How to calculate your room’s nodes and standing waves, and why you should care
By Bob Ross
Getting your music room to sound good can be a confounding and frustrating experience if you don’t take some basic physics into consideration. There are some fundamental principles of acoustics that apply to all small rooms with parallel walls. Once you are aware of the behavior of sound waves in these spaces, it becomes much easier to understand why music sounds the way it does when played in a certain room. It also can help predict how music will sound in that room, and how this “room sound” might affect your recordings
Two terms, which you will invariably encounter in any discussion of studio acoustics, are Room Modes and Standing Waves. The two terms are very closely related, practically synonymous. Moreover, they are inextricably related to the physical dimensions of your room.
A room mode is essentially a resonance, an area of increased amplitude that results when a sound wave reflects off a boundary surface (wall, floor, or ceiling) and combines in phase with the original direct sound wave. What causes the direct and reflected waves to combine in phase is simply a whole-number correlation between the length of the sound wave and the length (or width, or height) of the room.
Sound waves are variations in air pressure, alternating plus or minus the static atmospheric pressure of a given space. (Here’s a thought experiment: what’s the difference between variations in barometric pressure—i.e., the weather—and really, really low bass notes? Talk amongst yourselves.) The amount by which it exceeds (plus or minus) the static atmospheric pressure is the amplitude of the wave (which corresponds to the Sound Pressure Level).
Antinodes and nodes
Areas of maximum amplitude, representing the maximum change in air pressure, are called Antinodes. Areas of minimum change in air pressure (essentially the zero-crossings between positive and negative halves of the pressure wave cycle) are called Nodes. The concept of nodes may be familiar to string instrument players: Natural harmonics, those pure tones that are played by lightly stopping a string at some fraction of its length, occur at the node of that vibrating string length.
If a sound wave is exactly one-half the length of any single room dimension, its nodes will be at that boundary surface [Fig. 1]. The reflection off of that parallel surface will be in phase with the initial sound wave, the antinodes will line up in space/time and reinforce one another, and there will be an increase in amplitude at that fundamental frequency.
This phenomenon not only occurs for frequencies whose wavelength is one-half a room dimension; it also occurs at any whole number multiples of that frequency… i.e., harmonics of that fundamental.
Double that fundamental frequency and a full wave cycle now fits in the space between boundaries [Fig. 2]. The nodes will still occur at the room boundaries, causing the reflections to be in phase. However, there is now an additional node in the middle of the room; the antinodes of the harmonics do not line up physically with the location of the fundamental’s antinodes. This has important consequences, as we shall see.
Standing waves
These sound waves whose wavelength is equal to one-half (or any whole number multiple of) a room dimension, and hence whose nodes occur at boundary surfaces, are called Standing Waves. Actually, they’re a very specific type of standing wave: Axial Mode is the name given to standing waves that exist between two parallel surfaces (front and back walls, left and right side walls, or floor and ceiling). Other types of standing waves include the Tangential Mode, where the sound wave bounces off of four distinct surfaces, and the Oblique Mode, where all six room boundaries are involved.
Since axial modes are the most troublesome standing waves in music studios, they are our primary concern. Incidentally, they are called standing waves because unlike most sound waves in an enclosed space, axial modes do not propagate through the room. Rather, they occupy a stationary location in that room, their amplitude peaks always occurring at a particular (and predictable) physical location in the space.
How to measure and calculate
To calculate the axial modes of a rectangular room, one uses the formula
1130 / 2L = f
1130 is the approximate speed of sound in feet per second, and L represents the length of a room dimension in feet. The result f is the frequency of the axial mode in Hertz. So for example, if your control room has an 8-foot ceiling…
1130 divided by (2 times 8), or
1130 divided by 16
equals 70.625
…there will be a standing wave (between floor and ceiling) at 70.6 Hz. There will also be a standing wave at the whole number multiples of 70.6, including 141.2 Hz (70.6 x 2), 211.8 Hz (70.6 x 3), 282.4 Hz (70.6 x 4), and so on.
For any rectangular room, there will be a standing wave at each of the three fundamental axial modes (corresponding to the room’s length, width, and height) as well as at the whole number multiples of those three frequencies.
This, in recording studio parlance, sucks.
Bad news
Picture this: There is a standing wave between your 8-foot ceiling and floor at 70.6 Hz. This means the antinode of this fundamental will be equidistant between the floor and ceiling. So there’s an area of maximum amplitude four feet off the ground…right about ear level when you’re sitting in front of the console.
There’s also a standing wave at 141.2 Hz, only because a complete wave cycle fits between those room boundaries, the node (area of least amplitude) of that wave is four feet off the ground. Regardless of what type of monitor speakers you use, or where in the room they are placed, your perception of 70.6 Hz and 141.2 Hz will be colored by the existence of these standing waves. Move your head to another place in the room (e.g., stand up) and the response at 70.6 Hz and 141.2 Hz is different, because now instead of sitting in the antinode and node of those respective waves, you’re in a place where they have a different amplitude relationship. No wonder it’s hard to fit that kick drum into the mix reliably.
And this is only the effect of the first two axial modes of one dimension we’re talking about! Imagine what happens when you add in a third axial mode [Fig. 3]. Now imagine what happens when you add in axial modes from the other two dimensions. Wait, don’t imagine it; calculate it.
Calculate the fundamental axial modes for each of the three dimensions of your room using the formula 1130 / 2L = f. Multiply each of those three frequencies by 2 through 8 to figure out the first eight multiples of each axial mode. (We’re only concerned with the first eight multiples because above 300–400 Hz, standing waves have less of a destructive influence on sound.)
You should wind up with a list (or a table, or a graph, if you’re industrious) of 24 frequencies. Yep, there’s a standing wave at each one of those frequencies. But don’t panic yet. If they’re fairly evenly spread out, you got lucky, and you should put down this magazine and go make some music now! Howev, if any of those standing waves are within 5 Hz of each other, these will be problem areas in your room’s frequency response.
My particular room
The control room I use as an example is 12′ x 10′ x 8′. Plotting the axial modes indicates potential problems around 141 Hz (the second multiple of the 8′ dimension is 141.3, and the third multiple of the 12′ dimension is 141.2… for all practical purposes those numbers are identical). It also pointed to the likelihood of serious problems at 282.5 Hz: the fourth multiple of the 8′ dimension, the sixth multiple of the 12′ dimension, and the fifth multiple of the 10′ dimension all come to exactly that number!
So what can you do about standing waves? Here’s what won’t help: Sticking foam, or fiberglass, or heavy drapes on the walls. Remember that standing waves occur as a result of the physical dimensions of the room. To a 141 Hz sound wave (whose wavelength happens to be eight feet long), a few inches of foam on the wall is essentially invisible. If you want to alter the standing waves in your room, you have to change the dimensions of the room. That’s right… knock a wall down, and move it closer or farther away. That’s the only way to change where the standing waves will occur.
Forewarned is forearmed
This is why it’s a good idea to plot the axial modes of a room before you decide to turn it into your recording studio; if you’ve got a choice between several rooms, choose the one whose dimensions yield the more evenly spaced axial modes. (Hint: Try to stay away from rooms where any two dimensions are multiples of one another, and avoid perfectly cubical rooms at all costs. Plot the axial modes and you’ll see why.) And if you’re planning on building any walls (say, to separate your control room from your recording space), calculate the axial modes to determine what the best dimensions for those spaces will be. And if possible, angle walls so they’re not parallel, which can help break up those modes.
That’s not to say there’s nothing you can do to improve the sound of a room already plagued by standing waves… but that’ll have to wait for another article!