I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism <http://arxiv.org/abs/1301.7540>


So it begins with Euler Equations of inviscid compressible fluid. Apply some constraints and then find a solution. The solution is a Bessel function:


$$\left.\begin{array}{rcl}
 \dfrac{\partial \mathbf{u}}{\partial t} \;+\; (\mathbf{u}.\nabla)\mathbf{u}& \; = \; -\: \dfrac{1}{\rho} \nabla P \\\
 \rho(\mathbf{x}, t)& \; \ll \; 1 & 
\end{array}\right\rbrace$$
$$\Rightarrow \xi =\: \psi\_o(t)\: R\_{mn}(\mathbf{x}) \;\;: \\\
\begin{cases}
\mathrm{Re}(\xi) &\overset{\underset{\mathrm{def}}{}}{=}\: \dfrac{\rho}{\rho\_o} - 1 \\\
 \psi\_o &\overset{\underset{\mathrm{def}}{}}{=}\; A \: e^{-i\omega\_ot} \\\
\displaystyle R\_{mn} &\overset{\underset{\mathrm{def}}{}}{=}\; \int\_{0}^{2\pi} e^{-i(m{\theta}'\,-\,n\phi )}j\_m(\kappa\_r\sigma)\kappa\_rR\_o \mathbf{d} \phi
\end{cases}
$$


My goal is to do a step-by-step proof of his derivation and learn somethings about such system. Later I would like to derive step-by-step how two such systems interact with each other, if possible. The article is rather dry on the derivations as it assumes these are rather uninteresting and unremarkable.


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**Update 1:** So far I have found online derivations to the Euler equation and a very attractive derivation of Bessel functions with gorgeous physical insights to it:


<http://galileo.phys.virginia.edu/classes/311/notes/fluids1/fluids11/node10.html>


<http://physics.ucsc.edu/~josh/116C.07/bessel/node1.html>


I can't apply the derivation of Bessel directly because it starts from the equation:
$\nabla^2\mathbf{u}(x, y, z) = 0$ . I don't know how to relate that to the Euler equation of the form $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}.\nabla)\mathbf{u} = - \frac{1}{\rho} \nabla P$. Does someone know how the two relate?


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**Update 2:**
Carlo Beenakker pointed out that the target solution ignores the effects of the convective term of Euler equation: $(\mathbf{u} \cdot \nabla)\mathbf{u}$


That relates to the reference article in that the author makes the assumption of "low amplitude", meaning $\mathbf{u} \ll 1$


Carlo Beenakker has also given a full answer I am still studying. I hope it is complete but I would appreciate anyone helping. I should take a couple of days.


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**Background:** I am not a professional mathematician or physicist. I know the proper way to do this would take a couple of semesters and do the proper college courses on differential equations, with much calculus background, which I don't have. As my interest is mostly only on this specific set of equations and I don't have a tutor or teacher to help me I would need some points on what would be the fastest way to finish this such that the math is rigorous.


I hope some of you have any interest for this curious approach too. Thank you for helping.


PS: Don't mind the article talks about Quantum Mechanics. Im not interested in that.
(I eliminated references to QM to avoid misundertandings.)



Derivation of Bessel functions
I'll make an attempt at providing the steps you are seeking to go "from Euler equation to Bessel function".


You start from the Euler equation, describing conservation of momentum,


$$\rho\frac{\partial \vec{u}}{\partial t}+\rho\vec{u}\cdot\nabla\vec{u}=-\nabla p$$


and the continuity equation, describing conservation of mass,


$$\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho \vec{u})$$


These are nonlinear equations, to make them tractable you'll want to linearize them, both in the velocity $\vec{u}$ and in the deviations $\delta\rho=\rho-\rho\_0$ of the density from the uniform density $\rho\_0$. This approximation throws away lots of interesting physics (shock waves, turbulence,...), but without it no simple solution exists.


The linearized equations read


$$\rho\_0\frac{\partial \vec{u}}{\partial t}=-\nabla p$$


$$\frac{\partial\delta\rho}{\partial t}=-\rho\_0\nabla\cdot\vec{u}$$


We may also assume a linear relation $p=p\_0+C^2\delta\rho$ between the pressure $p$ and the density variations. (This is a socalled adiabatic equation of state, the coefficient $C^2$ must be positive for mechanical stability.) We define $\xi=\delta\rho/\rho\_0$, take the divergence of the first equation and the time derivative of the second equation,


$$\nabla\cdot\frac{\partial \vec{u}}{\partial t}=-C^2\nabla^2 \xi$$


$$\frac{\partial^2\xi}{\partial t^2}=-\frac{\partial}{\partial t}\nabla\cdot\vec{u}$$


Finally, we substitute the first equation into the second one, exchanging the order of differentiation with respect to time and space, to arrive at a wave equation for $\xi$,


$$\frac{\partial^2\xi}{\partial t^2}=C^2\nabla^2 \xi$$


The quantity $C>0$ represents the speed of sound.


We seek a solution of this equation that is a harmonic function of time, so it oscillates with frequency $\omega$. Rather than working with sines or cosines, it is more convenient to use a complex notation, writing


$$\xi(\vec{r},t)={\rm Re}\;e^{-i\omega t}f(\vec{r})$$


The complex function $f$ satisfies the Poisson equation,


$$C^2\nabla^2 f=-\omega^2 f$$


Let's seek a solution with cylindrical symmetry, so $f(R)$ depends only on the radial coordinate $R=\sqrt{x^2+y^2}$. The Poisson equation in cylindrical coordinates takes the form


$$\frac{d^2}{dR^2}f(R)+\frac{1}{R}\frac{d}{dR}f(R)=-(\omega/C)^2f$$


The solution is a Bessel function


$$f(R)={\rm constant}\times J\_0(\omega R/C)$$


The full solution thus becomes


$$\delta\rho/\rho\_0=A\cos(\omega t+B)J\_0(\omega R/C)$$


where $A$ and $B$ are arbitrary coeffients.


And we're done :)


