3D Wave Equation

If ${\mathbb{S}}^{\mathbb{2}}$ denotes the unit sphere in ${\mathbb{R}}^{\mathbb{3}}$, we define the spherical mean of $f$ over the sphere of radius $t$ centred at $x$ by

where $d\sigma (\gamma )$ is the element of surface area for ${\mathbb{S}}^{\mathbb{2}}$. Since $4\pi$ is the area of the unit sphere, we can interpret $M_t(f)$ as the average value of $f$ over the sphere centred at $x$ with radius $t$

We can also write this in another form

where $S(x,t)$ denotes the sphere of centre $x$ and radius $t$ and $|S(x,t)$ its area.

Lemma: If $f\in \mathcal{S}({\mathbb{R}}^{\mathbb{3}})$ and $t$ is fixed, then $M_t(f)\in \mathcal{S}({\mathbb{R}}^{\mathbb{3}})$. Moreover $M_t(f)$ is infinitely differentiable in $t$, and each $t$-derivative also belongs to $\mathcal{S}({\mathbb{R}}^{\mathbb{3}})$

Lemma: $\frac{1}{4\pi }{\int }_{{\mathbb{S}}^{\mathbb{2}}}{e}^{-2\pi i\omega \cdot \gamma }d\sigma (\gamma )=\frac{\sin (2\pi \parallel \omega \parallel )}{2\pi \parallel \omega \parallel }$

By the defining formula for the spherical mean, we may interpret $M_t(f)$ as a convolution of the function $f$ with the element $d\sigma$, and since the Fourier transform interchanges convolutions with products, we are lead to believe that $\widehat{M_t(f)}$ is the product of the corresponding Fourier transforms.

Th: The solution when $n=3$ of the Cauchy problem wave equation

We can get another way to solve to write the solution, using the other way to write the spherical mean of $f$, getting

This alternate expression for the solution of the wave equation is sometimes called Kirckchoff's formula