**Room Acoustics**

The advent of high speed digital signal processing has raised the possibility of correcting room acoustics with a digital filter. Some of the available systems are noted in the section on signal processing.

Both measured and calculated results are presented in this section. Calculations are based on image analysis. For all calculations the "subwoofer" itself is idealized; it is assumed to generate sound at an absolutely flat level from 0 Hz to 200 Hz, so we can focus attention on the room itself. The measured data unavoidably includes the effects of a loudspeaker. Measurements were made using the CLIO system.

A lot of the literature on this subject is mainly directed towards the design of professional sound studios, where the acoustics should be fairly uniform over the entire room. As far as I am concerned, a "sweet spot" where my ears will be located is (literally) the focal point of my design - although it is generally desirable to make this spot as large as possible. The ideal frequency response is flat with no peaks and valleys. This could only be obtained inside a perfect anechoic chamber - a room where the walls absorb sound without any reflections. This is not a feasible solution for most of us (actually it would probably sound pretty weird), and for almost any sound system the room will pretty much determine the frequency response at low frequencies. Room effects virtually always swamp out imperfections in drivers, enclosures, crossovers, and yes, even cables (joke).

- The nature of room resonances is reviewed (immediately below).
- The standard modal approach for "optimizing" room dimensions is presented.
- Measured and calculated results are compared.
- The best location for a subwoofer is discussed.
- Examples of spatial variation within a room are presented.
- Examples of temporal variation are presented.
- Calculations are presented comparing an optimum room and a horrible room.

A major result is that the difference between a room with optimum dimensions and horrible dimensions is not as much as you might think from all of the attention devoted to the subject - all rooms look horrible. Given this, we end by asking: is all this analysis worth anything?

The physics of room resonances is derived and discussed in detail in the physics section. Any room (including rooms with odd shapes) will resonate at many frequencies. The bass response is sharply boosted for a narrow frequency band near resonance, and then is depressed between resonances. A figure derived in the physics section [5.44 kb] shows the response for one series of room modes. The sharpness and height of the resonant peak depends on the sound absorbing properties of the room. A room with a lot of soft furniture, heavy carpeting and drapes will be relatively "dead," and the peaks and valleys of the frequency response typically vary by 5-10 dB. A room with bare walls and floor will be very "live," and the peaks and valleys vary 10-20 dB or more. The absorption coefficient of 0.2 used to produce the figure referred to above is about average, corresponding to a reverberation time of about 1/2 sec. A web site with a handy calculator for resonances for a rectangular room, and for reverb time, can be found at the here.

A common misconception is that resonances can be eliminated by curving the walls. As an extreme example, this figure [43 kb] compares the resonant frequencies of a cubical room, and a spherical room, of the same volume. The frequencies are similar. For more details see Resonances in a Spherical Cavity

The standard modal approach for designing a room with good acoustics is to create as many different resonances as possible, and to spread them as evenly as possible across the frequency spectrum, as discussed in the Handbook for Sound Engineers, Chapter 3. There is even a complicated "Bonello Criterion" to evaluate the spread. The lowest resonance is determined by the largest dimension of the room. (Technically there is also a resonance at zero Hz for all rooms, but this is generally not considered a true resonance). In general, the lower the better for the first resonant frequency, because this region is where the frequency response is most variable. Bigger rooms also reduce the spacing between resonances. The limiting factor here is usually cost. For a 19-foot long room the first resonance is about 30 Hz. Every harmonic of this frequency (60, 90, 120, etc.) is also a resonance. The width and height of the room each give rise to another series of resonances. These are the primary "axial" resonances, involving reflections from two opposing surfaces. Additional resonances are created by reflections that ricochet off four different surfaces. These "tangential" resonances are generally weaker, because energy is lost at each reflection. Finally there are "oblique" resonances which ricochet off all six surfaces. Each resonance gives rise to a "mode" with a characteristic spatial pressure variation. The mathematical description of a mode is given in the physics section, and some graphical examples are illustrated below. To spread these resonances as uniformly as possible, various ratios between the room height, width, and length have been proposed. Three such sets from the Handbook for Sound Engineers are shown in the table below.__"Optimum" Room Dimensions__

Dimension |
Design #1 |
Design #2 |
Design #3 |

Width |
1.14 x Height |
1.28 x Height |
1.60 x Height |

Length |
1.39 x Height |
1.54 x Height |
2.33 x Height |

According to the modal design theory, the worst possible room shape is a cube. The next worst is a room where all dimensions are multiples of the height. A pretty horrible example is a room 8-ft high, 16-ft wide, and 16 ft long. The resonances for an optimum room (design #3) and for the latter horrible example illustrate the difference in the resonances [10.7 kb]. The two rooms have the same total volume. The horizontal frequency scale varies from 0 to 200 Hz. Each vertical line represents a resonance. There are three tiers of lines; the highest tier represents axial modes, the middle tier tangential, and the lower tier oblique. The resonances for the horrible room are less dense, because many occur at exactly the same frequency, and there is a fairly large gap between the second and third resonances, at about 60 Hz. The blue, green, and red lines represent resonances related to the room length, width, and height, respectively.

