Quibbles and Bits

Dispersion

“Knowledge increases by diffusion, and grows by dispersion”

– Daniel J. Boorstin

One thing that audiophiles regularly fail to grasp adequately is how a sound wave propagates away from a loudspeaker – or from anything else for that matter.  The light from your average flash lamp can usually be focused into a beam by twisting the housing, so it is easy to visualize how it is dispersed (the “dispersion pattern”), and also how we are able to control it.  In particular, we have all seen that although the light can be more or less focused on the target, there is definitely a limit to how much we can focus down that spot.  As we twist the focus ring to make the spot smaller, eventually it stops getting smaller and starts getting larger again.  Yet lasers, for some reason, are apparently able to emit in pencil-thin beams [although I have news for you – some of them don’t!].  These are all physical manifestations of the nature of dispersion, which governs how anything with a wave-like nature propagates.  And as with light waves, so it goes with sound waves.  And yes, it is possible to do very similar party tricks with sound waves, although that is not the subject for the present column.

When you sit in front of your loudspeaker, the sound projects straight out at you from the various drive units.  But some of it also projects out in other directions.  If it didn’t, then as you moved sideways from your listening chair the sound would tend to disappear altogether.  Think of it like a light bulb illuminating your listening room.  You’d prefer the light to come out in all directions.  If it didn’t, and instead projected a laser-like pencil beam straight down onto the floor, it wouldn’t do much of a job of illuminating the room.  In the same way, you’d prefer the sound from your loudspeakers to also spread out and fill the room.  So the question arises, how exactly does sound spread out – disperse from – your loudspeakers?  Let’s take a peek at the science involved.

The way most loudspeakers work is that they have a surface, usually referred to as the “diaphragm” which is made to vibrate back and forth.  In doing so, it causes variations in sound pressure to build up in the air immediately adjacent to the diaphragm.  Those variations in air pressure in turn give rise to sound waves that propagate away from the speaker, and the question we want to ask is “Where exactly do they propagate to?”  To answer that, we start by treating the overall surface of the diaphragm as a separate bunch of tiny little diaphragms which all vibrate forwards and backwards in unison.  Each of these tiny little diaphragms generates its own individual pressure wave (i.e. sound wave) which propagates away from it.  And if the area of the tiny diaphragm is small enough, then that pressure wave will propagate away from its surface equally in all directions.  In other words, just as much sound will radiate directly forwards as in any other direction.  It is the perfect unidirectional sound source.  You will often hear of such concepts described as an “ideal point source”.

This is the jumping-off point for dispersion theory.  Suppose we are sitting directly in front of a loudspeaker driver whose surface area we are considering as just such a collection of ideal point sources.  Ignoring room reflections, we will only hear those sound waves which radiate straight ahead in our direction.  And because we are directly in front of the speaker, the individual sound waves from all of those point sources will take the exact same time to travel from the speaker to your ears.  Now, you all know about the propensity of waves to cancel each other out.  If the peak of one wave arrives at the same time as the trough of another, then each will cancel the other out through a mechanism called interference.  But in our case, all those ideal point sources are operating in unison, and so they are all generating peaks at the same time and troughs at the same time.  And because they all arrive at our ears in perfect synch, all the peaks will arrive at the same time, and all the troughs at the same time, so nothing will cancel out at all.  Consequently, if you stand directly in front of the diaphragm of a conventional loudspeaker, the sound will always be at its maximum.

Now let’s move away from the center and stand off to one side.  In this case, one side of the drive unit’s diaphragm will be closer to you than the other side.  So the sound waves originating from the ideal point sources located at the nearer side of the diaphragm will arrive before those originating from the farther side.  This time delay means that the peaks from the nearer side can arrive at the same time as the troughs from the farther side, and so we have the possibility that they can cancel each other out.  In other words, if we are standing off to one side of the speaker, it is quite possible that we will hear less sound than we heard standing right in front of it.  This is dispersion at work.

