$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 6.3: Examples

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Now let me look at two examples

Example $$\PageIndex{1}$$

Find graphically a solution to

\begin{aligned} \dfrac{\partial^2}{\partial t^2} u &= \dfrac{\partial^2}{\partial x^2} u\;\; (c=1 \text{m/s}) \nonumber\\[4pt] u(x,0) & = \begin{cases} 2x& \text{if 0 \leq x \leq 2} \\ 24/5-2x/5 & \text{if 2 \leq x \leq 12} \end{cases} \quad.\nonumber\\[4pt] \dfrac{\partial}{\partial t} u (x,0) &= 0\nonumber\\[4pt] u(0,t) &=u(12,t) = 0\end{aligned}

Solution

We need to continue $$f$$ as an odd function, and we can take $$\Gamma=0$$. We then have to add the left-moving wave $$\dfrac{1}{2} f(x+t)$$ and the right-moving wave $$\dfrac{1}{2} f(x-t)$$, as we have done in Figs. ???

Example $$\PageIndex{2}$$

Find graphically a solution to

\begin{aligned} \dfrac{\partial^2}{\partial t^2} u &= \dfrac{\partial^2}{\partial x^2} u\;\; (c=1 \text{m/s}) \nonumber\\[4pt] u(x,0) & = 0\nonumber\\[4pt] \dfrac{\partial}{\partial t} u (x,0) &= \begin{cases} 1& \text{if 4 \leq x \leq 6} \\ 0 & \text{elsewhere} \end{cases} \quad.\nonumber\\[4pt] u(0,t) &=u(12,t) = 0.\end{aligned}

Solution

In this case $$f=0$$. We find \begin{aligned} \Gamma(x) &= \int_0^x g(x') dx'\nonumber\\[4pt] &= \begin{cases} 0 & \text{if 0<x<4}\\ -4+x & \text{if 4<x<6}\\ 2 & \text{if 6<x<12} \end{cases}.\end{aligned} This needs to be continued as an even function.