Main: The Wave equation
The one dimensional wave equation (with no source term) is
where c is the wave speed. (From purely dimensional
arguments, c must have dimensions
m s-1 and hence must represent a
speed.)
The general solution, known as d'Alembert's solution, is
which can be easily verified, using the chain rule. It is, however, so
easy to derive that it is worthwhile going through the derivation:
Introduce new variables
Then the wave equation can be written
Now, this immediately integrates to
and then
This can also be written in the form
which is slightly more convenient in what follows.
The important points are that:
- F(x-ct) represents a wave of shape F(x) (at time t=0)
moving in the direction of increasing x at constant speed c. That
is, F(x-ct) represents the same shape as F(x), but with the origin
moved to x=ct, ie the same shape as F(x) translated by ct units
in the positive x direction.
- E(x+ct)
represents a wave of shape E(x) (at time t=0) moving in
the direction of decreasing x at constant speed c. That is,
E(x+ct) represents the same shape as E(x), but with the origin
translated to x=-ct.
Main: The Wave equation
