– Maths Wizards – What is/are Spherical Harmonics?
In: Mathematics
If you pluck a guitar string, and watch it while it vibrates, the string moves too fast for you to see and it looks all blurry.
The shape of that blur is a nice little curve. (It’s actually a sine curve). There are two points that aren’t vibrating, those are the two ends of the string.
You might know, if you play guitar, that you can make the string vibrate at a higher pitch, by gently touching the middle of the string when you pluck it.
Then, the blur has a different shape. It’s still a sine curve, but now there are three points that aren’t vibrating: the two ends of the string, and the middle.
By gently touching different points on the string when it’s plucked, you can get it to ring at higher and higher pitches, and each pitch gives a different shape to the blur.
These special shapes are the harmonic functions for the string. They are the natural ways the string vibrates, and they are important in a whole lot of other areas of physics, for example if you wanted to model heat flow along the string, or quantum mechanics in a quantum particle confined to the string.
The same principle applies to higher dimensional shapes, like the surface of a drum. Or the surface of a sphere.
If you could somehow hear the vibrations of a soap bubble, you could get different pitches by having it vibrate in different patterns. The patterns of vibration for these pure tones are called “spherical harmonics”.
Just like for the guitar string, spherical harmonics are also useful when analyzing heat flow on the surface of a sphere, or the behaviour of spherical quantum objects such as hydrogen atoms.
Edit: just for fun, here’s a cool video showing the harmonics of various Tetris pieces: https://youtu.be/Qh3lRZFnUJ4?si=LhnFqGOpiqzZ0VKg
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