Science and Math for Audio Humans – Audio Collisions

by Danny Maland

Once more, my disclaimer: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's just flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.

Last time, we got into talking through the basic "inverse-square" behavior of sound pressure waves. The short-form recap of that is: Without obstructions or intentional intervention, sound pressure waves propagate in a sphere. The intensity of sound that an observer experiences is inversely proportional to their distance from the sound source. If the observer doubles their distance from the source, the apparent sound pressure wave intensity drops to one-quarter of the previous intensity.

Righto. That's great, and all, but...

If you set up a loudspeaker in a room, dig out your SPL (Sound Pressure Level) meter, find yourself a tape measure, and then start taking readings against, say, broadband noise, you will probably not observe inverse-square intensity falloff. What you are likely to get are readings that are higher than you would expect. Why?

A really big factor (amongst other, potentially major factors) is that whole bit about being inside a room. In addition to the section of the sound pressure wave that propagates to you, and strikes you directly, you are also in the way of sound pressure waves that are propagating to you indirectly. Unlike an “idealized” model of pressure wave propagation, all that energy that begins by radiating away from you is not “lost.” A pretty significant portion of it does end up reaching you – it's just that it's a bit late.

Figure 1 is a visual illustration of this. The top part of the picture shows a loudspeaker producing sound pressure waves that strike an observer. With no boundaries present, any part of a sound pressure wave traveling away from the listener never gets a chance to interact with that person. The contrasting part is the lower piece of the diagram. Inside an enclosed space, the portions of the sound pressure waves not traveling directly to the observer are (at least partially) redirected into the space. Because of this, that “indirect” energy does have a chance to eventually meet with the observer.

  • The reason you are likely to get readings higher than expected is because that indirect energy is added to the direct energy that the simple inverse-square model expects.
  • Now, if you want to be brutally frank, the bottom part of Figure 1 is highly simplified. In real life, there's a lot more to sound bouncing around a room (and everything in the room) than what's depicted above. The good news is that everything basically comes down to three factors: Reflection, absorption, and diffraction. (Sound can be made to refract, but acoustic refraction in the practical reality of most audio humans is due to temperature gradients – not interactions with solid objects.)
  • Simple acoustical reflection occurs when a sound pressure wave encounters a smooth, flat, rigid object with a surface area that is large in comparison to the sound's wavelength. The object must also have sufficient mass to prevent the pressure wave from simply passing through it. With all of this in place, you get a behavior like that shown in Figure 2. The pressure waves strike the object at an angle, and then change direction to continue on at an angle opposite that of their arrival.

If you've ever been in a room dominated by simple acoustical reflection, you would probably describe the sound of the room as very “live,” and probably “springy.” This is because the sounds reaching you in an indirect fashion are a large number of still-somewhat-distinguishable-from-one-another echoes, and in many cases, those echoes may be able to bounce back and forth between surfaces repeatedly (flutter echo). One could argue that the “liveness” of a space is directly proportional to the sound pressure level of indirect reflections versus that of the direct sound. You could also say that the “springiness” of a room is directly proportional to the sound pressure level of the echoes and amount of time that the flutter echo can be sustained.

If you introduce a rigid, sufficiently massive, and sufficiently sized object to a space, but give it a varied surface instead of a smooth one, what you're likely to get is diffuse reflection. Diffuse reflection occurs when there is sufficient surface variation of an object to cause a sound pressure wave to apparently reflect in multiple directions at once. This isn't some sort of magic, of course – what you really have is just a lot of simple reflections. It's really a question of scale. If all of those echoes happen very closely together in time from a human perspective, then we can't tell them apart as “separate” acoustical experiences. Instead, they form a smooth “wash” of echoes that we otherwise call reverberation. Figure 3 is a visual example of diffuse reflection.

Aesthetically, humans have a tendency to prefer diffuse reverberation to simple “slap” echo – when it comes to room design, that is – and you may find that reverberation is more pleasant when its high frequency content is “rolled off” to some extent.

Absorption occurs when a sound pressure wave strikes an object and a large portion of the associated energy is converted into heat. Like reflectors, absorbers need to have a large surface area compared to a sound's wavelength, but in contrast, absorbers usually need to be a soft, porous material. (It is possible to create absorbers from non-porous materials, but examples of those absorbers are quite rare in terms of room acoustics.) If all other factors are unchanged, absorbers with greater mass will outperform an absorber of lesser mass.

Rooms with a great deal of absorption tend to be referred to as “dead.” A sufficiently large amount of absorption in a room can create what is called an anechoic chamber, where a human's perception of reflected sound is negligible. Because any sound pressure wave you encounter in such a room is direct sound, you are very likely to observe sound pressure level decay that is consistent with the inverse-square law.

Humans make fairly decent absorbers, which is why so much is made of being ready for the difference in the sound of a room when it is empty, versus when it is filled with people.

The final topic for this chunk of audio math and science is diffraction. Diffraction is the phenomenon of sound pressure waves bending around obstacles and through openings. Diffraction occurs when the obstacle or opening in question is small relative to a sound's wavelength. In the case of a small obstacle in front of a sound pressure wave, you will still be able to hear the sound. The wave simply reforms itself on the other side of the obstacle. In the case of sound encountering a small opening, a more curious effect occurs. From an observer's reference point, the small opening acts as though it is the original emitter of the sound pressure wave. Figure 4 shows the “small object” condition in its top portion, whereas the bottom portion depicts a small opening.

As this topic comes to a close, I think you can already see the utility of having this information stashed somewhere in your mind. Although real acoustical work requires detailed analysis and planning, just having an idea of how sound waves are rocketing around in a room can make you much more able to identify associated problems – and much more capable of participating well in solving them.

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