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The most general solution of the wave equation (10) is of
the form
|
u(x,t) = uR(x-ct) + uL(x+ct).
|
(14) |
The contribution uR is a function of x that maintains a fixed shape
but propagates towards the right with speed c, while uL
propagates to the left with that speed. In other words, u(x,t)
always consists of two travelling wave moving in opposite directions.
In case the initial velocity is zero, half of the initial displacement
will move toward the right and half toward the left.
It is possible to take the wave equation apart into two equations,
each of which is a one-way wave equation. One of these will
describe the left going wave and the other the right going one.
Next: Smoothness of the
Up: The wave equation.
Previous: The initial conditions.