What, Physically is a Spherical Harmonic?

[Fyrefly]

What, "Physically" is a Spherical Harmonic?

I'm trying to use spherical harmonics to get an equation to fit a set of data I have. I'm fine with that, I've found a derivation of what the general form is and I crunch that into MATLAB. My problem is derivations online really don't help me understand what this thing is, not really. I'm an engineer and my grasp of abstract math concepts really isn't that good.

I have a bunch of points now that my model churned out that are complex. I need to be able to compare the results of my model to the actual data before I use my model for other purposes, and I have no idea if I can just use the real part or not because I physically do not understand what this equation I have represents.

It's easy enough to tell me that Fourier is the idea you can approximate ANY equation with a sum of sines and cosines. That physically makes sense to me. Somehow a spherical should be Fourier in two dimensions from what I've gathered. But now we're complex and I just have no idea in what situations you use the real part, imaginary part, both parts?

 

[alexfloo]

I'm not intimately familiar with them, but I can hopefully describe the intuition.

On a vibrating string, we can talk about the different natural modes of oscillation. These modes describe how energy might (theoretically, will) move through the string. If there is mechanical energy in the string, then it will vibrate, and it's motion will be a combination of that natural modes (in theory).

That only describes the case of a line (string) vibrating, however. Spherical harmonics are the analogous idea for a spherical vibrating body. You could do the same for a circular vibrating body (a thin vibrating membrane, like a drum head). This is easier to envision, because its a two-dimensional object vibrating in three-dimensional space. Spherical harmonics actually describe a three-dimensional body vibrating in four-dimensions. (The fourth dimension may or may not be spatial: it might be something like time, or the strength of a magnetic field, or density of a fluid at a point.)

 

[Fyrefly]

Alex, thanks for your response. That makes sense, but then I can't figure out where the complex numbers come into play. If I have pairs of two variables and I know for example that they are coordinates in Cartesian 3D space and real numbers (with another value attached to them which I am modeling which is also real), why will a spherical harmonic then return a model which, when those coordinates are placed back in, returns a complex number? I can't understand where this complex comes from, why we stick an i in that equation to begin with if the data isn't even complex.

 

[alexfloo]

Could you post the particular formula you're using? Complex exponentials are a common alternative representation of sine/cosine functions, so that may be what you're referring to. Again, I don't know much about spherical harmonics beyond the intuition, but hopefully I can explain a bit of the formula.

 

[Fyrefly]

I'm using equation 590 as seen on this page: http://farside.ph.utexas.edu/teaching/qmech/lectures/node75.html

Thanks again!