I’ve endeavored to model the scattering of an acoustical plane wave off of the surface of a rigid sphere. As with most computational models, we have no “reference” with which to compare the results–how can we know they’re accurate? And even though analytical solutions exist for the problem, they are (1) not perfectly implementable since they require infinite sums (which must be truncated) and (2) they are quite challenging to program, and we have no intuition for how they should look. An analysis of limiting cases and of the actual physical scenario are therefore needed to determine whether or not the results are accurate.

With regards to point (1) above, in order to practically solve the problem at hand we must truncate an infinite sum. So how do we know where to truncate? One possible method involves the addition of more and more components until the solution stops changing noticeably (for this project results are plotted and changes found by inspection), or actually begins getting worse (for more on this see my most recent post on pages.vassar.edu, specifically where aliasing errors are discussed). Keeping only the first sqrt(M) terms in the sum, where M is the number of points on the sphere for which the scattering was computed (or the number of sampling points), was found to be sufficient. This number was also found in the literature cited for this project.

For point (2), the examination of limiting cases is crucial. First, however, let’s consider what variables we’re dealing with. The primary independent variables of our scattering problem are sphere radius and sound frequency. If one imagines the relative sizes of the wavelength, though, which is essentially interchangeable with frequency, and the sphere radius, it becomes clear that there is really only one independent variable: the ratio of wavelength to sphere radius. In other words, increasing the sphere radius is the same as decreasing the wavelength of incident sound, and vice versa; the physical scenario remains essentially the same.

We therefore focus on the effect of frequency on the scattering pattern. For very low frequencies, when the wavelength is much bigger than the size of the sphere, we should expect very little scattering, since such a huge wavelength effectively “ignores” the sphere. What scattering does occur should be relatively uniform, since the sound field varies little with respect to the size of the sphere. For high frequencies, when the wavelength is much smaller than the radius of the sphere, we expect the sphere to act like a “wall” and reflect a good portion of the sound back towards the angle of incidence. For wavelengths comparable to the radius of the sphere, the results become quite interesting. We expect some sort of complex scattering pattern that is neither uniform nor immediately predictable, though we still expect the majority of the energy to be scattered in the direction of incidence.

With a look at the figure below, which shows cross-sectional scattering patterns for 200, 300, 341, 400, 600, 1000, 2000, and 4000 Hz, respetively, it is clear that these expectations are met. I think we can rest assured that these results are accurate. Note that 341 Hz corresponds to a ratio of exactly 1 between the wavelength and radius of the sphere, as the radius of the sphere was chosen to be 1 m and the speed of sound 341 m/s.

For sake of space and visualization, each graph has been normalized. In practice, the low frequencies scatter very little energy in comparison with the higher frequencies. Note how uniform the lower frequencies are, and how the higher frequencies approach a delta function in the direction of incident sound, just as expected. 600 Hz looks especially interesting; I would never have guessed such behavior at the beginning of this project.

Finally, why do we care about this work? One of the main reasons is that it provides the basis for measurements of sound energy using a spherical microphone (a rigid sphere with many microphones on it). In order to measure the sound on the sphere, we need to know how sound behaves on the sphere. A spherical microphone, using a technique called beamforming, allows us to extract directional information from a sound field (i.e. to determine from which angle the majority of sound energy is coming). This is the what I’ve attempted to do in my thesis. In addition, a spherical microphone also allows one to “reconstruct” a sound field (think virtual reality or sound maps) whereas a normal microphone does not. Future work for this project could include developing beamforming techniques, creating a system that can “intelligently” determine direction of incoming sound, and extending the scattering analysis to other shapes of interest, perhaps a hemisphere or 3D polygon of some sort.