###[Problems](id:sect-2.6.P) > 1. [Problem 1](#problem-2.6.P.1) > 2. [Problem 2](#problem-2.6.P.2) > 3. [Problem 3](#problem-2.6.P.3) > 4. [Problem 4](#problem-2.6.P.4) > 5. [Problem 5](#problem-2.6.P.5) > 6. [Problem 6](#problem-2.6.P.6) > 7. [Problem 7](#problem-2.6.P.7) > 8. [Problem 8](#problem-2.6.P.8) > 9. [Problem 9](#problem-2.6.P.9) **[Problem 1.](id:problem-2.6.P.1)** a. Find solution \begin{align\*} & u\_{tt}-c^2u\_{xx}=0, &&t\>0, x\>0, \\\\\ &u|\_{t=0}= \phi (x), &&x\>0, \\\\ &u\_t|\_{t=0}= c\phi'(x), &&x\>0, \\\\ &u|\_{x=0}=\chi(t), &&t\>0. \end{align\*} (separately in $x \> ct$ and $0 \< x \< ct$). b. Find solution \begin{align\*} & u\_{tt}-c^2u\_{xx}=0, &&t\>0, x\>0, \\\\\ &u|\_{t=0}= \phi (x), &&x\>0, \\\\ &u\_t|\_{t=0}= c\phi'(x), &&x\>0, \\\\ &u\_x|\_{x=0}=\chi(t), &&t\>0. \end{align\*} separately in $x \> ct$ and $0 \< x \< ct$. **[Problem 2.](id:problem-2.6.P.2)** a. Find solution \begin{align\*} & u\_{tt}-c\_1^2u\_{xx}=0, &&&t\>0, x\>0, \\\\ & u\_{tt}-c\_2^2u\_{xx}=0, &&&t\>0, x\<0, \\\\ &u|\_{t=0}= \phi (x), &&u\_t|\_{t=0}= c\_1\phi'(x) &x\>0, \\\\ &u|\_{t=0}=0, &&u\_t|\_{t=0}=0, &x\<0,\\\\ &u|\_{x=+0}=\alpha u|\_{x=-0}, &&u\_x|\_{x=+0}=\beta u\_x|\_{x=-0} &t\>0 \end{align\*} (separately in $x \> c\_1 t$, $0 \< x \< c\_1 t$, $-c\_2t \< x \< 0$ and $x \< -c\_2t$). b. Discuss *reflected wave* and *refracted wave*. In particular, consider $\phi(x)=e^{ik x}$. **[Problem 3.](id:problem-2.6.P.3)** a. 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 $0 \< x \< c\_2t$). b. Discuss two *reflected waves*. In particular, consider $\phi(x)=e^{ik x}$. c. Discuss connection to [Problem 2](#problem-2.6.P.2). **[Problem 4.](id:problem-2.6.P.4)** a. Find solution \begin{align\*} & u\_{tt}-c^2u\_{xx}=0, &&&t\>0, x\>0, \\\\ &u|\_{t=0}= \phi (x), &&u\_t|\_{t=0}= c\_1\phi'(x) &x\>0, \\\\ &(u\_x+\alpha u)|\_{x=0}=0, &&&t\>0 \end{align\*} (separately in $x \> c\_1 t$, and $0 \< x \< c\_1 t$). b. Discuss *reflected wave*. In particular, consider $\phi(x)=e^{ik x}$. **[Problem 5.](id:problem-2.6.P.5)** a. Find solution \begin{align\*} & u\_{tt}-c^2u\_{xx}=0, &&&t\>0, x\>0, \\\\ &u|\_{t=0}= \phi (x), &&u\_t|\_{t=0}= c\_1\phi'(x) &x\>0, \\\\ &(u\_x+\alpha u\_{t})|\_{x=0}=0, &&&t\>0 \end{align\*} (separately in $x \> c\_1 t$, and $0 \< x \< c\_1 t$). b. Discuss *reflected wave*. In particular, consider $\phi(x)=e^{ik x}$. **[Problem 6.](id:problem-2.6.P.6)** a. Find solution \begin{align\*} & u\_{tt}-c\_1^2u\_{xx}=0, &&&t\>0, x\>0, \\\\ &u|\_{t=0}= \phi (x), &&u\_t|\_{t=0}= c\_1\phi'(x) &x\>0, \\\\ &(u\_x+\alpha u\_{tt})|\_{x=0}=0, &&&t\>0 \end{align\*} (separately in $x \> c\_1 t$, and $0 \< x \< c\_1 t$). b. Discuss *reflected wave*. In particular, consider $\phi(x)=e^{ik x}$. **[Problem 7.](id:problem-2.6.P.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} a. 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: a. None, b. $u|\_{x=vt}=0$ ($t\>0$), c. $(\alpha u_x +\beta u\_t)|\_{x=vt}=0$ ($t\>0$), d. $u|\_{x=vt}=u\_x|\_{x=vt}=0$ ($t\>0$).
Consider cases $v\>c$, $-c \< v \< c$ and $v \< -c$. In the case condition (3) find necessary restrictions to $\alpha,\beta$. b. 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.](id:problem-2.6.P.8)** By method of continuation combined with D'Alembert formula solve each of the following four problems (a)--(d). a. \begin{equation\*} \left\\{\begin{aligned} &u\_{tt}-9u\_{xx}=0, \qquad &&x\>0,\\\\ &u|\_{t=0}=0, \qquad &&x\>0,\\\\ &u\_t|\_{t=0}=\cos (x), \qquad &&x\>0,\\\\ &u|\_{x=0}=0, \qquad &&t\>0. \end{aligned}\right. \end{equation\*} b. \begin{equation\*} \left\\{\begin{aligned} &u\_{tt}-9u\_{xx}=0, \qquad &&x\>0,\\\\ &u|\_{t=0}=0, \qquad &&x\>0,\\\\ &u\_t|\_{t=0}=\cos (x), \qquad &&x\>0,\\\\ &u\_x|\_{x=0}=0, \qquad &&t\>0. \end{aligned}\right. \end{equation\*} c. \begin{equation\*} \left\\{\begin{aligned} &u\_{tt}-9u\_{xx}=0, \qquad &&x\>0,\\\\ &u|\_{t=0}=0, \qquad &&x\>0,\\\\ &u\_t|\_{t=0}=\sin(x), \qquad &&x\>0,\\\\ &u|\_{x=0}=0, \qquad &&t\>0. \end{aligned}\right. \end{equation\*} d. \begin{equation\*} \left\\{\begin{aligned} &u\_{tt}-9u\_{xx}=0, \qquad &&x\>0,\\\\ &u|\_{t=0}=0, \qquad &&x\>0,\\\\ &u\_t|\_{t=0}=\sin(x), \qquad &&x\>0,\\\\ &u\_x|\_{x=0}=0, \qquad &&t\>0. \end{aligned}\right. \end{equation\*} **[Problem 9.](id:problem-2.6.P.9)** Solve \begin{align\*} &(t^2+1)u\_{tt}+tu\_t-u\_{xx}=0,\\\\[3pt] &u|\_{t=0}=0, \qquad u\_t|\_{t=0}=1. \end{align\*} *Hint*: Make a change of variables $x=\frac{1}{2}(\xi+\eta)$, $t=\sinh (\frac{1}{2}(\xi-\eta))$ and calculate $u\_\xi$, $u\_\eta$, $u\_{\xi\eta}$. ______________ [$\Uparrow$](../contents.html)  [$\uparrow$](./S2.6.html)  [$\Rightarrow$](./S2.7.html)