A rapid computational method to investigate the directivities of quasi-omnidirectional sources of sound

Jeshua H. Mortensen and Timothy W. LeishmanAcoustics Research Group, Dept. Physics & Astronomy, Brigham Young University

1. BACKGROUNDRegular Polyhedron loudspeakers (RPLs) have been widely used in room acoustics as omnidirectional sources of sound. This research investigates the sound directivity of the platonic solid loudspeakers via the boundary value method (BVM), using the spherical caps approach [1] with an axially oscillating cap, by distributing the caps over a sphere according to the platonic geometries.

3. METHODOLOGY AND DISCUSSION

4. RESULTS

5. FUTURE WORK

6. REFERENCES

2. MOTIVATIONS

BRIEF ARTICLE

THE AUTHOR

(1)

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1

Prior to this research, the Platonic solid loudspeakers have been used as approximate omnidirectional sound sources in room acoustics. This poster presents a rapid computational method for predicting the directivities of the Platonic solid loudspeakers. Of the five Platonic solid loudspeakers, the dodecahedron, while most commonly used in acoustical measurements as a quasi-omnidirectional sound source, may not be the best overall; other Platonic solid geometries may be better suited for this purpose.

In the case of the single spherical cap model, the pressure can be calculated using the Helmholtz equation and boundary conditions.Since we have symmetry about the z-axis, the ϕ dependance vanishes, dimensionally leaving only θ and r dependance. Then the coefficients can be computed from the boundary conditions. The coefficients are then expressed as functions of frequency f, cap size θ0, and sphere radius a, in terms of the Hankel function and Legendre Polynomials.

By the superposition principle we can take the solution of a single cap, and by rotating it we can superimpose it and obtain an interference pattern. The following side by side images are shown to illustrate this concept.

It is significant to note that the computational algorithm requires that we transpose the rotation matrix so that the reference poles stay put, while the function undergoes the transformation, as is demonstrated here with the two matrix operations.

The time that it takes to compute similar models using the boundary element method were on the order of several hours for a single frequency. Here we have been able to compute a model for 1600 different frequencies all at once with computation times (in MATLAB) ranging from 6-10 seconds.

Icos. Model 4000 Hz 5750 Hz

Experimental Data

3000 Hz

Icos. Model Icos. Model

Experimental Data Experimental Data

The figure to the left is a comparison of the area-weighted standard deviation of two models with different sphere radii blue and green, compared to the experimental data in red. The green uses the mid radius of the icosahedral (RPL), while the blue sets the sphere radius where the driver edge would be.

We want to look at varying the cap size θ0, and sphere radius a, and see which (RPL) might have better omnidirectionality.

[1] E. Skudrzyk, The Foundations of Acoustics (Springer, New York, 1971), pp 399-400

SINGLE CAP

TETRAHEDRON HEXAHEDRON OCTAHEDRON DODECAHEDRON ICOSAHEDRON

Thursday, June 25, 15