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Curle Analogy Function object

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Old   October 22, 2021, 05:33
Default Curle Analogy Function object
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Alberto Artoni
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I was trying to understand why the Curle analogy is implement as it has been implemented:

p' = \frac{1}{4\pi} \frac{\mathbf{r}}{|\mathbf{r}|^2}\large( \frac{\mathbf{F}}{|\mathbf{r}|}   + \frac{1}{c_0}\frac{d\mathbf{F}}{dt} \large) .

The actual formulation by Curle can be seen as:

p' = \frac{1}{4\pi}\nabla\cdot\nabla\cdot \int_{V} \frac{\mathbf{T}}{r} dV - \frac{1}{4\pi}\nabla\cdot \int_{\partial V} (\frac{p'\mathbf{I}}{r}) \cdot \mathbf{n} d\partial V

I can see that if we are far enough, we can replace the divergence with -\frac{1}{c_0}\frac{\mathbf{r}}{|\mathbf{r}|}\frac{\partial}{\partial t}.

I don't understand:
1) why was the volume term omitted
2) from where the term \frac{\mathbf{F}}{|\mathbf{r}|} comes from.

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aeroacoustics, curle

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