Acoustics

Chapter 4. From Digital Audio to Analog Sound

4.3. Acoustics

We now have speakers to recreate airwaves from the transduced signal coming out of the DAW. Awesome. We are sitting in our room, listening, and it sounds awful. Why? Because the sound interacts with the room walls, ceiling and floor, creating interferences and all sorts of artefacts that are on a mission to destroy the flat response that your $10’000 speakers are sending to your ears.

Three phenomena describe what happens when sound bounces around your room:

–       Diffraction: what happens when sound passes through openings or around a barrier

–       Refraction: what happens when the sound changes medium

–       Reflection: what happens when the sound bounces off room surfaces

Diffraction

From general wave theory, we know that diffraction is what helps waves bend around corners and openings. The amount of diffraction is proportional to wavelength; this means that it is inversely proportional to frequency: the lower the frequency, the larger the diffraction effect. This is the reason why owls can be heard through large distances in a forest, while other birds with higher-pitched chirps struggle to get their voice heard over such distances: diffraction counters the attenuation effect described in Section 1.1.

When the wavelength is smaller than the opening, there is no observable effect: the airwave is “too small”. An open window might have an opening dimension of about 1 meter (~3 feet); a sound with a much smaller wavelength (say 20 cm or about 8 inches) has a frequency of about 1715 Hz: anything around or above that will not exhibit any diffraction effect.

For our purposes, diffraction effects are important basically only in microphones: how does the microphone enclosure diffract the incoming sound? We want as little diffraction as possible so that no interferences can be created between the incoming and diffracted sound, even though you could argue that diffraction patterns are what is giving some microphones their “color”.

Refraction / Absorption

Refraction is what happens when a wave changes medium, for example when the sound produced by one of your speakers hits a wall. Wave theory tells us that when refraction happens, the speed of propagation and the wavelength change but not the frequency (see Equation 2 to remind yourself why). The logical consequence is that the sound waves hitting the wall will keep its frequency (would it not be fun if it were not the case?) and continue its path through a much denser medium; with what we know from Section 1.1, this means that the sound will be attenuated “into the wall” and mostly not come out on the other side (except for those pesky low frequencies). This process is also called absorption and plays a very large part in our rooms and studios.

Reflection

The last phenomenon we want to discuss is the most important one: reflection is what happens when a wave bounces off a surface.

Why do I care about sound reflection?

A wave bouncing off room walls will behave differently depending on the size of the room. I will take walls as my generic example, but everything is also true for ceilings, floors and any other obstacle. If the wavelength is right around the distance between our walls or any of its multiples, reflections will create a standing wave by bouncing off the wall and recombining with the original sound. If, on the other hand, the wavelength is much shorter than the room dimensions, then we can consider airwaves to be very directional like rays of light and use basic geometry to figure out where the sound goes.

Imagine a 34.3 cm long air wave; its frequency is 1 kHz (from Equation 2, we have 343 / 0.343 = 1000): this bounces nicely in a standard 3 by 4 meter room (roughly a 1 ft. wave in a 10×13 ft. Room). Now go to 100 Hz: the wave is 3.43 meters long, which is almost exactly the size of the room I just mentioned – how does a wave bounce back and forth in those conditions? Note that when people say that bass frequencies are not directional, that is what they mean: low frequencies cannot bounce around because of their size, while higher frequencies can (and do).

Why do people say that bass frequencies build up in corners?

To understand why, we must remember that a sound wave represents the various changes in air pressure; basic physics tell us that at a boundary (e.g. a wall), the speed of those changes will be zero (the airwave speed is zero at the wall; this sounds logical, the wall is not moving) while the pressure there will be maximal. An angle is the junction of two boundaries (or surfaces), and a corner is the junction of three boundaries: pressure will be respectively double and triple that of the surface! That is where the buildup comes from.

Sound as Standing Waves

What is a room mode?

Let us go back to standing waves. Because the formation of standing waves in a room depends on the room’s dimensions, you could say that the room has acoustical properties which are going to induce those standing waves; that is why standing waves are sometimes called room modes. Technically speaking, a mode is an energy satisfying the standing waves boundary condition of the general wave equation, but it is easier to speak about room modes. Let us imagine a wave bouncing between two walls (see Figure 3): the first standing wave will have nulls at the wall positions; the second standing wave will have nulls at the walls and right in the middle (second harmonic), and so on.

