Using MATLAB, I’ve created several functions to visualize scattering from a rigid sphere. I’ve included three pictures below, as well as the MATLAB code (Note, code is still in rough form. Commenting and overall readability shall be improved). In all Images, the sound is incident from the left. In the next post, I’ll go into more detail with regard to the actual mathematics; during the past week my greatest priority has been to simply get the code running. For now, here’s the equation describing the sound field due to plane wave scattering from the rigid sphere:

where jn and hn are the Spherical Bessel and Hankel functions of the first kind, respectively, Pn are the associated Legendre Polynomials, x is simply the x coordinate, k is the wave number, and i is the imaginary unit. Notice that since we have an infinite sum, in practice we must truncate and approximate this solution.

The first–a two dimensional cross section of the pressure incident on the sphere from a plane wave (frequency of ~54 Hz). The red curve is simply for visualization of the sphere; it has no other significance in this plot. Note how the greatest pressure is on the face of the sphere, and the least is on the sides and around on the directly opposite side.

Next is the same graph, but extended to 3D. Note the similar shape, but a little more pronounced as this is for 100 Hz and the previous was for ~54 Hz.

Finally, the intensity of the sound field; this includes both incident and scattered sound energy, and one can therefore see the constructive and destructive interference patterns. This is for a plane wave of ~54 Hz.

CODE (note I’ve removed large comments from the front of the code for the purposes of space)

Note: all code to run these simulations should be present. If you can’t get the code to run, please let me know.

Here is a link to the code in Google Drive:

https://drive.google.com/a/vassar.edu/folderview?id=0B3UNwnkM9fLrfjNyTDJEc0tQeklHTTBueFEtMDRZMUVJWnpvV2p4MFpLbHlpVTJ4QXJCXzg&usp=sharing