About Dirac function

Question

In Vladimirov's book "A Collection of Problems on the Equations of Mathematical Physics", p129, 11.16, there is a equality about Dirac function, which is the fundamental solution of three dimensional wave equation, $$\frac{\theta(t)}{4\pi a^2t}\delta_{S_{at}}(x)=\frac{\theta(t)}{2\pi a}\delta(a^2t^2-|x|^2),$$ where $(t,x)\in\mathbb{R}\times\mathbb{R}^3$, $\theta$ is Heaviside function, $S_{at}:\{x\in\mathbb{R}^3;|x|=at\}.$

It confuses me, since I always regard $\delta_{S_{at}}(x)$ the same as $\delta(at-|x|)$, $\delta(a^2t^2-|x|^2)$.

Answer 1

Your notion and the equality in the book are consistent, since $$ \delta (a^2 t^2 - |x|^2 ) = \frac{1}{2|at|} (\delta (at-|x|) + \delta (at+|x|) ) $$ (I suppose an assumption is being made that $at>0$).