Derivation of Bessel functions

Question

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.

---

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?

---

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.

---

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.)

Answer 1

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 :)