Spherical Harmonic on Constant Latitude or Longitude

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View the oscillating radial amplitude of the spherical harmonic (relative to the radial offset of 1.0) on a horizontal plane taken at constant latitude or on a vertical plane taken at constant longitude. You can set the degree and order of the harmonic ({, } with ), the latitude () when the "latitude" button is selected, or the longitude () when the "longitude" button is selected. The relative amplitude (0 to 1) of the harmonic, when added to the constant radial offset of 1.0, is controlled by the slider labeled "scale". Use the "" control to change the oscillation speed, to start or stop it.

Contributed by: David von Seggern (University of Nevada) (March 2011)
Open content licensed under CC BY-NC-SA



Spherical harmonics (normalized) in the radial direction are expressed by the equation:


where and are the degree and order, respectively, of the generalized Legendre function . The function is expressed on a sphere with polar angle (starting from 0 at the positive axis) and azimuth (counterclockwise from the axis in the equatorial plane as viewed from the positive axis).

For display purposes in this Demonstration, is multiplied by the time function , scaled by a constant [0,1] and then added to 1.0. The radial motion in a plane given by constant or is then shown as a function of time, with a complete oscillation every units of time.

There are three common classes of radial spherical harmonics. The first, zonal harmonics of degree , is given when . For this class, there is no longitudinal variation and there are zeros along parallels of latitude. The second, sectoral harmonics of degree and order , has no latitudinal zeros but has zeros along longitudes. The third, tesseral harmonics of degree and order (), has zeros along parallels of latitude and zeros along longitudes.

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