Frequency response of a speaker cabinet on the rear axis
Last updated: April 25, 2018
Approximation of the frequency response of a speaker cabinet on the rear axis with a first-order low-pass filter.
In Rear Wall Reflection Simulator and Room Boundary Simulator I've used a first-order low-pass filter to calculate the frequency response behind the speaker cabinet. The cutoff frequency of the low-pass filter is shifted according to the baffle width. Of course, there is an obvious question: what is the validity of this simple calculation?
So I compared my rear-axis frequency response calculations with some measurements. These rear response graphs are normalized to the (front) on-axis response (in other words you can see the difference between the on-axis response and the rear radiation in dB).
Before I show the results I had to make some additional remarks:
- In the far field the frequency response of a speaker behind the cabinet varies according to the angle of the measurement axis.
- In the near field the exact frequency response behind the speaker cabinet depends on both the distance of the measurement point from the cabinet and both the angle of the measurement axis.
- In the far field the minimum amount of rear radiation (where the frequency response has the steepest slope) is on the 150-degree axis and not on the 180 degree rear axis. This can be seen on the polar graphs of speaker cabinets.
The first measured curve is exported from the polar charts of a pro PA speaker (JBL AC2212-95). The speaker has a 12" woofer and the width of the cabinet is 355 mm. Unfortunately, there is no measured data below 200 Hz.
The second one is an impulse response measurement of a small, 9 cm wide and 16 cm tall "multimedia" speaker. I've set the gate time to 8 msec, and the microphone distance from the baffle is 45 cm. (In the impulse response the first 8 msec is reflection free, this gives a 125 Hz low frequency limit.)
As can be seen on the graphs the first-order low-pass filter approximation gives acceptable results. The maximum error is +- 2 dB between 0 dB and -15 dB, and only becomes larger where the response falls below -15 dB. The cutoff frequency of the low-pass filter can be shifted down or up according to the baffle width. If we need a more accurate calculation, then the geometric theory of diffraction, or modeling true wave propagation is a better choice.