For two signals to cancel each other they have to be correlated, i.e., their waveforms (as seen on an oscilloscope) have to match, whereas uncorrelated signals are immune to both polarity reversals and time offsets.
Example 1 in Figure 1 shows two sine waves (pure tone) whose amplitudes are matched, and where one instance (B) has been polarity reversed.
Reversing the polarity of signal effectively turns addition into subtraction when combined. Obviously, when we subtract two equal parts from each other the remainder is zero, where zero on a linear scale becomes minus infinity in decibels on a logarithmic scale.
Example 2 in Figure 1 shows the same sine waves as before where instance (B) has been attenuated by 1 dB. Clearly, the remainder is no longer zero and cancellation has gone from minus infinity to -20 dB. That mere one dB offset made an infinite difference in cancellation!
So what happens if we are not perfectly out of phase even though levels are perfectly matched?
Example 2 in Figure 2 shows the same sine waves where instance (B) has been shifted to the right by 10°. Again, the remainder is no longer zero and cancellation has gone from minus infinity to -15 dB. That mere 10° offset made an infinite difference in cancellation!
So far, we can conclude that, in order to achieve perfect cancellation, i.e., minus infinity (in dB), signals need to be correlated, perfectly level matched, and perfectly out of phase (reversed).
It is uncommon for any pair of brand new loudspeakers (or systems), of the same make and model, to be perfectly level-matched out of the box, or remain matched over time. After all, a 10% tolerance in a resistor equals one dB, and that is just one example of the many parts used in loudspeaker systems. Products with a potentiometer are even more prone to subtle level differences.
Enter the Storm
I have yet to experience perfect cancellation in the field, using actual loudspeakers, and doubt if it can be done. So what if we lower our expectations, and aim for something more moderate such as at least -15 dB of cancellation or more.
Figure 3 shows the "funnel" which kinda resembles a tornado. Purportedly, at the center of an tornado, in the eye of the storm, there's tranquility, i.e., clear blue sky and sun shining. At the base of the funnel in Figure 3, there is also "tranquility" which happens to be silence (perfect cancellation).
In order to live at the base of the funnel where maximum cancellation occurs, levels need to be perfectly matched and out of phase (reversed). Once either condition is no longer met, you go up the funnel (along its walls), where cancellation is less and things get louder.
If we bisect the funnel (roughly) in half at -15 dB of cancellation, the challenges becomes to end in the bottom part of the funnel, below the bisecting (grey) plane.
Figure 4 allows us to look into the funnel from above, notice the oval shape where the grey plane intersects the funnel. This is our destination, and in order to get there, we can derive the following (stringent) conditions:
- matched levels ± 1,5 dB or less
- out of phase ± 10° or less
When either condition is not met, you go up the funnel (along its walls), where cancellation is less and things get louder.
One dB is the Just Noticeable Level (JND) or smallest audible step for sound levels, and ten degrees at, e.g., 100 Hz, equals (10° / 360°) x 10 ms = 0,3 ms. Suffices to say, these rules‑of‑engagement are quite stringent, even at lower frequencies...
15 dB is a shedload
Fifteen dB of cancellation might sound like very little, but the Equal Loudness Contours (Figure 4), which admittedly only apply to pure tones, suggest that our hearing sense is more sensitive to relative pressure differences at low frequencies.
It takes a 10 dB increase in SPL for all frequencies above 1 kHz to achieve a 10 phon increase (subjective doubling) in perceived loudness, whereas at low frequencies only 5 dB is needed to achieve the same 10 phon increase.
Therefore, 15 dB less SPL at low frequencies is 30 phon less loud and 23 = 8 times quieter subjectively!
The first time I became aware of this was when I visited the late Siegfried Linkwitz's website many years ago. Click here to read his article that covers this.
When I am asked to design solutions that cancel (typically subwoofer arrays), I aim for at least 15 dB of cancellation. If, on site, it turns out to be more, I consider it a good day. Anything less, requires me to go back to the drafting table.
Fifteen dB is doable, and frankly as good as it typically gets. More often that not, are there no time and resources to tweak every element in a system by fractions of a decibel to account for tolerances, or physically position the elements with 5 cm accuracy or less to minimize timing errors.