**Measured and Calculated
****Frequency Response of My Music Room
**

Calculations in this section are obtained using a computer program based on image analysis. To indicate how closely the results correspond to the real world, here we compare computed and measured data. My music room is not totally rectangular (see the floor plan [4.2 kb]), it contains furniture, and the absorption coefficient of the floor and walls is different. The image calculations assume an empty rectangular room, and the same absorption coefficient at all surfaces. The measured data involves a real loudspeaker; the image calculation assumes a perfect speaker. So one does not anticipate terrific agreement between measured and calculated response curves. The frequency response was calculated at the listening location shown in the floor plan, at the elevation of my calibrated mike. The room has a measured reverberation time of 1/2 sec at 125 Hz, and using the table in the image analysis section, this corresponds to an absorption coefficient between .20 and .25. The agreement between the results [42.8 kb] is not too bad; the red curve is computed, and the black measured. Some features are shifted around, but I would say that the flavor of the real response is well captured by the computation. The agreement is actually quite a bit better than I expected. The lines across the top show the modal resonances of the room. The calculated results show a peak at the 1st and 4th resonances. The measured results have lower peaks at somewhat shifted frequencies. Why aren't there peaks at the 2nd and 3rd resonances? Mainly because these modes have a dip in their response at the sweet spot location where the response is calculated. Examples of modal spatial variation are shown below. A comparison of the measured and computed time-domain response [8.4 kb] indicates a similar degree of agreement. The blue curve is the measured response, and the red "+" marks are the computed response. (I removed my sound absorbing panels for this measurement. A similar measurement with the panels [5.8 kb] in place shows the reduction in the first wall reflection produced by the absorbing panels - the black curve compared to the red. This latter measurement used a different microphone placement, and has a different time scale). The reflection surfaces for the first six echoes are identified in the measured/computed figure. Most measured echoes arrive pretty close to the computed time. The timing discrepancies are less than 0.2 milliseconds, which corresponds to a path length difference of about 3-inches. I don't think my measurements were off by this much, but it is still quite reasonable agreement. There are a few stray reflections in the measured data that probably come from furniture in the room. One significant difference is that the levels of the computed echoes, relative to the direct path, are much higher than the measured values. This is apparently due to two factors: (1) the directionality of the microphone, which was pointing directly at the real speaker, lowers the echo responses, and (2) all of the calculated echoes are perfect impulses; the real-world reflected pulses are smeared out in time compared to the direct path pulse, reducing the relative peak value. This smearing is too small to effect the computed frequency response at low frequencies.

**Where Should the Subwoofer Be?**

Modal theory tells us that a subwoofer in a room corner will excite all modes. In contrast, for a subwoofer perfectly centered along one wall, several modes will not be excited at all. So that's helpful information, but it really doesn't give much insight into what the response will actually be. Using image analysis, the frequency response can be calculated at any point in the room, for any subwoofer location. It turns out that on paper the smoothest low-frequency response is obtained when the subwoofer is as close as possible to my ears - like sticking my head in the cone! The reason for this is that the sound I hear is then dominated by the direct radiation from the subwoofer, rather than by room resonances. However, when I experimented with this in the real world, I found that I was painfully aware that the subwoofer was behind my head, badly spoiling the musical illusion. This is most likely due to the crossover letting some high-frequency stuff through the subwoofer. In any case, I threw out this option. However it is an interesting example of how engineering calculations always need a reality check! The next best location I found is the corner of the room, consistent with the modal theory. Moving it a few feet one way or another doesn't make much difference.

