File:Spherical harmonics.png
Summary
Spherical harmonics in 3D.
Plot of the spherical <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38959a8eacb9848ad5556036ec8d01daac5151ab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.036ex; height:2.843ex;" alt="{\displaystyle P_{n}^{m}(\cos \theta )\exp(im\phi )}"> harmonic of degree n=5, and orders m=5,4,3,2,1,0,-4,-5 When the spherical harmonic order m is zero, the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. When <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780e4684ab101f7fc04b856044493800a9deb624" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle n=|m|}">, there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral. Spherical harmonics of negative order have the same appearance, but much smaller amplitude and sign (-1)^m
Plotted with Matlab.
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 07:26, 9 January 2017 | | 1,243 × 2,791 (424 KB) | 127.0.0.1 (talk) | Spherical harmonics in 3D. <p>Plot of the spherical <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msubsup><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>cos</mi><mo><!-- --></mo><mi>θ<!-- θ --></mi><mo stretchy="false">)</mo><mi>exp</mi><mo><!-- --></mo><mo stretchy="false">(</mo><mi>i</mi><mi>m</mi><mi>ϕ<!-- ϕ --></mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle P_{n}^{m}(\cos \theta )\exp(im\phi )}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38959a8eacb9848ad5556036ec8d01daac5151ab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.036ex; height:2.843ex;" alt="{\displaystyle P_{n}^{m}(\cos \theta )\exp(im\phi )}"></span> harmonic of degree n=5, and orders m=5,4,3,2,1,0,-4,-5 When the spherical harmonic order m is zero, the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. When <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>n</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle n=|m|}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/780e4684ab101f7fc04b856044493800a9deb624" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle n=|m|}"></span>, there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral. Spherical harmonics of negative order have the same appearance, but much smaller amplitude and sign (-1)^m </p> Plotted with Matlab. |
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