We can calculate the frequency of these standing waves  with Equation 2 for a room with distance  between two surfaces, the  standing wave frequencies will be:

Equation11 Acoustics standing wave

Equation 11 Standing Wave Frequencies

where c is the speed of sound. The factor 2 accounts for the fact that a full wave cycle is from one surface to the other and back, not just from one surface to the other. For example, a room with one side measuring 4 m (about 13 feet) will have a first harmonic room mode (n=1) at about 43 Hz, a second harmonic mode at about 86 Hz, etc. A much larger room of 8 m (about 26 feet) will have its first mode at 21.5 Hz, the second at 43 Hz, etc. You already see that larger rooms are better in the sense that their first room mode will be below our hearing threshold, but since larger is usually more expensive, you must strike a balance between size and cost for your listening space.

Which of my room modes has the lowest frequency?

Looking at Equation 11, we see that the room dimension is in the denominator; this means that if we take the largest side of our room for d, we will find our lowest mode. That is what I have calculated above for a room with a 4 meter (13 foot) length: the first mode will be at 43 Hz. This means that if you would be interested in playing 20 Hz sounds (from a subwoofer, for example), you would not need to worry about your room’s dimensions. Why? 20 Hz corresponds to a wave with a wavelength of about 17 meters; our room would have to be about 8.5 meters long for this to become a problem.

Why do I care about room modes?

Do these modes complicate our lives all the way up to our 20-kHz hearing limit? They do but they don’t, and here is why. Our listening space is three-dimensional; thus, a room will have different modes along each direction depending on how the sound bounces around. First, the most important and more intense, the axial modes; these are modes where sound is reflected between two opposing surfaces (similar to what is shown in Figure 3. Second, tangential modes: they involve two sets of parallel surfaces (four surfaces) and are about half as intense as the axial modes. Finally, the oblique modes are the least intense of the three, where sound bounces off all three sets of opposing surfaces. Let us see what kind of frequencies these modes have.

We can generalize Equation 11 into an equation valid in case of real space in three dimensions:

Equation12 Acoustics standing wave 3D

Equation 12 Standing Wave Frequencies for a Three-dimensional Space

where c is the speed of sound, l & w & h are the numbers representing the modes (like n in the one-dimensional case) and L & W & H are the room dimensions. Let us take a 4 by 3 by 2 meter room and calculate the lowest mode (13 by 10 by 6.5 feet). My largest dimension here is L, and we know that axial modes are the more intense (the more annoying); this means that l=1, w=0, h=0 . We nicely fall back on our previous calculation: 43 Hz! What you see is that you can very easily calculate your room modes in an excel spreadsheet. If you do, you will get all your room modes, up to whatever  you want. The trick is that room modes are only relevant up to a certain range of frequencies because past that region, airwaves start to behave like light and bounce around like light rays. I say “region” because the change in behavior (from standing wave to light ray) does not happen dramatically.

What is the Davis frequency?

If you want to know up to what point you need to calculate your room modes, pick l = w = h = 5 . This normally coincides with the Davis frequency:

Equation13 Acoustics Davis Frequency

Equation 13 Davis Frequency

where  is the smallest dimension of your room (see reference [4]). A lot of people mistakenly reference the Schroeder frequency here; this is wrong because Schroeder’s work specifically targeted large acoustic spaces; we are not in a large acoustic space, unless you live in a mansion, that is. In the case we are discussing, νDavis = 515 Hz and the l = w = h = 5 mode is at 520 Hz: both give us approximately the same limit.

There are lots of nice room mode calculators on the internet (see this one for a nice graphical representation), so head over there and try out your room: where are your main modes? If it so happens that your room is square, look out: each axial mode will ring twice as loud! That is the reason why cinema and studio rooms are never square by design. Another thing to look for is modes bunched up together forming clusters: their intensities will add up and become more troublesome. How close? Closer to 5% from each other; I read the 5% number in several places but could not figure out an accurate source.

Sound as Light Rays

If we now turn to the higher frequency case (say anything beyond 500 Hz for our test case), sound waves behave like rays of light: we are in the realm of geometry. Sound will bounce off walls multiple times before fading out, having lost some energy after each hit. Because of the way our ears work (see section 4.4), we want the sound coming out of the speakers to reach our ears before any of the pesky reflections we just talked about.

