Delta distributions with nonlinear arguments I am confused by the use of nonlinear arguments with the Dirac $\delta$ distribution that I am encountering in the literature. This looks like a widespread use, but for concreteness let us focus on a single example. In this paper by Tataru and Geba, the very first formula is
$$\tag{1} K(t, x)=c_n 1_{t>0} \begin{cases} (t^2-x^2)_+^{-\frac{n-1}{2}} & n \text{ even} \\
\delta^{\left(\frac{n-3}{2}\right)} (t^2-x^2) & n\text{ odd}
\end{cases}$$

Big question. How to interpret this formula?


According to the linked paper, formula (1) is the solution to this problem: 
\begin{equation}
\begin{cases} 
\frac{\partial^2 u}{\partial t^2}-\Delta u =0 \\
u(0)=0,\ \frac{\partial u}{\partial t}(0)=\delta(x).
\end{cases}
\end{equation}
To gain some insight, I tried to restrict myself to the $n=3$ case, where Kirchhoff's formula tells us that 
$$K(t, x)=c_3 1_{t>0} \frac{1}{t} d\sigma_t, $$
where $d\sigma_t$ is the surface measure on the sphere of radius $t$. 

Small question. Equating Kirchhoff's formula and formula (1) I infer that the following identity should hold true: $$\tag{?} \delta(t^2-\lvert x\rvert^2)=\frac{1}{t}d\sigma_t, \qquad x \in \mathbb{R}^3,\ t>0.$$
  Can we prove (?) directly?

Understanding the identity (?) could be a good start towards the understanding of the more general formula (1), in which the superscript ${n-3\over 2}$ does not vanish. 
P.S. The answer to the questions in this post might lie somewhere in Chapter 5 of the book "The Fourier transform and its applications" by Bracewell. 

I suppose (key word here is suppose) that if we have the function $f(t,x)$ on $\Bbb R\times \Bbb R^d$ such that the equation $f(t,x)$ gives a manifold $M$ of dimension $d$ without border (problems arise when we have degenerations like $t^2-x^2=0$ in the point $(0,0)$), then we can define $\delta(f(t,x))$ as a measure on this manifold $M$.
If we need $\delta^{(k)}(f(t,x))$, then we can define it as $(\delta^{(k)}(f(t,x)),\phi) =(-1)^k (\delta (f(t,x)),\frac{\partial^k\phi}{\partial n^k}) $ with $\frac{\partial^k\phi}{\partial n^k}$ being the derivative of $\phi$ in the direction of normal vector to $M$.
