what is a waveguide?: The term waveguide is a generic term used to describe an acoustic device that transforms the radiation characteristics of a speaker driver. The best known example is the exponential horn. Horns have traditionally been used to both control radiation characteristics and to increase efficiency. A speaker diaphragm coupled directly to the surrounding air represents a poor impedance match. A horn acting as an acoustic transformer can be used to more effectively couple the energy of the driver. The horn acts as an impedance matching transformer between the small speaker diaphragm and the much larger air load. However for domestic or studio use the efficiency gain of a horn is of less importance than it is for PA applications. If the requirement of maximum efficiency is relaxed, the designer is freed to pursue other approaches to horn design and can concentrate more on the aspect of radiation control. The rate of expansion of a horn can be radically increased so that it adds very little to efficiency. At this point the acoustic device is no longer considered a horn and becomes known as a waveguide. Another key difference between a horn and waveguide is that a horn is usually used with a compression driver whilst a waveguide is usually coupled to a more conventional speaker diaphragm such as a dome tweeter without a compression chamber.

aspects of a waveguide: A horn or waveguide has several basic design parameters that can be varied. The region close to the speaker diaphragm is the throat, the other end is the mouth and depth related to both the throat and mouth is the rate of expansion. The expansion can be straight or curved according to various geometric series. Examples include exponential, hyperbolic and tractrix expansions. A waveguide has a very rapid expansion from throat to mouth, often appearing as more of a slight recess in the baffle where the speaker driver is mounted. In the extreme, a flat baffle is a waveguide with the most rapid expansion. Here the radiation pattern is hemispherical for wavelengths where the baffle is large compared to the wavelength of sound. In any case, for a waveguide to be effective, the dimensions of the mouth must be large compared to the wavelength of the lowest frequency of operation. For frequencies lower than this, the waveguide becomes invisible and the driver operates as if the waveguide wasn't even there. A some point the mouth of the waveguide will be attached to the front baffle of the speaker system and the transition region needs to be smoothly terminated in order to avoid sound waves reflecting back down the waveguide and causing interference effects. The region around the throat is even more critical. The air pressure is relatively high in the throat region and the velocity low. A smooth transition from diaphragm to throat is essential to avoid the creation of cavities that can resonate and reflect energy back towards the diaphragm. Because of the small wavelengths involved at very high frequencies, the dimensions around the throat coupling need to be carefully considered.

 

waveguide design: For a waveguide, the size of the throat will be fixed by the size of the diaphragm. A horn, on the other hand, uses compression at the throat to raise efficiency. A typical 25mm tweeter coupled to a waveguide therefore needs a 25mm throat aperture. The size of the mouth is fixed by the lowest frequency of operation of the waveguide. I have empirically determined that the diameter of the mouth needs to be approximately the same as the longest wavelength to give good pattern control. This is somewhat dependant on the shape of the expansion, the included angle and termination of the mouth to the front baffle, however this rule is a good first approximation. So to be effective down to 2kHz, which is comfortable lower limit for a typical 25mm dome tweeter and an included angle of about 120 degrees requires a waveguide with a diameter of about 180mm. The radiation pattern will be similar to the included angle for most waveguide designs. Typical designs use included angles of 120 to 150 degrees. This corresponds to a directivity increase of about 3 to 4dB over a flat baffle. This is a nice efficiency gain that can translate into lower distortion for the same SPL or alternatively, a higher maximum SPL at the listening position. Smaller included angles will increase the efficiency further. At some point as frequency increases, the natural narrowing of the beamwidth of a piston diaphragm will mean the waveguide is no longer fully illuminated and directivity will increase further. This effect has little to do with a well designed waveguide and maintaining wide dispersion within the confines of the included angle of the waveguide requires a speaker with a suitably small diaphragm. For example, a 25mm tweeter loading a 120 degree waveguide will begin to lose pattern control above about 8kHz. The above parameters will combine to fix the total depth of the waveguide. This depth can have a fortunate side effect because it allows the acoustic centre of the tweeter to more closely match that of a midrange or bass driver relative to the front baffle of the speaker system.

