Imagine a Room Full of Mirrors

How loud is your audio system when you play music? Keith Howard recently carried out an extensive investigation into the amplifier powers you need to deliver transient peaks comparable with real-world examples like a full orchestra. The results he obtained indicate that you might well need an amplifier able to deliver peak powers of many hundreds of Watts, and a speaker able to handle these high levels. I found his results fascinating. However I did also have a nagging doubt about what they implied. Put simply, when I listen I find that even when playing orchestral music ‘loud’ I don’t use such high peak powers. So I started to wonder. Are such very high powers needed for a domestic audio system? What other factors might affect the loudness of what we hear?

It is perhaps ironic that I should have started asking myself these questions. My last involvement with working in the ‘audio biz’ was in the late 1970’s and early 1980’s as a designer of amplifiers. During that time I developed an amp that could deliver peak powers well over 400 Watts per channel. Far above the norm at that time in the UK. During that work, and on many occasions since then, I have monitored the levels my power amps were applying to the speakers. I found that the actual peaks were nothing like 400 Watts. Indeed, since I have tended to use Quad Electrostatics of various vintages it would have been a problem to find I was trying to apply such high signal levels!

Thinking about this I felt that two factors might be significant. These were the effects of the acoustics of a domestic listening room, and the properties of human perception that influence how ‘loud’ a sound seems to be.

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Figure 1

To explain what I mean, lets start with the situation illustrated in Figure 1. This shows a listener and a sound source (i.e. loudspeaker) in a room. The red circle represents the loudspeaker and the diamond shape represents the listener. For simplicity I’ll assume the room is a rectangular box and that the speaker radiates in all directions. As is well-known, when you listen in a room the sound doesn’t only reach you via the direct path from loudspeaker to listener. It can also arrive via being reflected by the walls, ceiling, or floor. This means we can expect the sound level at our ears to tend to be greater than in “free space” – i.e. out in the open – because of the added contributions from the reflections. Although I’ve only drawn a few reflection paths in Figure 1 some sound from the speaker will bounce repeatedly from wall to wall before reaching the listener. This behaviour is usually described in terms of the room ‘acoustic’ producing a reverberant sound field which can surround the listener. It’s also often described in terms of room modes or resonances when people are concerned by how it can seriously affect the frequency response. But for simplicity lets concentrate on the basic idea of reverberation.

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Figure 2

A serious problem with acoustics is that working out the details can be fiendishly difficult for any real room. The behaviour depends on all kinds of variables. The room shape. The contents of the room – i.e. what furniture there is and where it is placed. The reflective behaviour of the walls, carpeted floor, windows, etc, etc. It also depends on the details of the sound waveforms. Here I just want to illustrate the general behaviour relevant to the question of perceived loudness. So for the sake of example I will consider some basic rectangular rooms with boundaries that have some average value of sound reflectivity.

Table of examples for room sizes

Room “Small” “Large” “Hall”
Length (m) 3·25 6·50 16·25
Width (m) 3·00 6·00 15·00
Height (m) 2·25 4·50 11.25

To represent a ‘small room’ I chose one whose width is 3m, length is 3·25m, and height 2·25m. These values were chosen on the highly scientific basis of looking at the plans in some brochures for ‘luxury apartments’ on-sale locally and settling on numbers that seemed typical. I deliberately made no attempt to choose a room that was either ideal or as bad as could be imagined. I then imagined two other rooms. These were based on scaling up this ‘small room’ shape to produce a ‘large room’ and a ‘hall’. Of course, in practice the shapes of rooms will vary and affect the results, but these examples are designed simply to indicate the effects of room size so I kept the shapes the same. In each case I assumed the speaker and listener were placed a reasonable distance from the walls.

Figure 2 shows how the sound level we’d hear varies with distance from the loudspeaker. The red line shows what we’d get in the open air, or in a room where all the surfaces were perfectly absorbing. I’ve scaled the values so that 0dB refers to the level we’d measure in this situation if we were 1 metre from the speaker – i.e. the sort of distance used for many of the values given for loudspeaker sensitivity. As you’d expect, the red line shows a sound level that falls away with the inverse square of the distance. This is the textbook behaviour you may have encountered if ‘sound’ was a topic you covered in school physics.

