In this topic, we will show you how can we use Fourier transform to solve initial value problem for wave equation in . Following is the problem
From the equation by taking Fourier transform to the both sides, we obtain
This is an ODE, the solution is given by
From the initial date we get
which implies that
.
From the initial date we see that
.
Thus, we obtain
.
Note that
.
Moreover,
.
Then
.
Since
and
then
.
Thus,
or equivalently,
.
This is the so-call D’ Alembert formula.