Editing Wave equation (section)

=== Solutions in ''D = 1, 2, 3'' ===
When <math>D=1</math>, the integrand in the Fourier transform is the [[sinc function]]<math display="block">\begin{aligned}
G_1(t, x) &= \frac{1}{2\pi} \int_\R \frac{\sin(|\omega| t)}{|\omega|} e^{i\omega x}d\omega \\
&= \frac{1}{2\pi} \int \operatorname{sinc}(\omega) e^{i \omega \frac xt} d\omega \\
&= \frac{\sgn(t-x) + \sgn(t+x)}{4} \\
&= \begin{cases}
\frac 12 \theta(t-|x|) \quad t > 0 \\
-\frac 12 \theta(-t-|x|) \quad t < 0
\end{cases}
\end{aligned}</math>
where <math>\sgn</math> is the [[sign function]] and <math>\theta</math> is the [[Heaviside step function|unit step function]]. One solution is the forward solution, the other is the backward solution.

The dimension can be raised to give the <math>D=3</math> case<math display="block">G_3(t, r) = \frac{\delta(t-r)}{4\pi r}</math>and similarly for the backward solution. This can be integrated down by one dimension to give the <math>D=2</math> case<math display="block">G_2(t, r) = \int_\R \frac{\delta(t - \sqrt{r^2 + z^2})}{4\pi \sqrt{r^2 + z^2}} dz 
= \frac{\theta(t - r)}{2\pi \sqrt{t^2 - r^2}}
</math>