Wave behavior (of various kinds) in three dimensions.
https://brilliant.org/wiki/spherical-harmonics/
https://brilliant.org/wiki/spherical-harmonics/
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Thank you Rayma for all those references, but a I am looking for a simplified mathematical and more intuitive description
If possible
If possible
Not sure the math can get simple, so just think of it as a computational technique,
sort of a generalization of Fourier analysis.
sort of a generalization of Fourier analysis.
Spherical harmonics are a set of basis functions for functions on the surface of the sphere.
Let’s say you are interested in the temperature everywhere on Earth. One way to represent that would be to say that each point (lat, long) has a temperature value, perhaps stored in a big table. That table represents the temperature everywhere on earth.
An alternative way to represent the temperature everywhere would be with a sum of increasingly high-frequency signals. Something like this (these numbers are made up)
Spherical harmonics are kinda like that, only instead of Country/State/City the shapes used are blobs in the shape of electron orbitals.
Let’s say you are interested in the temperature everywhere on Earth. One way to represent that would be to say that each point (lat, long) has a temperature value, perhaps stored in a big table. That table represents the temperature everywhere on earth.
An alternative way to represent the temperature everywhere would be with a sum of increasingly high-frequency signals. Something like this (these numbers are made up)
- The average temperature of earth is 25 C.
- The average temperature of the US is 4 degrees lower than the average temperature of the earth,
- The average temperature of the California is 6 degrees warmer than the average temperature of the US.
- The average temperature of Palo Alto is 8 degrees cooler than the average temperature of California.
- …
Spherical harmonics are kinda like that, only instead of Country/State/City the shapes used are blobs in the shape of electron orbitals.
They are the equivalent on a sphere of a Fourier series on a line. If you are a bit unsure about how and why Fourier series work (and the related transforms and discrete versions) then I would suggest getting up to speed with this first since it will then leave only a small amount of knowledge related to the increase in dimensionality of the components to pickup.
The first link tackles the subject directly and seems to be concisely presented. The maths is the maths and would be wrong if simplified.
The acoustic Helmholtz equation is similar except it has a second partial derivative with time leading to requiring two quantities on the surface (typically pressure and the normal component of particle velocity) rather than one. However for spherical waves the two are related and can be calculated from each other and so we usually need only the spherical harmonics of one quantity (typically pressure or velocity potential) to know what is going on acoustically everywhere. The spherical harmonics are mathematically the same as that presented. It is only the values that will change with the boundary conditions and/or governing equation.
A Fourier series is actually living on a circle and not a line. It assumes the signal is periodic