Fields with limited order

The spherical basis solutions can be used to represent sound fields entirely only when the indices n and m are complete. In practice, it is convenient, sometimes numerically necessary to limit the index n to a finite order. The following animations show plane-wave and point-source sound fields truncated at different orders n.

The table shows several animations of an N-truncated representation of a plane wave and a Green's function. A rough rule of thumb:

We need an increase of the order by 6 for to increase the radius of accurate representation by one wave-length

A very interesting fact when designing appropriate playback for a big sweet-spot.

plane waveGreen's function at z/lambda=2
N=0 planewave_0.gif note that this column uses all two kinds of base solutions
  • regular base solutions (red/blue)
  • singular base solutions (magenta/cyan)
N=1 planewave_1.gif greenwave_2_1.gif
N=3 planewave_3.gif greenwave_2_3.gif
N=6 planewave_6.gif greenwave_2_6.gif
N=13 planewave_13.gif greenwave_2_13.gif

Franz Zotter, 2008.