Using my image analysis computer program I calculated the response in a plane 39 inches above the floor, at four frequencies, for a room 8.2 x 13.1 x 19.1 feet (design #3 of the above table), to illustrate spatial variations at low frequencies [18.4 kb]. There is one subwoofer in the lower left-hand corner illustrated by the red circle. Many modes are present at each frequency, but at the lower frequencies often a single mode dominates. The lowest axial mode has a resonance of 29.6 Hz, and the pattern at 25 Hz is similar to the classic pattern for this mode. The lowest tangential mode has a resonance of 52.2 Hz, and the pattern at 50 Hz is quite close to this mode pattern. The next highest tangential mode has a resonance at 73.2 Hz, and the pattern at 75 Hz is very close to this mode pattern. These results also add to the credibility of the image analysis computer program, which contains absolutely no information regarding modes. As the frequency increases, more modes tend to be excited and the spatial variation becomes more complex, resembling the pattern of a single mode less and less. If your listening position happens to be at a location where there is a null of one of these mode patterns, at the resonant frequency of that mode you will hear no response instead of hearing a strong resonant response. A graph of the measured early time response [same 8.4 kb figure as above], and the comparison with the computed echoes, has been discussed above. The time segment in this figure is 20 milliseconds. Room reflections are significant for the reverberation time, typically 1/2 second or so. The number of reflections grows geometrically with time. The computed echoes for the first 100 milliseconds demonstrate the temporal behavior [5.3 kb]. Each dot in the figure represents a discrete echo (or equivalently a single loudspeaker image). The time of arrival at the listening position is shown by the horizontal axis, and the strength of the echo by the vertical axis. There are 1,482 echo-dots in the first 100 milliseconds for this 8 x 13 x 19 foot room. An absorption coefficient of 0.25 was used in this calculation. The dots at the top of the figure represent reflections from the opposing walls with the largest separation, which have the lowest reflection loss. The lowest dots represent reflections involving all six surfaces, and thus the highest reflection loss. When the echoes are added incoherently the resulting decay in dB is very nearly linear vs. time, and the reverberation time can be calculated from the decay rate. Jumping ahead a bit, I thought there might be a distinct difference in the echo magnitude/time plots for an optimum room and a horrible room. No. They look very similar.Using the same program, I calculated the frequency response of an optimum and a horrible room. First, as an additional check on my computer program, I calculated the response of the horrible room with a speaker in one corner, and a listening point in the opposing corner, and a zero absorption coefficient. Theoretically, with a zero absorption coefficient, an infinite number of reflections should be used, but obviously this is impossible. A limit of 11 reflections was imposed. All modes should be excited and picked up with this corner-corner geometry, and the room resonances should be evident. They are indeed evident in the computed results; the red curve is the computed response, and the vertical black lines are the resonances. There is a response peak at every resonance. The image program does not contain any equations regarding resonances, so it is a nice independent test of the code. This result also illustrates the logic of evenly spreading out the resonances to minimize the gaps between them. For a more realistic calculation, an absorption coefficient of 0.2 was used, and two woofers were placed near the corners. The response was calculated at a point 2/3 of a room length away from the woofers, centered in width, and 3-feet above the floor. I also made calculations at points 2-feet away from this point, in various directions. The geometry is shown here [4.4 kb]. The large red circles are the two woofers. The 5 small circles are the points where the response was calculated. The color of the listening points match the color of the curves in the computed frequency response[24kb]. The top graph is for the horrible room, and it is pretty horrible. The second result is for the optimum room (design #3), with equal interior volume. It is also horrible. The lines across the top show the room resonances, and unlike the cooked-up corner-corner test case, the response no longer neatly follows the resonances. The response also changes a lot when the listening point is moved two feet in any direction. The reference sound level of 0 dB is the level 1 meter in front of a theoretical point source in an anechoic chamber. For a speaker mounted flush in a wall, the image program creates an image at the same location as the real speaker, increasing the sound level by 6 dB. The center listening point is 12.7 feet away from the woofer, which in free space would reduce the level about 12 dB. So if there were no "room gain," the level of the black curve would be -6 dB. The resonances and images provide a room gain which boosts the response substantially above -6 dB, a big help in achieving a big bass response, but at quite a cost in smoothness of response. And remember this is with absolutely perfect speakers. Maybe knowing about room response is like knowing how sausage is made - its better to be ignorant. But being an engineer I want the truth, even if it is ugly. I played around with various room dimensions, speaker placement, etc, without much success. The one thing that did appear to smooth out the response was to increase the absorption coefficient, and particularly the absorption of the wall facing the speakers. At some point I plan on looking into this further.

**So What Use is All This Acoustics
Stuff?**

- It does make sense to locate the subwoofer near the corner of the room.
- The optimum room frequency response is a little better than the horrible room, so if you have a choice, pick the optimum dimensions.
- Don't take low frequency engineering results too seriously.
- Don't take a speaker manufacturer's claim of ±1 dB frequency response too seriously either.

My guess is that the best approach to coping with this situation is to use your ears and play around with your listening position and absorbing stuff. And if your room dimensions aren't optimum, don't feel too bad.