But it gets more elaborate than this if we look closely enough.  The time delay alone is not enough to determine whether two waves will arrive in phase or out of phase.  You need to know the wavelength (or the frequency – the two are directly related).  For a long wavelength (low frequency) sound to end up out of phase you need a correspondingly longer time delay than a shorter wavelength (higher frequency) sound.  So for a high-frequency sound you only need to go a little bit off-center to encounter cancellation, whereas for a longer-wavelength sound you need to go further.  Eventually, when the frequency gets low enough that the wavelength of the sound is more than twice the diameter of the diaphragm, you can never build up enough of a delay to generate complete cancellation.

Of course the picture above is too simplistic.  It is an incomplete description to talk about sounds propagating from the farther side arriving later than those from the nearer side.  In reality the sounds propagating from all of the ideal point sources distributed across the whole surface area of the diaphragm will all arrive with their own individual delays, and the net result involves the summation of all of those individual sound waves, some of which will tend to reinforce each other, while others will tend to provide a cancellation effect.  It is only at the straight-ahead point where the sound waves originating from all of the ideal point sources will arrive in synch.  At all other positions there will be some degree or other of cancellation in play, and that degree of cancellation will be strongly frequency dependent.

There are, broadly speaking, three regimes that come into play.  At low frequencies, where the wavelength of sound is significantly larger than the diameter of the diaphragm, the driver will tend to approximate an ideal point source, and will spread the sound uniformly away from the speaker, regardless of frequency.  At high frequencies, where the wavelength of sound is significantly smaller than the diameter of the diaphragm, the dispersion pattern will tend to ‘beam’, like a flash lamp.  Then there will be the transition zone, where the dispersion pattern remains quite broad across the frequency band, but it neither ‘beams’ nor approximates a point source.

As it happens, these three zones also correspond very well to other zones which constrain the design of a loudspeaker.  In the low frequency zone, the ability to generate a good volume of sound requires that the vibration of the speaker diaphragm displaces a large volume of air.  If the diaphragm itself has a small area, then it has to be able to move back and forth by a correspondingly larger amount.  This creates its own problems, as there are practical limits on just how much displacement a given drive unit design can accommodate.  At high frequencies, where beaming gets to be a concern, we run into issues where either the diaphragm’s high mass or its tendency to flex get to be problematic, and in effect hinder the driver from functioning as a set of ‘ideal point sources’ vibrating in unison.  So practical drive unit design issues end up mandating that – at least for midrange and tweeter drivers – drive units end up operating in the region where there are strong frequency-dependent dispersion signatures.

There is one final dispersion-related loudspeaker performance issue to consider – the crossover region.  Consider, for example, the range of frequencies over which the midrange unit hands over to the tweeter.  The loudspeaker’s crossover handles this, divvying up the input signal so that each drive unit only sees those frequencies that it is intended to handle.  We like to say that there is a “crossover frequency”, above which all frequencies are directed to the tweeter, and below which all frequencies are directed to the midrange unit.  In practice, there is a gradual transition between the two, but that is not the key point here.  The crossover frequency, by its very nature, will be at the lower end of the frequency band which the tweeter is designed to handle, but at the higher end of frequency band that the midrange unit is designed to handle.  Therefore, to the tweeter it will be a “low” frequency, and will disperse widely into the room.  However, to the midrange driver that same frequency will be a “high” frequency, and will tend to project more directly forward.

To a first approximation, the total amount of sound energy generated by a drive unit will be the same, regardless of whether that sound is all beamed straight ahead, or distributed evenly in all directions.  So a beamed frequency will be more intense when measured straight-on than another frequency which is more widely dispersed.  Consequently, if a speaker has a nice flat frequency response measured directly in front of it, then it almost certainly won’t have the same flat frequency response in terms of the total sound energy projected into the room.  To the hapless speaker designer, this all adds up to a thorny problem, because the poor guy started off thinking all he wanted to do was design a speaker with a flat frequency response.  But which is more important – a flat frequency response measured in front of the speaker, or a flat frequency response in terms of the totality of the emitted sound energy?  And if there is a correct answer to that, how will it be impacted by the room the speaker is used in?

These are just some of the considerations that make loudspeaker design such an inexact science, and mean that its most skilled practitioners remain artists and craftsmen first and foremost.