Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

6.3: Examples

Now let me look at two examples

Example \(\PageIndex{1}\)

Find graphically a solution to

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

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 \(\PageIndex{1}\)

Find graphically a solution to

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

Solution

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