3D Wave Equation

Subjects: Fourier Analysis, Partial Differential Equations
Links: The Wave equation, Fourier Transform in Rn, Scalar Surface Integral

If ${\mathbb{S}}^2$ denotes the unit sphere in ${\mathbb{R}}^3$, we define the spherical mean of $f$ over the sphere of radius $t$ centred at $x$ by $$M_t(f)(x) = \frac{1}{4\pi} \int_{\Bbb S^2} f(x-t\gamma), d\sigma(\gamma)$$where $d\sigma (\gamma )$ is the element of surface area for ${\mathbb{S}}^2$. 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 $$M_t(f)(x) ) \frac1{|S(x,t)|} \int_{S(x, t)} f(\gamma),d\sigma(\gamma)$$where $S(x,t)$ denotes the sphere of centre $x$ and radius $t$ and $|S(x,t)$ its area.

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

Lemma: $$\frac1{4\pi} \int_{\Bbb S^2} e^{-2\pi i \omega\cdot \gamma}, d\sigma(\gamma) = \frac{\sin(2\pi|\omega|)}{2\pi|\omega|}$$
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. $$\widehat{M_t(f)}(\omega) = \hat f(\omega) \frac{\sin(2\pi |\omega| t)}{2\pi |\omega| t}$$
Th: The solution when $n=3$ of the Cauchy problem wave equation $$\Delta u = \frac{\partial^2 u}{\partial t^2} \quad \text{subject to} \quad u(x, 0) = f(x) \quad \text{and}\quad \frac{\partial u}{\partial t} (x, 0)=g(x)$$
is given by $$u(x,t) = \frac{\partial}{\partial t}(t M_t(f)(x)) + t M_t(g)(x)$$
We can get another way to solve to write the solution, using the other way to write the spherical mean of $f$, getting $$u(x, t) = \frac1{|S(x,t)}\int_{S(x,t)} [tg(y)+f(y)+ \nabla f(y) \cdot (y-x)], d\sigma(y)$$
This alternate expression for the solution of the wave equation is sometimes called Kirckchoff's formula