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The solution of
the wave equation, written in spherical coordinates (r,
![]() ![]() ![]() can be written,
for the solutions with separated variables of the form
![]() ![]() ![]() where the
functions jn and nn
are respectively the spherical Bessel functions of the 1st kind and the
spherical Neumann functions, nth-order, and where the functions Pnm(cos
![]() ![]() The functions
![]() ![]() Hence, for any given n and m, any given solution coming from the separation of variable methods has a structure which is well determined on the unit sphere (parameters ![]() ![]() ( ![]() ![]() These solutions express a complex directivity of the fields which are emitted in an infinite space, by a set of "localised" sources with complex properties. |
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The
animations below present the variations as a fucntion of time, of the
first spherical harmonics ![]() ![]() |
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"Zonal"
harmonics: m=0 "Sectoral" harmonics: n=m |
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"Tesseral"
harmonics: 0<m<n |
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