By Perrin Meyer
The high-resolution polar data used by Meyer Sound MAPP Online is acquired at 1 degree spatial resolution using an automated positioning device in an anechoic chamber. The measurement computer uses a SIM-type algorithm to acquire 1/24th octave complex (magnitude and phase) data from 0 Hz to 20 kHz.
However, the wedges used in our anechoic chamber are only rated to 100 Hz. Below 100 Hz, the data in our anechoic chamber, while pretty good, is not accurate enough to give good results in MAPP Online. For this reason, until now, we have not allowed any predictions in MAPP Online below 100 Hz.
Building an anechoic chamber that would give a good measurement of the free-field (anechoic) response of a subwoofer would be very difficult and expensive. The wavelength of sound at 30 Hz is 11.4 meters (38 feet), and therefore the anechoic wedges would need to be on the order of 10 to 20 feet long in order to absorb, and not reflect, low-frequency energy.
However, measuring outdoors has its problems as well. Cars and trucks create low-frequency rumble that can contaminate measurements. Wind noise can also be a big problem. But if done carefully, outdoor measurements correlate well with indoor anechoic chamber measurements.
Figure 1 shows an on-axis frequency response measurement of a single Meyer Sound M2D loudspeaker at 4 meters, measured on a flat surface with the microphone placed directly on the ground. The ground is known as a half plane, and on axis to a single loudspeaker it creates a very accurate measurement of the frequency response compared with a single loudspeaker measured in an anechoic chamber. The half plane causes the magnitude of the response to increase by 6 dB compared with the anechoic chamber measurement.
The blue trace in Figure 1 shows the outdoor SIM half plane measurement of a single M2D, minus 6 dB. The red trace shows the anechoic chamber measurement of a single M2D. Note the excellent correlation between the two measurements between 100 Hz and 10 kHz. Above 10 kHz, ground plane measurements are inaccurate due to the size of the microphone compared with the small wavelengths of the sound. We have found through extensive testing that our anechoic chamber is very accurate above 10 kHz.
Below 100 Hz, as expected, the two traces differ. Since our chamber wedges are only rated to 100 Hz, we have found that ground plane measurements are more accurate. Especially visible in the red trace is a 10 dB hole centered at around 80 Hz. This acoustic hole in our anechoic chamber below 100 Hz is the reason that we have not allowed predictions in MAPP Online below 100 Hz.
Boundary Element Modeling
In order to correct our anechoic chamber measurements below 100 Hz, we could measure outdoors at 4 meters every 1 degree. However, this is tedious and prone to errors.
Instead, we have chosen to simulate the polar response of our loudspeakers below 100 Hz using a mathematical technique called Boundary Elements.
Boundary Element Methods in computational acoustics are similar to Finite Element methods in mechanics. A mesh is created of the object to model, and the physics of wave phenomena are solved for each element in order to provide a solution of the wave equation. Figure 2 shows the model mesh of an M2D loudspeaker. Even though this mesh might look boxy, it is more than sufficient for accuracy at low frequencies. A general rule of thumb used in Boundary Element Modeling is to use a mesh grid that corresponds to 1/10th of a wavelength of the highest frequency predicted. At 100 Hz, the wavelength of sound is approximately 3 meters (about 12 feet), and so 1/10th of a wavelength is about 1 foot, which is larger than the model mesh used for the M2D. This mesh would not yield accurate results for mid or high frequencies, but for these frequency ranges we use the very accurate data from our anechoic chamber.
By specifying the velocity boundary conditions of the model loudspeaker mesh, we can inform the model of where the vibrating structure corresponds to the dual ten-inch woofers of an M2D. The other mesh elements are then assumed to be rigid (reflecting).
Once we specify the velocity boundary conditions, the software solves the Kirchhoff-Helmholtz Integral Equation in order to find the surface pressure on the loudspeaker mesh. Figure 3 shows a colormap of the surface pressure (at 90 Hz).
Once we know both the velocity boundary conditions and the surface pressure on the loudspeaker mesh, the software can solve the integral equation to find the pressure at any point in free space. In our case, we have asked the software to solve for the pressure on a 4-meter circular radius, which corresponds to the 4-meter radius in our anechoic chamber.
Figures 4 and 5 show how we calculate the complex pressure at 4 meters every 1 degree. In our software package (Sysnoise), this is called Directivity.
By combining the on-axis 4-meter outdoor half plane measurement and the Boundary Element 1-degree Directivity calculations, we can supply the accurate complex polar data that MAPP Online requires below 100 Hz. Figures 6, 7 and 8 show a MAPP Online prediction of a single M2D with a microphone at 1 meter. Notice how the Frequency Response and the Spectrum calculations now extend (accurately) below 100 Hz.
Not all of the loudspeakers in MAPP Online have been corrected at this time. MAPP Online will automatically display the spectrum and frequency response below 100 Hz if the polar data file has been corrected. If information below 100 Hz does not display, the data have not yet been corrected.
More information about boundary element methods in acoustics can be found in the book, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition, David Colton and Rainer Kress, Springer-Verlag, Berlin, 1998.