Loudspeaker Technology Part 4: The Frequency Domain and Human Hearing

In Part 3 of this series on speaker technology, we saw that accurate loudspeakers need to consider the time, space and frequency domains. Now it is useful to consider what that means in terms of arriving at some kind of specification for a real loudspeaker.

The development of digital audio was a tremendous boon for audio quality because by recording data using error correction, the sound quality became independent of the medium. Audio could be stored on tapes, hard drives, RAM and optical discs or transmitted down wires, radio links or optical fibres without any loss of quality beyond that due to the initial conversion. Basically digital audio is time accurate because the sound waveform can be preserved. Microphones can, and often are, made time accurate and audio power amplifiers traditionally have been tested with square waves to prove it.

This led to the bizarre situation in which sound waveforms could be captured, stored and amplified very accurately and delivered to a legacy loudspeaker that would destroy them. There seemed to be a schism between manufacturers of microphones, recorders, desks and amplifiers who regarded phase linearity as important and loudspeaker manufacturers who said it didn’t matter or it couldn’t be done. Clearly both points of view could not be correct.

The physics do matter

One of the ways in which one knows one has become a cynic is when the realisation dawns that someone who claims something is unnecessary or impossible is doing so simply to avoid having to admit that they don’t now how to do it. Since that realisation, I only accept impossibility when the laws of physics need to be violated.

There is thus a marvellous symmetry whereby manufacturers, who hold to be impossible things that physics does permit, sell products to hi-fi enthusiasts, who hold views that physics does not permit.

All of the evidence suggests it is those who hold time accuracy to be important that are correct. A modern understanding of human hearing suggests that it is theoretically important and the dramatic increase in realism that is obvious to any unbiased listener when the original sound waveform accurately traverses the entire reproduction chain confirms that it is practically important.

Photo 1. There appears to be no shortage of creativity in the field, hence the wide range of cabinetry designs offered in today’s loudspeakers. The above system is called Pnoe and is produced by Arcadian Audio. Cost $25,000.

Photo 1. There appears to be no shortage of creativity in the field, hence the wide range of cabinetry designs offered in today’s loudspeakers. The above system is called Pnoe and is produced by Arcadian Audio. Cost $25,000.

Those who are familiar with digital imaging know that the smaller the pixels are the sharper the image becomes. In audio the equivalent is that the smaller the acoustic source is, the sharper the image. Acoustic source has an idiomatic meaning in the context of loudspeakers: it is the place from which the sound produced by a loudspeaker appears to come, in three dimensions. Ideally the acoustic source should be a fixed and vanishingly small point. In most legacy designs it is neither, for reasons which we shall explore.

There is another aspect of the spatial domain that is important. This is that the frequency response should be the same in all the directions in which sound is radiated. That is the same as saying the directivity pattern is independent of frequency. If this is not the case, the speaker may fail to excite reverberation that can be identified as such by the ear because it will be coloured.

In some respects meeting an advanced specification such as that outlined here is difficult. But in other respects it is easier, because if all of the technologies, architectures and components that cannot meet the specification are discounted, the final choice must be made from a smaller list.

Speaker diaphragm

Out of many ideas that have been tried, the most successful way of reproducing sound is the moving diaphragm. As was explained in an earlier article, air cannot sustain a local pressure difference. It leaks away and the more time there is available and the smaller the source, the more powerful the leakage. Thus a naked diaphragm, what we call a dipole, of moderate size oscillating at low frequency radiates next to no sound because the pressure increase on one side and the reduction on the other side are cancelled by air moving around the edge.

To reproduce the lowest audible frequencies, the dipole has to be tens of feet across and this is not feasible. Thus all practical reproduction of the lowest audible frequencies requires some baffle or enclosure that prevents the radiation from the two sides of the diaphragm from cancelling. That forms a subject in itself.

For a given diaphragm area, the tendency for air pressure to leak away means that in order to obtain a flat frequency response, the amplitude of motion of the diaphragm must rise as frequency falls, at 12dB per octave to be precise.

Figure 1 shows the peak displacement needed by a diaphragm of a given diameter to radiate 1 Watt at a given frequency. Note how the amplitude rises as frequency falls.

Figure 1 shows the peak displacement needed by a diaphragm of a given diameter to radiate 1 Watt at a given frequency. Note how the amplitude rises as frequency falls.

A good speaker needs a large diaphragm as this chart illustrates. As the frequency falls, the speaker needs either a large diaphragm or large radiator displacement.

