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The first solution we can discern by noting that the wave equation
equates a second derivative in time to a second derivative in space.
Suppose we write the solution as where is an unknown function
of and and substitute it into the differential equation and use
the chain rule:
|
(108) |
or
|
(109) |
|
(110) |
with a simple solution:
|
(111) |
What this tells us is that any function
|
(112) |
satisfies the wave equation. Any shape of wave created on the
string and propagating to the right or left is a solution to the wave
equation, although not all of these waves will vanish at the ends of a
string.
Next: Harmonic Waveforms Propagating to
Up: Solutions to the Wave
Previous: An Important Property of
  Contents
Robert G. Brown
2004-04-12