1. A rapid computational method to investigate the directivities of quasi-
omnidirectional sources of sound
Jeshua H. Mortensen and Timothy W. Leishman
Acoustics Research Group, Dept. Physics & Astronomy, Brigham Young University
1. BACKGROUND
Regular 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)
Vm =
8
>>>>>>>>><
>>>>>>>>>:
u0
4 sin2
(✓0) , m = 0
u0
2 1 cos3 (✓0) , m = 1
u0
2
h⇣
m
2m 1
⌘
Pm 2 (cos ✓0)
⇣
2m+1
4M2+4m 3
⌘
Pm (cos ✓0)
⇣
m+1
2m+3
⌘
Pm+2 (cos ✓0)
i
, m = 2, 3, 4 · · ·
9
>>>>>>>>>=
>>>>>>>>>;
(2) Am (r) =
⇢
⇢0c(2m+1)h
(2)
m 1(kr)
i
h
mh
(2)
m 1(ka) (m+1)h
(2)
m 1(ka)
i
(3) k =
2⇡f
c
(4) bp (r, ✓) =
1X
m=0
WmPm (cos ✓)
(5) Wm = VmAm (r)
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