As an example of a hyperbolic equation we study the wave equation. One of the systems it can describe is a transmission line for high frequency signals, 40m long.
Separate variables,
(5.19) |
We find
(5.20) |
Which in turn shows that
We can also separate most of the initial and boundary conditions; we find
(5.22) |
Once again distinguish the three cases
,
, and
:
(almost identical to previous problem)
,
,
. We find that
(5.23) |
implies , and taking both together we find (for )
(5.24) |
. due to the boundary conditions. We find that , and is 0 due to initial condition. We conclude that
(5.25) |
No solution.
Taking everything together we find that
(5.26) |
The one remaining initial condition gives
(5.27) |
Use the Fourier cosine series (even continuation of ) to find