The blue and green lines show what we’d get in the small or large rooms. Comparing these with the plot for open space you can immediately see that the reverberation has a quite significant effect. In this example I assumed the reflectivity of the walls was 80%. The result is that if you sit at a reasonable distance from the speaker then the sound powers you hear may be between 10dB and 15dB higher than if the wall reflections weren’t present! Now it is important to bear in mind that these results do vary as we change various values such as the reflectivity of the room surfaces. Hence the effect will differ from one domestic system to another. But it should be clear from this that the effect of the room can be quite dramatic. A change of 10dB means you could use a 50W amplifier in the room to get the same perceived sound level as using a 500W one out in the open!

The main problem with the above for our present purposes is that we aren’t comparing like-with-like. Keith’s series of articles were looking at the powers required for transient peaks in the music. Whereas the above is the effect of room reverberation upon sustained sounds, not a brief transient peak. When the sound is a brief transient, the echos turn up after the direct sound. This means the echos arrive too late to change the level of the initial transient. So we now need to switch our attention to the transient sounds.

Most of the descriptions and models you see of room effects tend to be based upon the reverberation/modal presentation which describes what is happening in terms of the steady levels produced by sustained sounds. But let’s look at the effect of the room in another way. Imagine yourself in a room full of mirrors.[1]

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Figure 3

Imagine what you might see if the room walls were all mirrors and the loudspeaker was a light. Looking at the walls you’d see reflections of the reflections of the... Well, in audio terms we can use a similar idea as illustrated in Figure 3. We surround the real room with a series of reflected images of the room. Each one contains an image of the speaker. We can now ‘unfold’ a real-world sound path from speaker to listener and straighten out each wall reflection to represent the signal as if it came via a straight line from one of the mirror images of the speaker. The example in Figure 3 shows a solid blue line for a path which reaches the listener after three wall reflections. The broken blue line is the equivalent path from the appropriate image of the speaker in the wall ‘mirrors’.

This way of looking at the room acoustics is useful for various reasons. We can now see how long each sound path will be, and how many reflections it undergoes. It makes it easier to predict the relative amplitudes of the reflections which reach the listener. It also makes clear that they arrive after a time set by the lengths of the paths. In fact, for a physicist who is familiar with topics like crystalline structures this approach is a fairly comfortable one. It is particularly useful when we realise that reflections can also reach the listener via the ceiling and/or floor as well as the walls. In effect our listener in the room is surrounded by a three-dimensional ‘crystal lattice’ of images of the speaker, all sending their contributions to arrive sometime after the direct sound.

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Figure 4

Figure 4 illustrates the effects of the room by showing the direct signal (red) reaching the listener, followed by the echos which seem to come from the images of the loudspeaker in the mirrors. For these calculations I also included the effect of a ceiling and floor, but for simplicity assumed they had the same reflectivity as the walls. The blue lines indicate the echos and represent both their relative times of arrival and power levels. I assumed the listener in the hall was near the back wall, and this means a few echos arrive soon after the direct sound even in the hall. Discussions of topics like reverberation time in books tend to talk in terms of reflectivity and surface areas, etc. But by looking at Figure 4 you can see that the reason a larger hall tends to have a longer reverberation time than a small room is that the reflections take longer to arrive as the path lengths are (mostly!) relatively long. This means a bigger room spreads out the reflected sound power over a long duration. Note that for the time/power plots of the rooms in Figure 4 I have shaded pink a period of 30 milliseconds starting with the time of arrival on the direct signal. The reason I did this is as follows...

