Generalized Multipole Numerical Method

In this section the generalized multipole technique is briefly described, and results presented to validate the method. A general discussion follows. As an addendum, some history of the technique is reviewed.

The technique

The method is simplicity itself, conceptually. A number of elementary sources, each radiating a simple spherical wave of the form of equations (45) and (46) are placed outside the boundary. (In some problems several spherical waves of ascending order are placed at each source point). For example, see a configuration [24kb] used for a cone driver mounted in a baffle. The red dots in this figure are points on the boundary, and the blue lines emanating from the dots are the surface normals. The blue dots to the right of the boundary are the sources. The contribution of each source to the velocity component normal to the boundary is computed. A matrix is then inverted (actually a pseudo-inverse) to obtain source excitations such that the boundary condition is matched in the least-squares sense. It is then a simple matter to compute the field anywhere. For problems with symmetry around the z-axis, the sources are arranged in rings, and a single source excitation coefficient is obtained for each ring. This greatly reduces the size of the matrix.

Validation

The technique (and software!) was primarily validated by comparing results to the exact plane wave spectrum solution for a flat piston. Plots of pressure at the baffle surface for ka=π/2 [27kb], ka=2π [30kb], ka=8π [30kb], and ka=16π {30kb], all show excellent agreement. In all plots the solid line is the exact solution, and the dashed line is the numerical solution. The agreement is so good it is sometimes difficult to see that there are two lines. There is also excellent agreement between the exact radiation impedance (solid lines), and the numerically calculated values (dots) [32kb] (the dashed curve is for a cone rather than a flat piston). The far-field pattern completely overlays the exact solution.

The secondary validation uses what is perhaps the most attractive feature of the technique: for each solution it is quite easy to compute the resulting velocity normal to the boundary, and compare it to the boundary condition. It is possible to use one set of boundary points to obtain the solution, and a completely independent set of points to check the solution. However, if a very dense point set on the boundary is used, so nothing odd can occur between points, it is really not necessary to check on a different set.

For a planar boundary, the error in the result can be directly computed from the error in the boundary match, as shown in the boundary integral section. As far as I know, there is no rigorous way to directly compute accuracy from the boundary condition match for other surfaces.

An example of the boundary condition check [23kb] for a flat piston shows a very nice match. The rectangular blue curve is the boundary condition. The red and black curves are the real and imaginary parts of the multipole solution for velocity, respectively. This degree of match corresponds to excellent agreement with the exact theoretical solution, as shown. It is not unreasonable to expect that in practice a similar match on another surface would lead to similar accuracy.

The boundary condition match for the cone [23kb] is not quite as pretty, but still good. The blue curve is again the boundary condition. From radius=0 to 0.1 the normal velocity component drops due to the curvature of the dust cap. Over the remainder of the region the velocity is less than 1.0 due to the cone angle. There are two spikes where the velocity is 1.0, where the normal was taken to be parallel to the z-axis, the direction of the cone motion.

Results for a Cone Loudspeaker

The pressure at the surface of the cone [25kb] exhibits a more complex pattern than for the flat piston. The blue curve is the cone velocity. The red and black curves are the real and imaginary parts of the multipole solution for pressure, respectively. (If anyone reading this wants to compare solutions, I will be happy to e-mail the exact dimensions of the cone). More interesting are the near field contour plots for pressure amplitude [60 kb], and phase [65kb]. The far-field pattern [27kb] is fairly close to the pattern of the flat piston.

Discussion

This is not a cookbook technique, and arranging the sources, and to a lesser degree, the boundary points, is the key to getting good results. A bad arrangement will lead to ghastly matrix conditioning and/or horrible boundary condition matches. It is quite important to have many more boundary points than source unknowns, and to use a pseudo-inverse solution, for numerical stability. If you choose a poor arrangement, you will find out very quickly by doing the boundary condition check as above. Regarding accuracy, I believe the results shown here speak for themselves.

History

My primary background is in electromagnetics (E/M), where the most common numerical techniques are finite-difference, for the time-domain , and "method of moments" for the frequency domain. The latter method numerically solves the exact boundary integral expression for the fields using a set of discrete basis functions on the boundary. I believe it is the equivalent of the "boundary element method" in acoustics (see the site created by Stephen Kirkup).

In the early 1980's a half-dozen people (including Christian Hafner, and myself, developed the technique equivalent to the one described above. As far as I know, it was not used in E/M prior to that time. In E/M this technique is most commonly known as the "Generalized Multipole Technique", or "Multiple Multipole." A good reference is the book by Hafner.

I posted a question regarding acoustic applications of this technique in the alt.sci.physics.acoustics newsgroup. I received a response from Eike Brechlin saying that the technique was also being studied in acoustics during the early 80's by Prof. Lothar Cremer at the Institute for Technical Acoustics from the Technical University in Berlin (the only references I have are two papers in German published in 1984 and 1988). The technique is known in acoustics under several names such as "multipole synthesis", "equivalent source method", "method of comparative sources" and others. I recently learned it is currently being used for acoustics problems by Dr. Peter Svensson at the Norwegian University of Science and Technology. Anyone knowing of other active users please inform me and I will post the information here.