Studio & Gear

Welcome to Sound Advice on Acoustics! Last time we introduced the idea of standing waves and modes in a room, and introduced a simple formula for calculating them. In this month’s installment, we’ll run the numbers for a few sample rooms and learn what sorts of relationships between room dimensions are best, and which ones will get you into trouble.To determine what specific modal frequencies will be present in a rectangular room, we can use the simple formula given last time (1130÷2L, where L is the dimension of the room you’re checking) for each room dimension (length, width, height) to find the primary axial modes and their first few harmonics, and list them in a chart—we’ll do that for three rooms.We’ll be looking for two main things: (1) to find and avoid coincidences and near-coincidences (where the same modal frequency develops between two or all three pairs of parallel surfaces), and (2) to achieve relatively even spacing and avoid wide gaps between the frequencies of the modes that are present.Number one is fairly obvious—if the same modal frequency occurs for, say, both height and width, then the imbalances at that frequency will be twice as bad. This will occur if two (or more) room dimensions are the same, or are multiples of each other (the worst-case scenario would be a cube, L=W=H)—one of the examples will illustrate this.Number two is based on two assumptions. First, if a lot of closely- and evenly- spaced modes are present in a room, the overall effect will be more of a general reinforcement of the low frequency range. Second, if a few widely-spaced modes are present, musical notes whose fundamentals and harmonics coincide with these modal frequencies will be altered in timbre and noticeably boosted or attenuated in level relative to other notes.In a bad room, this can be very obvious—imagine a scale played evenly on the bass, with some notes almost dropping out and others booming excessively, depending on the listening position.

There’s no absolute consensus as to what the best distribution of modal frequencies might be. Even spacing is preferable, and it’s been suggested that modal frequency spacing of greater than ~20 Hz will result in audible unevenness, to be avoided or minimized to whatever degree possible.

With this in mind, let’s look at a few simple examples of room mode charts (feel free to analyze your own room this way as well).

We’ll look at the axial modes for three different rooms, first listing the first four axials under the room dimensions, then listing the first twelve axials for each room in ascending order. We’ll arrange the dimensions from greatest (L) to smallest (H) because this makes it easy to spot the numerical relationships.

As you can see in Figure 4, Room A is not at all ideal: there are wide gaps between modal frequencies, and there are coincidences. Since the 16′ long wall is twice the dimension of the 8′ ceiling, the 2nd (harmonic) mode of the length (70.6 Hz) coincides with the 1st mode between floor and ceiling, also at 70.6 Hz. Since 8, 12, and 16 are all multiples of 4, at around 141 Hz a three-way coincidence occurs, which will be sure to make the imbalance of any notes/harmonics at that frequency really stand out!Room B is somewhat better: there are still some uneven, wide spacings, but there is only one coincidence, at around 141 Hz, and it only involves two modes rather than all three.Room C is even better—the spacings are more even, and there are no exact coincidences.This last set of room dimensions, 15’5″ L x 12’10” W x 10′ H, was based on one of a group of recommended “Golden Mean” room ratios; these ratios have been analyzed to provide the most even modal distribution (of course, in addition to the Axial modes they also take into account Tangential and Oblique modes).Here are a few of these Golden Mean room ratios, from various sources. In theory, it doesn’t matter which number applies to which dimension of the actual room, but building practicalities will mean that the shortest is usually the height; since many control rooms are wider than deep, the other two dimensions could interchangeably be width or length, but for consistency’s sake, let’s list the middle dimension as the width and the greatest dimension as the length of the room.

For a quick idea of how this translates into the real world, assume a room with a 10′ ceiling, and apply the ratios; the formula at the top of the list, for example, yields a room of 10′ x 11’5″ x 13′ 11″ (H x W x L).

We’ll talk more about these Golden Ratios, and what you can do about room modes in terms of practical room treatment, next time. See you then!

The reinforcements (maximum pressure) of the standing wave occur at the walls (the points of reflection), while a cancellation occurs midway between the walls. This happens between every pair of parallel walls in the room (there are three pairs in a rectangular room, front/rear walls, side walls, and floor/ceiling).

The locations in the room where the sound wave is canceled are called nodes (in this example the midway point); the areas of maximum reinforcement are called antinodes (in this example at the walls). If you generated a sine wave test tone at that frequency, stood at one wall, and slowly walked to the other wall, you would hear that tone drop in level as you passed the midway point and increase in level again as you approached the opposite wall.

So far we’ve established the first modal frequency in that room, and obviously there are at least three, one for each pair of parallel surfaces. That’s only the beginning—for each room dimension, there are additional modal frequencies.

The second harmonic of this first mode will be twice that frequency and therefore half the wavelength—a sound wave of this frequency will also form a standing wave with discrete node and antinode areas. Since the wavelength of the second harmonic is half that of the fundamental’s, the peaks and nulls will be more closely spaced (by half), as illustrated in Figure 1B.

Once again (as always), the standing wave’s reinforcements occur at the walls, but now another reinforcement (Antinode) occurs halfway between, with cancellations (Nodes) a quarter of the way out from each wall. Again, remember this happens for each of the three parallel surfaces in the room.

The same effect happens at the third harmonic (3 x the frequency, 1/3 the wavelength) of that initial modal frequency, (Figure 1C), and so on up, with more closely spaced nodes and antinodes. Eventually the wavelengths will become short enough that the reduced strength, greater density, and closer spacing of the various nodes and antinodes will tend to average out rather than be audible at specific spots in the room; this will happen gradually, beginning at around 300–400 Hz.

Figure 2 shows a plot of where the nodes and antinodes of the first three of these modes occur in the room. If you plotted each of the three pairs of parallel surfaces and overlaid them, you’d see the specific layout and intersections of the various nodes and antinodes. The particular frequencies affected would of course depend on the room dimensions.