In the last post we looked at interference between multiple coherent sources. We saw how a difference in path length from the listener to the different sources creates a relative delay between the arrival of wavefronts from each, which has a corresponding phase delay at a given frequency. Ultimately we derived equations for calculating the resultant phase and amplitude of the superposition of the two sources at specified frequencies and listening positions.
I want to consider now what these effects look like when we take a broader view, considering a) frequency response across the audible range at a given listening position, and b) level of a given frequency as we adjust the listening position.
Let’s consider first a scenario in which we have two sources and a fixed listener position. In the previous post we examined this scenario at a specific frequency, now we are going to extend this to look at the effects of a broadband source.
As we recall from the previous post; with two sources it is likely that there will be a path length difference between the two sources and the listener (unless the listener happens to be situated on the plane that is exactly equi-distant from the two sources. There is a time delay between the arrival of wavefronts from the two sources corresponding to this path length distance. This time delay results in a corresponding phase angle between wavefronts from the two sources at a given frequency.
It is key to appreciate that the phase angle created is dependent upon frequency.
Frequency is directly related to wave length:
Where c is the speed of sound, f is frequency, and λ is the wavelength.
As c is a constant we can see that as f increases λ must decrease, so for higher frequencies the wavelength is shorter. One wavelength corresponds to a phase angle of 2π radians (or 360°), so a given difference in path length from the listener to two sources will therefore result in a bigger phase angle the higher the frequency.
The graph below demonstrates how a 5ms delay results in a different phase shift for waves of different frequencies. Notice how the 2kHz wave has shifted an entire wave cycle, whilst the 1kHz wave has only shifted 180°, and the 500Hz wave only 90°:
Calculating the phase angle at a given frequency and known path length difference becomes quite straightforward:
If we now consider a variable frequency scenario it is evident that the phase angle will vary for different frequencies. We understand that the resultant interference is dependent upon the phase angle, so this will also vary based upon frequency. At certain frequencies interference will be constructive as the two sources will be in phase, at other frequencies it will be destructive as the two sources will be in anti-phase.
The resultant pattern is known as comb-filtering; as you will see from the example graph below, we get a series of notches in the frequency response which create a shape resembling a comb:
The precise frequencies of the various notches in the filtering pattern will vary depending upon the difference in path lengths, however the basic pattern of a series of alternating sharp notches and shallow peaks is consistent regardless of the relative positions of the sources and the listener.
In a world where we are extremely demanding of the quality of the frequency response from the majority of our components, putting multiple coherent sources into our environment feels like a extremely poor design choice. The comb-filtering effect seems to be pretty damning for the prospects of our audio quality.
It is worth noting that the example above is pretty extreme. It presumes only two sources (the absolute worst case scenario) and no reverberation. Adding additional sources does not fix the problem, although it will at least partially smooth out the worst effects; when any two of the sources might be in anti-phase, odds are that the remaining sources will be slightly more favourably aligned. If we put our system in a reverberant environment then reflected wavefronts will also help with this. The result will still not be a linear frequency response, but will at least smooth over the most severe notches.
Let’s now consider an example where we allow the listening position to vary. This is important because whilst it is easy to analyse comb-filtering at a specific position, this is not representative of a real-life listening environment. In reality we have to design our system not just to serve a single fixed listener position, but an entire listener plane (or even multiple planes) distributed throughout the venue.
We already know that there will be comb-filtering in the frequency domain at any given listener position. Whilst the pattern remains the same, the position of notches and peaks in the frequency domain will shift as listener position changes. This may appear random, but if we return to considering just a single frequency it becomes more predictable.
The following graphs look at the level of two frequencies as we adjust the listener position. In this example we are looking at a typical two speaker stereo system and the listener position is moving from left to right across the venue in front of the speakers:
We can see that at both frequencies there is a similar pattern, although with some important differences. Both examples are symmetrical about the centre-line of the venue. In both examples there are a series of troughs in the level as we move off axis. For the lower frequency 100Hz example these are wider but less frequent, whilst for the higher frequency example these become much narrower but also more frequent.
This is the general pattern: The higher the frequency the more frequent and narrower the troughs. The lower the frequency the wider and less frequent the troughs. In fact, if we went low enough in the frequency range we will eventually reach a point at which there are no more troughs.
Why would this be? It occurs because the wavelength is such that regarless of listening position the distance between the two speakers is never enough to put the two sources into anti-phase. In this scenario we would say that the two sources are coupled.
Let’s consider the 100Hz example. In the example above the two speakers are clearly separated by enough that they will interfere negatively at some angles. However if we move them closer together this will cease to be the case.
At 100Hz sound has a wavelength of roughly 3.4m. Half a wavelength is 1.7m. If we position the two speakers 1.7m apart we will only get one zone of negative interference, this will be at its most significant when we are exactly 90° off axis as the wavefront from the second speaker will arrive precisely half a wavelength behind and so be in anti-phase.
If we move the speakers closer it is now no longer possible for the two speakers to be in perfect anti-phase; regardless of the position of the listener, the path length difference cannot be enough (at 100Hz or lower) to create a complete cancellation.
At 1.1m the two speakers are now within a third of a wave length. Remember that at a path length difference of a third of a wavelength or less the interference is always positive. This is the point at which I classify the two speakers to be coupled. Regardless of the listener position, we always get positive interference of some degree between the two.
This coupling is frequency dependent as it is based upon wavelength, so two speakers will couple at lower frequencies earlier than they will couple at high frequencies. At 1.1m the speakers are coupled at 100Hz and everything below, but are not yet coupled at higher frequencies. At 1kHz the speakers need to be much closer (around 11cm), and at 10kHz closer still (11mm). This obviously creates some challenges due to the physical size of loudspeakers. We cannot get the centre point of most loudspeakers 11mm or even 11cm apart due to the physical body of the cabinet, so whilst it is relatively easy to create coupling at lower frequencies we cannot acheive it (at least not by physical positioning of loudspeaker cabinets alone) at higher frequencies.