What is interesting is that sound delayed by less than 1 ms helps our brain determine the location of the source of the sound, so those reflections are in fact useful: source location is related to sound imaging, also called stereo imaging. If we move up the ladder to between 1 and 50 ms, our brain will lump those reflections with the original sound, so they are no danger because we will not perceive them. Reflections coming between 50 and 400 ms after the initial sound will give the sound a cavernous quality; after 400 ms, reflections will be heard as separate sounds, also called echoes (see here). First conclusion: any reflection with more than 50 ms delay with respect to the original sound needs to be eliminated.

What is the sweet spot?

Because of reflection, the sound travels further. Some waves coming back from reflection will cross the path of waves moving in the opposite direction; this will produce interferences and reshape the sound: in some locations, frequencies will add up, and in some others, they will cancel out. This means that depending on where you are listening from, the sound will not be the same.

This is where the notion of sweet spot comes from: taking the geometry of the room and speaker placement into account, there are better spots than others for a good listening experience. If the sweet spot is large, moving around it will not dramatically alter the listening experience; if it is smaller, the sound will change even if you move your head a few inches. This is of course true for any listening environment, not just a home studio: in front of your TV, in your car, etc. You can measure the acoustic properties of your room; it takes a bit of time and money but is well worth it; you can check this article for details.

Dealing with Reflection: Absorption

How does absorption work?

We are going to use absorption to deal with the reflection we want to get rid of. How? Remember Equation 7? measures the time it takes for sound to decrease its intensity by 60 dB;  is the volume of the room,  is the total surface area of the room and  is the average absorption coefficient of the material of the room surfaces. With everything we just discovered, we want to be as small as possible, which means that we want the absorption to be as large as possible.

Airwaves absorption depends on the frequency of the sound, as demonstrated by the absorption tables you can find on the internet. Do not fret, I have done the work for you: for general material absorption, see here; for rock & glass wool, see here. Thus, putting blankets up in your recording space will tame high frequency reflections (yes it will, see the tables I just mentioned), but will do almost nothing to prevent lower frequencies creating happy standing waves, causing this booming sound feared by all home studio owners. This means that if you want to tame the lower room modes (typically our 43 Hz mode for example), you must put a fair amount of material in its path.

How do I get rid of unwanted reflections?

So what should we do? The standard answer is: “It depends”. And it does. Your room is not my room. However, what I can tell you is that if you move around your room clapping, you will hear reflections. If you play around with positioning your speakers differently, you will hear differences.

You can even go one step further and get a free software like Room EQ Wizard (see here), a mic, and start studying your room hard: where are the bad spots? Where are the room modes? After what time delay do the first reflections reach my ears in this location? How about that location? Then build or buy some absorption panels, place them, and continue testing your room.

There are literally hundreds of pages and videos on how to build good absorbers; if I can do it, trust me, you can too. You will be amazed at what a couple of absorbers can change in terms of the quality of the listening experience.

I am short on resources, how do I get rid of unwanted reflections?

But I can hear you grumbling: you do not have the time, it is too hard, etc. so I will start you off with the basics. We already know that your room is rectangular and that no one dimension is a multiple of another. The first idea is to get rid of the low-frequency modes closest to you, accumulating in angles and corners; this means panels first against corners with three walls (wall-wall-ceiling corners) then against corners with two walls (wall-wall angles); you can have a look at see this very common Q&A from this gearslutz.com forum. The next idea is to tame the earliest reflections. This means panels at the first reflection points for your speakers (both on the walls and on the ceiling!), then behind the speakers. If I had to choose your surface material for you, I would do plywood floors, mount 10 cm Foil Reinforced Kraft (FRK) rigid fiberglass panels at reflection points, then draperies or carpet on concrete at selected spots. Do not put too much carpet on your walls or your room will sound as dead as empty space and you then must spend money on reverb plugins for your DAW!

There is a lot of literature both in books and on the internet which goes into a lot of detail about these topics (see reference [1], herehere and here); my goal here is only to draw your attention to the fact that the listening environment is the most important factor to consider when building or analyzing a listening room. Yes, it is more important than which speakers or which cables are used in the signal chain.

This is of course also true when you are recording the sound: where you are recording it and how you are recording it is much more important than the microphones or the preamps used; one factor is even more important than where and how you record something: it is what you are recording; the performance quality trumps all the other factors. A great musician with a crappy mic will still always sound better than me on a Neumann.

 

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