
Separation of Variables
Our goal is to solve the wave equation, which
is of the form:
Let us assume that the solution can be written
as a product of a function of time (t) and a function of
space (x). W(x,t) = f(t)g(x). Then we can easily recompute
the partials and rewrite the wave equation like this:
Let us now divide everything by f(t)g(x). We now get this:
We see that we have two terms, where one depends only on x,
and the other depends only on t. Since the equation has to be
true at all times and at any point in space, we conclude that
these two terms are constant (let's call this
constant D)  otherwise we could vary time
and change the first term, but the second term would not
change if we didn't change x, and then the two terms would not
add up to zero. So from this we get two equations,
which are now ordinary differential equations because each
depends only on one variable:
 BACK 
