6.3 Examples

Now let me look at two examples

Example 6.1: 

Find graphically a solution to

\begin{eqnarray} {{∂}^{2}u\over ∂{t}^{2}} & =& {{∂}^{2}u\over ∂{x}^{2}}\kern 2.77695pt \kern 2.77695pt (c = 1\text{m/s}) %& \\ u(x,0)& =& \left \{\array{ 2x \quad &\text{if $0 ≤ x ≤ 2$} \cr 24∕5 − 2x∕5\quad &\text{if $2 ≤ x ≤ 12$}} \right .\quad .%& \\ {∂u\over ∂t} (x,0)& =& 0 %& \\ u(0,t)& =& u(12,t) = 0 %&(6.18)\\ \end{eqnarray}

Solution: 

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

Example 6.2: 

Find graphically a solution to

\begin{eqnarray} {{∂}^{2}u\over ∂{t}^{2}} & =& {{∂}^{2}u\over ∂{x}^{2}}\kern 2.77695pt \kern 2.77695pt (c = 1\text{m/s}) %& \\ u(x,0)& =& 0 %& \\ {∂u\over ∂t} (x,0)& =& \left \{\array{ 1\quad &\text{if $4 ≤ x ≤ 6$}\cr 0\quad &\text{elsewhere}} \right .\quad .%& \\ u(0,t)& =& u(12,t) = 0. %&(6.19)\\ \end{eqnarray}

Solution: 

In this case f = 0. We find

\eqalignno{ Γ(x) & ={\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{x}g(x')dx' & & \cr & = \left \{\array{ 0 \quad &\text{if $0 < x < 4$} \cr −4 + x\quad &\text{if $4 < x < 6$} \cr 2 \quad &\text{if $6 < x < 12$}} \right .. &\text{(6.20)} }

This needs to be continued as an even function.