 

testing, testing

I decided the easiest way to find out how well waveguides actually work and which design parameters matter was to look at work that others had already done. There are a number of commercially available speaker systems that use waveguide loaded tweeters such as nearfield monitors made by Genelec, JBL, Mackie, Behringrer and others. I was able to approximately measure the waveguides on a few of these systems and the basic designs all seem quite similar; a conical section from the throat termination with a generous radius transitioning from mouth to front baffle. I was able to acquire a waveguide baffle and tweeter assembly from a pair of commercially made monitors and begin looking at its behaviour...

Off-axis behaviour of the commercially manufactured tweeter/ waveguide assembly:

This an unequalised response, and clearly shows reducing amplitude as frequency increases. This is the natural behaviour of a waveguide loaded by a speaker driver with a nominally flat amplitude response when mounted on a flat panel. All similar waveguide loaded systems require electrical equalisation to render the amplitude response flat.

The power response, which is the main focus of this graph, looks excellent. The amplitude drops consistently as the test microphone is moved off-axis. In other words the tweeter will sound the same as the listener moves off-axis, just with a reduced SPL. Not much useful information is available below 2kHz as the natural tweeter response drops away; but the waveguide appears to be doing its job down to at least 2.5kHz. At around 15kHz there is a peak that appears to be due to a resonance within the tweeter diaphragm and has nothing to do with the waveguide. The on-axis amplitude response shows an unusual dip at about 13kHz that's unrelated to the off-axis plots. The reason for this will become clearer later. 

The small ripples are due to various recesses and features moulded into the waveguide for fasteners and indicators. With more careful attention to mounting, the ripples would disappear.

So it looks like waveguide loading does indeed work as advertised, providing even off-axis performance, and so a power response that follows the on-axis response very well.

horn loading a dome tweeter

expansions

As already mentioned, commercial waveguide implementations appear very similar to a conical expansion for most of their depth with a large radius transition region from mouth to baffle and a throat region that sometimes also has a small radius that couples the waveguide to the speaker diaphragm. The available literature from Earl Geddes, Peavey etc suggest suitable mathematical contours that can be used in successful waveguide designs. For the most part the literature is concerned with waveguides loaded by compression drivers and with relatively small included angles. In these cases the exact mathematics behind a given expansion are very important. But for the case of waveguide with a very wide included angle and direct coupling to a speaker diaphragm, the practical implementation limitations mean that most of the mathematical detail can be side-stepped.

In practice, all that is needed is a simple conical section that determines the radiation angle and a mouth with a diameter approximately equal to the lowest frequency of operation. The radius of the transition region at the mouth is well served with a radius approximately equal to about 1/4 to 1/2 the diameter. The transition region at the throat requires the most care. Given that a conventional dome tweeter will be loading the waveguide, then there must be a clean termination of the throat of the waveguide to the tweeter face-plate without a cavity being formed. So either a tweeter must be selected that has a dome that sits completely proud of the face-plate, or one that has a small radius to the face-plate. Available tweeters are all recessed to some degree so it becomes a matter of selecting one with as small a recess as possible and one with a smooth radius.

A close-up view of a commercial waveguide loaded tweeter and throat termination:

Note the insert that provides a smooth throat transition from diaphragm to the rest of the waveguide.

more testing

So let's go back to the commercial waveguide examined earlier and mate this with a standard dome tweeter.

Off-axis behaviour of equalised 25mm tweeter on a commercial waveguide:

Using a popular 25mm tweeter with a very small recess from faceplate to dome, the commercial waveguide shows an excellent power response from 3kHz up to 8kHz. Even above 8kHz the roll-off is less severe than the same tweeter mounted on a plain baffle. There is a serious dip in just the on-axis response at 7.5kHz. It's serious not only because of the size of the dip, but also because it's only observed in the on-axis response and not in the off-axis response. It can therefore not be electrically corrected without causing a complementary peak in the off-axis response. The dip appears to be due to a cavity resonance and reflection caused by the less than perfect throat termination.

