Three dimensional Wiener-Khinchin formula for incompressible isotropic random field

When I am reading Uriel Frisch's book "Turbulence", on page 55 he claimed that the Wiener-Khinchin formula for an incompressible isotropic random field, such as the velocity field of incompressible fluid based on the Navier-Stokes equation, is in the form of

$$E(k)=4\pi k^2 E_{3D}(k)=\frac{1}{\pi}\int_0^\infty k\rho \Gamma(\rho)\sin(k\rho)d\rho$$

where $$\Gamma(\rho)=<\pmb v(\pmb r)\cdot \pmb v(\pmb r')>$$ is the correlation function of two points in this random field and $$\rho=|\pmb r-\pmb r'|$$ denotes the distance between these two points. $$\pmb r$$ is the coordinates, $$k$$ is the wavenumber and $$E(k)$$ is the energy spectrum.

However, there is no reference or derivation procedure for this equation. Could you please shed light on me or give me some references for the following questions?

1. Is there an elementary derivation for this equation?
2. When I tried to link this equation to the one-dimensional Wiener-Khinchin formula $$E(f)=\frac{1}{2\pi}\int^\infty_{-\infty}e^{ifs}\Gamma(s)ds\,,$$ I found it difficult to explain why there is a $$\rho$$ in the integration of the three-dimensional version?
3. Continuing with the last question, why the three-dimensional version uses $$\sin$$ instead of $$e$$ in the integration?

I shall preface this by saying I am not an expert in these topics so feel free to correct any mistakes I might have made. I will try and answer your questions one by one:

1. Is there an elementary derivation?

The derivation of the formula stated in your question is not at all elementary and depends on the theory of homogeneous isotropic turbulence. One common way to derive it would be to start with the Fourier transform of the velocity field and then calculate the auto-correlation function in Fourier space. This will lead to a relationship between the energy spectrum and the auto-correlation function.

1. Why is there a $$\rho$$ in the integration?

The $$\rho$$ term in the integral is related to the spherical symmetry of the problem. In three dimensions, the auto-correlation function depends on the distance between the points, not their individual positions, hence the use of $$\rho=|\pmb r-\pmb r'|$$. The integration is over all possible separations, hence the $$\rho$$ in the integral.

When performing the integration, one integrates over a spherical shell in 3D space, which has a volume proportional to $$\rho^2$$ (or $$k^2$$ in wavenumber space). This is the source of the $$k$$ in the integral, and the $$k^2$$ in the $$4\pi k^2 E_{3D}(k)$$ term.

1. Why use $$\sin$$ instead of $$e$$?

The $$\sin(k\rho)$$ function appears because of the properties of the Fourier transform in three dimensions. In one dimension, the Fourier transform involves the complex exponential function $$e^{ikx}$$ as its basis or as it is more familiar to EEs, $$e^{j\omega t}$$. In three dimensions, however, the Fourier transform of a spherically symmetric function involves the spherical Bessel function, which for the zeroth order is proportional to $$\sin(k\rho)$$.

In more technical terms, this has to do with the fact that the Fourier transform of a radial function in 3D coordinates yields a function involving the spherical Bessel function, which behaves like $$\sin(k\rho)$$ for sufficiently large arguments.

For more detailed discussions and complex derivations, you may want to refer to more advanced textbooks on turbulence theory. For example, Turbulent Flows by Stephen B. Pope provides a comprehensive treatise on these topics.