The question that follows has to do with the effects of a turbulent atmosphere on wave propagation. The structure function, $D(\vec{r})$, which is defined as,
\begin{equation} D(\vec{r}) = \left\langle \left[ n(\vec{x} - \vec{d}) - n(\vec{x}) \right]^2\right\rangle \, , \end{equation}
and is what our instruments observe, is related to the power spectrum, $\Phi(\vec{k})$, thought
\begin{equation} D(\vec{r}) = 2 \int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \Phi(\vec{k}) [1 - \cos(\vec{k} \cdot \vec{r})] d^3k \quad (1) \end{equation}
According to the textbook I am reading on the topic, I can obtain the inverse of (1) by operating with a Laplacian on it, which results in,
\begin{equation} \Phi(\vec{k}) = \frac{1}{16 \pi^3 k^2} \int_0^{\infty} \int_0^{\infty} \int_0^{\infty} \cos(\vec{k} \cdot \vec{r}) \, \Delta D(\vec{r}) d^3r \, , \quad (2) \end{equation}
where $\Delta$ is the Laplace operator. Under the assumption of a locally isotropic field, I can introduce spherical coordinates and integrate over the angular variables to obtain
\begin{equation} \Phi(k) = \frac{1}{4 \pi^2 k^2} \int_0^{\infty} \frac{\sin(kr)}{kr} \frac{d}{dr} \left[ r^2 \frac{d}{dr} D(r) \right] dr \, . \quad (3) \end{equation}
Objective: Derive the equivalent formula to Eq. (3) in the 2D case.
Question: My calculus is a bit rusty so I am struggling to work out how to derive Eq. (2) from Eq. (1). I assume Eq. (2) above is for a 3D field and the relation will be different in the 2D case (correct me if I am wrong on this). If I have Eq. (2) in 2D, I can easily derive Eq. (2) by introducing polar coordinates.