Figure 1 illustrates the link between frequency, displacement and amplitude for constant power. It will be evident that as frequency falls, the designer is pushed towards large diaphragm area or large displacement. It should immediately be obvious why we must not expect much low frequency sound from iPhones or tablets. It is also clear why we can see woofers moving, but not tweeters.

Doubling the diameter of the diaphragm quadruples the area, and so the displacement can be reduced by a factor of four. This trade-off gives rise to the concept of volume velocity, which is the product of the diaphragm area and the velocity. All combinations that have the same volume velocity radiate the same power. Volume velocity is a misnomer, because it is not a vector quantity.

Another difficulty is that the sensitivity of the HAS (Human Auditory System) to low frequencies is not very good, so there is no point in having an extended low frequency response if sufficient level cannot be created.

Self-evidently, for good ability to radiate power, a large diaphragm is a good thing.

As noted, the tendency of air to leak away is frequency dependent, so any diaphragm moving with constant velocity would display a rising frequency response which would be no good for sound reproduction. More precisely, the level radiated would be proportional to frequency, or rise at 6dB per octave, and thus the power radiated would be proportional to the square of the frequency, or rise at 12db per octave.

Early speaker design

The solution to this problem was one of the most seminal discoveries in the history of audio which set out the principle on which the great majority of loudspeakers work to this day. In the 1920s Edward Kellogg and Chester Rice were working at General Electric on the problem of getting enough sound level from radio receivers. They understood the physics of moving masses. 

Figure 2. Basic Rice-Kellogg moving coil speaker.

Figure 2. Basic Rice-Kellogg moving coil speaker.

It is sufficient to describe the position of a diaphragm with respect to time. The rate of change of position (the first derivative) is the velocity and the rate of change of velocity (the second derivative) is the acceleration. Position, velocity and acceleration with respect to frequency are linked by 6db per octave functions. What Rice and Kellogg did was to realise that if the acceleration of their diaphragm was held constant with frequency, the velocity would fall at 6db per octave which would cancel out the rising radiation efficiency and result in a flat overall frequency response.

It follows from Newton’s Laws that the acceleration of masses, diaphragms included, is proportional to the applied force. What Rice and Kellogg needed was a motor that applied a force proportional to the audio input waveform. They found that solution in the moving coil motor, shown in Figure 2, which is a subject into itself.

Interestingly the Rice-Kellogg loudspeaker launched in 1926 as the Radiola model 104 was also an active loudspeaker as it necessarily contained an audio amplifier powerful enough to drive the moving coil.

The significance of the Rice-Kellogg speaker is more than academic. By making it possible to reproduce sound at realistic levels, they essentially created the audio industry.

With the exception of inhospitable locations, the speed of sound where anyone would want to live is about 340 metres per second: the equivalent of about one foot per millisecond, which is easier to remember, or a mile in five seconds. In contrast the speed of light in vacuo is about a million times faster, or one foot per nanosecond.

The lowest frequency anyone can hear is about 20Hz, and oddly, hi-fi enthusiasts don’t challenge it. 20Hz corresponds to a wavelength of about 17 metres or about 56 feet. The highest frequency a young person can hear is about 20kHz, although some hi-fi enthusiasts believe, and it is a belief, that higher frequencies can be heard. 20kHz corresponds to a wavelength of about 17mm or about three quarters of an inch.

John Watkinson, Consultant, publisher, London (UK)

John Watkinson, Consultant, publisher, London (UK)

This extraordinary range of wavelength, spanning some ten octaves, ranges from wavelengths that are considerably larger than most everyday objects to wavelengths that are considerably smaller. We must expect some change in behaviour because of that.

In contrast, visible light exists over a range of less than an octave and the wavelengths are always significantly smaller than everyday objects, making visible light behave much more consistently than sound. For example it is possible to obtain deep shadows when light encounters an obstacle. That simply doesn’t happen with sound.

The ten-octave range of wavelength fundamentally affects loudspeaker design and it will be necessary to consider wavelength related effects to see why. We will do that in the next instalment.

Parts 1, 2 and 3 of this loudspeaker series can be found at the links below.

Editor’s note;

John Watkinson has a new book readers may wish to view.

In The Art of Flight, John Watkinson chronicles the disciplines and major technologies that allow heavier-than-air machines to take flight. The book is available from Waterstones Book Store.

In The Art of Flight, John Watkinson chronicles the disciplines and major technologies that allow heavier-than-air machines to take flight. The book is available from Waterstones Book Store.

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