## Equation of the Day #8: Bessel functions

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Equation of the Day
**Akano Toa of Electricity**, in Math/Physics Feb 15 2014 · 173 views**My very first Equation of the Day was about the wave equation, a differential equation that governs wave behavior. It doesn't matter whether you have linear waves (sine and cosine functions), cylindrical waves, or spherical waves, the wave equation governs them. Today I will focus on the second, the so-called cylindrical harmonics, or Bessel functions.**

A harmonic function is defined as one that satisfies Laplace's equation,

A harmonic function is defined as one that satisfies Laplace's equation,

**For cylindrical symmetry, the Laplacian (the operator represented by the top-heavy triangle squared) takes the following form:**

**This is where a neat trick is used. We make an assumption that the amplitude of the wave, denoted here by**

*ψ*

**, can be represented as a product of three separate functions which each only depend on one coordinate. To be more explicit,**

**This technique is known as "separation of variables." We claim that the function,**

*ψ*, can be separated into a product of functions each with their own unique variable. The results of this mathematical magic are astounding, since it greatly simplifies the problem at hand. When you go through the rigamarole of plugging this separated function back in, you get three simpler equations, each with its own variable.

**Notice that the partial derivatives have become total derivatives, since these functions only depend on one variable. These are well-known differential equations in the mathematical world; the**Φ

**function is a linear combination of**sin(

*nϕ*)

**and**cos(

*nϕ*)

**(this azimuthal angle,**

*ϕ*

**goes from 0 to 2π and cycles, so this isn't terribly surprising) with***,**n*

**being an integer, and the**cosh(

*Z*function is a linear combination of*kz*)

**and**sinh(

*kz*)

**, which are the hyperbolic functions. These equations are not what I want to focus on; what we've really been working so hard to get is the radial equation:**

**This is Bessel's differential equation. The solutions to this equation are transcendental (meaning that you can't write them as a finite sum of polynomials; the sine and cosine functions are also transcendental). We write them as**

**The**

*J*

_{n}**are finite at the origin (**

*J*

_{0}

**is 1 at the origin, all other**

*J*

_{n}**are 0), and the**

*Y*

_{n}**are singular (undefined) at the origin. They look something like this:**

**The**

*J*

_{n}**are much more common to work with because they don't have infinities going on, but the**

*Y*

_{n}**are used when the origin is inaccessible (like a drum head that has a hole cut in the middle). These harmonic functions are used to model (but are not limited to)**

**Vibrational resonances of a circular drum head****Radial wave functions for potentials with cylindrical symmetry in quantum mechanics****Heat conduction in a cylindrical object****Light traveling in a cylindrical waveguide**

**Note that, while they kinda look sinusoidal, they don't have a set period, so the places where they cross the x-axis are have different intervals and are irrational; thus, they must be computed. This results in some weird harmonic series for instruments like xylophones, drums, timpani, and so on. I got into them because I'm a trumpet player, and the resonances of the surface of the bell of a trumpet are related to the Bessel functions.**

There are some cool videos (this one has a strobe effect during it) showing them in action. There are also some cool Mathematica Demonstrations related to them as well. There are also orthogonality relationships with them, but I'll save that for another day.

There are some cool videos (this one has a strobe effect during it) showing them in action. There are also some cool Mathematica Demonstrations related to them as well. There are also orthogonality relationships with them, but I'll save that for another day.