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Also BTW, the earlier bold assertion from M.F. that
For each
frequency, a spherical wave can be represented as a linear
superposition (3-D Fourier transform) of plane waves with the same
frequency.
is completely untrue.
It is untrue in even in D=1 ...
If this is not obvious, please take a look at the following
snapshot of a radially-symmetric outbound traveling wave in
one dimension.
http://www.av8n.com/physics/img48/radially-symmetric-wave-1d.png
The wave is emitted by the point source at the origin. The
wavefunction is φ = sin(k|r| - ωt)
If you take the Fourier transform of the whole thing, you
get contributions from infinitely many different frequencies.