Sound fields in terms of Spherical Basis Solutions

spherical harmonics, spherical Hankel/Bessel functions, truncation of the Fourier-Bessel series
The spherical basis solutions are the result of solving the Helmholtz-equation (lossless, linear wave equation in the frequency domain) in the spherical coordinate system. They consist of spherical harmonics, spherical Bessel and Hankel functions, and form modes of radiation and modes of the homogeneous field.

Spherical basis-solutions

The complete set of solutions contains infinitely many functions. Therefore, a selection of the basis functions for n<5 is shown, the parameter |m|<=n. These solutions are orthogonal with regard to the their value on a spherical surface. For the following visualizations, the two families of solutions are multiplied by the complex exponential exp(i omega t) and only iso-surfaces are shown, where the real part lies above some threshold. The color indicates if you see a compression or decompression. (This type of images has been inspired by N.A.Gumerov and R.Duraiswami, The Fast Multipole-Method...)
view animations for interior and exterior solutions


The spherical basis-solutions splits up into two independent parts that are only connected by the index n in which they appear together.

The angular dependency: real-valued spherical harmonics

The shape of the spherical harmonics and how they look like are given in the following
animated plot.

The radial dependency: spherical Hankel and Bessel functions and their derivatives

The radial part of the basis solutions describe how the value of one spherical harmonic changes with the radius. The corresponding functions are
animated for interior and exterior fields.

Decomposing a sound field with finite n

As mentioned before, the index n has something to do with the spatial, to be specific: the angular resolution. The
animations show a couple of sound fields decomposed with an increasing number of n.