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Denker is correct. Waves act fundamentally differently in
odd-dimensional geometries than they do in even-dimensional
geometries. For example, Huygens' principle can be used with the same
general physical conclusions in dimensions 1, 3, 5, ... (though for
technical reasons one usually says it's not used in dimension 1 --
see Farlow's "Partial Differential Equations for Scientists and
Engineers, Lesson 24), but it leads to fundamentally different
physical behavior in odd-dimensional geometries.
The technical term is "wake formation". Morse and Feshbach's Section
7.3 gives all the math (in terms of Green's functions) one would ever
want.
The *physics* results are that in 1D systems, and 3D systems, there
is no wake formed (plucked strings return to their pre-plucked
positions; sound waves pass us by and we don't continue to hear their
ringing). The shape of the wave _may_ be unchanged (barring
dispersion); in 1D systems a given displacement (with zero initial
velocity) will be unchanged in shape. In 3D systems, a given pulse
shape is unchanged if the initial *displacement* is zero and the
initial *velocity* is imposed on the system. No wakes are formed. In
2D systems (membranes), there is a wake formed, no matter if the
medium is dispersive or dissipative or not. If you were to "hear" a
cylindrical pulse on a large drumhead, an initial impact would be
followed by distorted copies for a long time, until dissipation
killed them.
For each
frequency, a spherical wave can be represented as a linear
superposition (3-D Fourier transform) of plane waves with the same
frequency.