Given that the throat termination can never be perfect, one way of dealing with throat resonances and reflections is to employ a phase-plug. A phase-plug is an acoustic device that equalises the path length of sound waves coming from different parts of the diaphragm. Physically it is usually constructed as one or more concentric or radial barriers. In the case of a waveguide it seems to break the throat region into a series of concentric radiators all operating synchronously, and thus help prevent the formation of resonances and reflections.

Off-axis behaviour of equalised 25mm tweeter on commercial waveguide with phase-plug:

The phase-plug employed here is simply a 10mm thin plastic disc held centred just in front of the diaphragm. It fixes the dip at 7.5kHz but causes a new problem at 12-18kHz, a small dip followed by a very steep dip due to cancellation in the top octave. This tends to be a characteristic of tweeters fitted with phase-plugs and but is only a major factor at larger off-axis angles.

Interesting, looking very carefully at the original tweeter with the waveguide revealed a similar invisible phase plug that I hadn't previously noticed. The dome is covered by a protective grille and beneath the the grille is a thin sheet of transparent plastic.

ring radiator

An alternative to a tweeter with a phase-plug is a ring radiator. A ring radiator has a ring shaped diaphragm which is fixed along its outer and inner edges. It solves most of the problems associated with fabricating phase-plugs and matching them to the speaker diaphragm and waveguide throat. It still exhibits off-axis cancellation at very high frequencies, but the rest of the operating range is smooth and consistent on and off-axis.

Off-axis behaviour of equalised 25mm ring radiator on the commercial waveguide:

This is really excellent performance. Directivity is superbly well controlled all the way from 3kHz to 9kHz. Even over the extended range of 2kHz to 12kHz, the results are excellent. There is a very slight peak in the on-axis amplitude response at 8kHz, but as this is evident in the off-axis response as well, it can be corrected electrically. It is also present when the tweeter is used without a waveguide. Even with the small peak, on-axis response is within +/- 1dB from 2kHz to 16kHz. At 60 degrees, which design limit of the waveguide, off-axis directivity control is beginning to falter. The drop in the off-axis performance in the top octave is a function of the driver diameter and can be improved by using a smaller diameter ring radiator.

eqalisation

As already discussed, equalistaion is required with a driver that exhibits a nominally flat on-axis amplitude response mounted on a flat baffle when that same driver is used with a waveguide. In general the equalisation needed to flatten the on-axis and power response is a simple HF boost of +6dB/oct. This can be realised with just a first order filter such as a series capacitor network. The 6dB drive reduction at lower frequencies created by the network also has the happy benefit of improving the low end distortion figures.

how it sounds

A single tweeter on its own doesn't really give much of an insight into how a system might sound. But listening to pink noise certainly confirms the relationship between on-axis and off-axis performance observed in the above tests. Instead of the character of the pink noise changing as the listening position moves further off-axis as happens with a conventionally mounted tweeter, the waveguide loaded tweeter simply gets less loud. It's an interesting experience after being used to the way a conventional tweeter sounds.

conclusions

I've tried to show how a relatively simple waveguide can improve the performance of a direct radiator tweeter. The important points include:

	Waveguides provide consistent power response
	Waveguides need to be flush mounted
	Careful attention is needed at throat and mouth terminations
	A simple conical expansion has the best performance
	Phase-plugs can eliminate mouth reflections caused by imperfect throat termination
	Ring radiators give the most effective throat coupling
	Simple electrical equalisation is needed to flatten the on-axis & power response
	Waveguides reduce distortion at lower frequencies
	Waveguides can assist with time alignment

further research

Given that throat termination is critical to achieving good waveguide performance and that ribbon tweeters tend to have poorer low frequency distortion characteristics, it will be interesting to see how ribbon tweeters fare when similarly configured.