Discuss reflected wave and refracted wave. In particular, consider $\phi(x)=e^{ik x}$.
Problem 3.
1. Find solution
\begin{align*}
& u_{tt}-c_1^2u_{xx}=0, &&&t > 0, x > 0, \\
& v_{tt}-c_2^2v_{xx}=0, &&&t > 0, x > 0, \\
&u|_{t=0}= \phi (x), &&u_t|_{t=0}= c_1\phi'(x) &x > 0, \\
&v|_{t=0}=0, &&v_t|_{t=0}=0, &x > 0,\\
&(u+\alpha v)|_{x=0}=0, &&(u_x+\beta v_x)|_{x=0}=0 &t > 0
\end{align*}
(for $u$ separately in $x > c_1t$, and for $v$ separately in $0 < x < c_2t$).
2. Discuss two reflected waves. In particular, consider $\phi(x)=e^{ik x}$.
3. Discuss connection to Problem 2.
Problem 4.
Find solution
\begin{align*}
& u_{tt}-c^2u_{xx}=0, &&&t > 0, x > 0, \\
&u|_{t=0}= \phi (x), &&u_t|_{t=0}= c\phi'(x) &x > 0, \\
&(u_x+\alpha u)|_{x=0}=0, &&&t > 0
\end{align*}
(separately in $x > ct$, and $0 < x < c t$).
Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.
Problem 5.
Find solution
\begin{align*}
& u_{tt}-c^2u_{xx}=0, &&&t > 0, x > 0, \\
&u|_{t=0}= \phi (x), &&u_t|_{t=0}= c\phi'(x) &x > 0, \\
&(u_x+\alpha u_{t})|_{x=0}=0, &&&t > 0
\end{align*}
(separately in $x > c t$, and $0 < x < c t$).
Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.
Problem 6.
Find solution
\begin{align*}
& u_{tt}-c ^2u_{xx}=0, &&&t > 0, x > 0, \\
&u|_{t=0}= \phi (x), &&u_t|_{t=0}= c \phi'(x) &x > 0, \\
&(u_x-\alpha u_{tt})|_{x=0}=0, &&&t > 0
\end{align*}
(separately in $x > c t$, and $0 < x < c t$).
Discuss reflected wave. In particular, consider $\phi(x)=e^{ik x}$.
Problem 7.
Consider equation with the initial conditions
\begin{align}
& u_{tt}-c^2 u_{xx}=0,\qquad &&t > 0, x > vt,\label{3a} \\
&u|_{t=0}= f(x), \qquad &&x > 0, \label{3b}\\
&u_t|_{t=0}= g(x), \qquad &&x>0.\label{3c}
\end{align}
Find which of these conditions (a)-(d) at $x=vt$,
$t > 0$ could be added to (\ref{3a})-(\ref{3b}) so that the
resulting problem would have a unique solution and solve the problem you deemed as a good one:
None,
$u|_{x=vt}=0$ ($t > 0$),
$(\alpha u_x +\beta u_t)|_{x=vt}=0$ ($t > 0$),
$u|_{x=vt}=u_x|_{x=vt}=0$ ($t>0$).
Consider cases $v > c$, $-c < v < c$ and $v < -c$. In the case condition (3) find necessary restrictions to $\alpha,\beta$.
Find solution in the cases when it exists and is uniquely determined; consider separately zones $x > ct$, $-ct < x < ct $ and $x > ct$ (intersected with $x > vt$).
Problem 8.
By method of continuation combined with D'Alembert formula solve each of
the following four problems (a)--(d).