)=g(x)=0$ for all $|x|>1000.$ The

$$

k(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_t^2 (t,x) dx

$$

and the

$$

p(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_x^2 (t,x) dx.

$$

Prove

problem 4 part (a)

\begin{equation*}

\left\{\begin{aligned}

& u_{tt}- u_{xx}= 0 ,\qquad&& ~{\mbox{in}} ~\mathbb{R} \times (0,\infty),\\[3pt]

&u (0,x) = f(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} ,\\[3pt]

&u_t(0,x)= g(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} .

\end{aligned}\right.

\end{equation*}

Suppose $f(x)=g(x)=0$ for all $|x|>1000.$ The

*kinetic energy*is$$

k(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_t^2 (t,x) dx

$$

and the

*potential energ*y is$$

p(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_x^2 (t,x) dx.

$$

Prove

- $k(t)+ p(t)$ is constant with $t$ (so does not change as $t$ changes),
- $k(t)=p(t)$ for all large enough times $t$.

problem 4 part (a)