6.3: Examples

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Now let me look at two examples

Example $\PageIndex{1}$

Find graphically a solution to

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

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

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

Solution

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