Research has found that echos which arrive within about 30 milliseconds of the initial sound aren’t normally heard as echos or reverberation. Instead they affect our perception of how loud the initial sound seems. Echos that arrive much later than 30 milliseconds tend to be perceived as coming ‘after’ the initial sound that caused them. This result has a very interesting implication when you are listening in a room or hall. In a large hall most of the echos arrive later than 30 milliseconds after an initial transient, so produce the effect we recognise hall reverberation. But in a small domestic listening room many more echos arrive within 30 milliseconds – as can be seen by looking at what happens during the pink-shaded portions of the graphs in Figure 4. These ‘early’ echos are heard as if the initial transient was more powerful. i.e. The transient seems distinctly louder in the domestic room than if we’d been out in the open and sitting at the same distance from the speaker.

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Figure 5

Figure 5 shows how many echos arrive during each 5 millisecond period when we are in each of the rooms/hall I’ve been using as examples. The times are plotted so that they all start at the instant the transient leaves the speaker. Note that in the small room many hundreds of echos arrive within 30 milliseconds. Doubling the size of the room has a dramatic effect and far fewer arrive in the same period. In a large hall only a small number arrive during this time. However Figure 5 doesn’t take into account the relative energies of the individual echos as they arrive at the listener. To see that we can examine the results plotted in Figure 6. For that I worked out the relative energies in the individual echos relative to that of the initial transient reaching the listener and then plotted the total sound energy arriving during each 5 millisecond interval. In this case the plots compare our small and large rooms with a ‘dead’ room (plot in red) which has surfaces which have almost no reflectivity. Thus the red plot approaches what we might get out in the open. No surprise that the result for a dead acoustic is that the sound level is essentially ‘1’ – i.e. just the energy of the directly arriving transient! As previously I’ve shaded in pink the first, crucial, 30 milliseconds.

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Figure 6

Figure 7 shows how the number of reflected echos varies if we scale up the room length whilst keeping the shape unchanged. You can see that with a room less than five or six metres long the number of reflected contributions that arrive during the first 30 crucial milliseconds is above a hundred. As the room size shrinks the number of echos within 30 milliseconds rises rapidly, showing that our experience would change a great deal if we altered the room size while keeping everything else the same. Of course, echos that have reflected from the room boundaries many times will be relatively weak. So the power reaching the listener does not rise so steeply as we reduce the room size.

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Figure 7

Figure 8 shows how the results of adding together the sound energy in the first 30 milliseconds varies with room size. In this case I have plotted the results in terms of decibels relative to the energy of the transient sound with no reflections. I have also plotted results for two different values of the reflectivity of the room boundaries. We can see that in a small, reflective room we can get values over 10dB above the level of the initial transient. A much larger room, or a lower reflectivity reduces this to around 5dB or less.

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Figure 8

These results have to be interpreted with caution for a number of reasons. The actual behaviour will vary a great deal from one room/speaker combination to another. But the results do show that we can expect a significant amount of enhancement of the loudness we perceive when listening to a transient sound in a small domestic room. If we assume a minimum value of around 6dB this would translate into only needing a 100 Watt amplifier in the room to replicate the same perceived sound level as a 400 Watt amplifier out in the open. For 10dB this falls to 40 Watts. So, particularly in small rooms which have less than idea acoustics we may find we don’t need very high amplifier powers. Indeed, the use of the higher powers may lead to excessive perceived sound levels which aren’t a faithful replication of what we’d experience at a real orchestra concert or similar even using acoustic instruments. In terms of sound power 6dB is a fairly modest amount, but it becomes significant when the choice is between a 400 Wpc amplifier/speaker combination and one for 100 Wpc!

Of course, Keith Howard was quite correct in what he showed about the powers needed to obtain a given peak transient sound pressure level in a room. The above does not change his results. But the human perception when room reflections are taken into account tends to boost the transient (and sustained) sound levels we hear. The good news is therefore that – depending on your circumstances – you may be just fine using lower powers. No need to disturb your neighbours or bank manager by buying a bigger amplifier! However I hope to investigate this further in future and in particular look at the interactions between the room acoustic and loudspeaker directionality to see what effects that can have on what we hear.
Jim Lesurf
2700 words
11th Apr 2008

[1]   OK. I admit I simply wanted here to be able to use a sentence that included one song-title by Lennon, and another by Hendrix!

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