### Author Topic: HA10 Problem 5  (Read 1984 times)

#### Victor Ivrii

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##### HA10 Problem 5
« on: March 26, 2015, 03:13:13 PM »
Consider wave equation

\rho u_{tt} - (k(x)u_x)_x=0 \label{eq-10.5}

on $-\infty < x < \infty$ where
\rho(x)=\left\{\begin{aligned}\rho_-(x)& &&x<0,\\ \rho_+(x)& &&x>0\end{aligned}\right. and k(x)=\left\{\begin{aligned}k_-(x)& &&x<0,\\ k_+(x)& &&x>0.\end{aligned}\right.

a. Write down equation (\ref{eq-10.5}) for $x< 0$ and $x > 0$ separately.
b. Find out *transmission conditions* (there must be 2 of them) linking $u(-0,t)$, $u(+0,t)$, $u_x(-0,t)$, $u_x(+0,t)$.

#### Victor Ivrii

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##### Re: HA10 Problem 5
« Reply #1 on: April 03, 2015, 03:56:08 PM »
a. Equations are easy

\rho_\pm u_{tt}- k_\pm u_{xx}=0\qquad \text{as } \pm x>0.

b. Transmission conditions are a bit more difficult. First $u_x$ must be a function, not a distribution and therefore

u_x(-0,t)=u_x(+0,t).
\label{P}

Next, $\lim _{x\to \pm 0}(k (x)u_x)=k_\pm u_x (\pm 0,t)$ and it must not to have jums so

k_-u_x(-0,t)=k_+ u_x(+0,t).
\label{Q}

(\ref{P}) and (\ref{Q}) are transmission conditions