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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.