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I am still working to check your derivation. I am explicitly stating most of the steps you made like: $\xi=\frac{\delta\rho}{\rho_0}$ rearranges to $delta\rho=\rho_0\xi$. And substitutes into $p=p_0+C^2\delta\rho$ : $p=p_0+C^2\rho_0\xi$ I will confirm your answer today.
@Carlo Beenakker, thank you for the observation. Could that neglect originate from the author's search for solutions of low limited amplitude? You might have deduced the assumption from the solution given, that is extremely skillful. Or are there more consequences if the assumption (from text): "At low amplitude this reduces to the wave equation: $\partial^2 \rho - c^2\nabla^2\rho = 0$ (2)" from page 1. Is this sentence correctly written as: $u \ll 1$? If the assumption is the cause I will update that into my question. And is it the wave equation the source of the Bessel function solution?
@S. Carnahan , thank you for the resources suggestions, I will take to heart. Do you mean the Bessel function is an approximate solution because they require extra assumptions to inviscid flow? Like small amplitudes? Or do you mean the Bessel function itself cannot be calculated with much accuracy?
See what I mean? Just because of the connection to QM someone made an idiotic vote that the question was not clear or useful. Or am I missing something here?
@The User I thought he was being a little sarcastic by the way he worded that if I "want to be taken seriously rather than viewed as a crank", I should fully understand Quantum Mechanics like him. But I have been arguing this topic a lot lately to know that my question doesn't entail me being "viewed as a crank". He is the one thats being a crank by bashing on a legitimate mathematical question because he thinks the long-term goals are not achievable. Or the philosophy of it is inconsistent in his view. I didn't delve in such topic in my question exactly because of this.
Not useful at all! Your comment is completely off topic. And not original at all. Really. This is not the place for this discussion. If you want to have it, send an email for Robert Brady. If you just want to bash him, join every one at: scottaaronson.com/blog/?p=1255 (Get cred with Scott Aahoronson while you do your rants.) Or if you want to know why this approach is worth it and probably right: vzn1.wordpress.com/2013/02/20/…
