Ever wonder what a guitar string and a microwave oven have in common? Both work with periodic, repeating waves. Tickle them with the right frequencies, and one will make beautiful music while the other heats up last night's leftovers. But tickle them at the wrong frequencies, and they do little or nothing.
Why are they so finicky? Well, they can't help it — they're built that way. Guitar strings, microwaves, and many other devices and instruments are subject to special properties of standing waves and resonance. In this column, I'll take a brief look into the secrets of the timbral universe — the key to sounds and the underlying vibrations of life. I'll provide a few tips on how to improve the sound in your recording studio and give you an idea of how your instrument's vibrations interact with the natural vibrations in different venues.
FIG. 1: A standing wave in a string produces stationary points called nodes. Between the nodes are points of maximum displacement called antinodes. The fundamental standing-wave frequency has a wavelength (λ) that is twice the length of the string. The string can also support integer multiples of that frequency.
Imagine two people holding a jump rope. First, they each shake their respective ends, producing pulses that move along the rope to the opposite end and reflect back. They start by shaking randomly, and the jump rope looks disordered. But pretty soon they sync up into a regular pattern, producing pulses at a rate (or frequency) such that their pulses meet at just the right times to piggyback onto each other. At this point, the rope's motion appears simple and repeating. Pulses do not appear to move forward or backward; rather, certain points along the rope (called nodes) remain motionless, while halfway between the nodes are big bouncing points (called antinodes), as shown in Fig. 1. These piggyback waveforms, with their nodes and antinodes, are called standing waves.
The two people holding the jump rope can produce a variety of standing-wave patterns. Fig. 1 shows standing waves at four different frequencies. Usually, people turning a jump rope use a frequency like the standing wave pictured at the top of Fig. 1, with one loop up or down at any given time. But if the people shake the rope twice as fast, three times as fast, and four times as fast, they can make the rope look like the other patterns shown in Fig. 1. The top wave is vibrating at the lowest frequency that the jump rope can support. That frequency is called the fundamental. (Frequencies are expressed in cycles per second, or hertz.) The rope, like a guitar string, can also produce frequencies that are at integer multiples (for example, 25, 35, and 45) of the fundamental frequency. Frequencies of this type are called harmonic frequencies or simply harmonics. What do all of the harmonics have in common? They all have nodes at the rope's end points.
Another characteristic of a standing wave that is worth considering is its wavelength. The wavelength is the distance between the beginning of the upward and the end of the downward parts of the wave cycle. It is expressed in feet or meters per second for sound and is commonly symbolized with the Greek letter lambda (λ). In Fig. 1, you can see that the fundamental has a wavelength twice as long as the rope, since only half of the cycle — the upswing or downswing — is visible at any moment. The second harmonic has a wavelength that is the full length of the rope — that is, half that of the fundamental. You can see the upswing and the downswing simultaneously. The third harmonic has a wavelength of one-third the fundamental, and so on.
Wavelength is the inverse of frequency. Longer wavelengths correspond to lower frequencies, and shorter wavelengths correspond to higher frequencies. The second harmonic's frequency is twice that of the fundamental, and its wavelength is one half that of the fundamental. The third harmonic has a frequency of three times the fundamental and a wavelength of one-third the fundamental, and so on.
Standing waves are at the heart of music because harmonic frequencies are the only stable vibrating modes for any object. But real objects are a little more complicated than the jump rope, and their vibrations are usually a composite of a number of standing waves. To understand why, I'll discuss resonance in the following section.
Press the damper pedal of a grand piano and hit a low note a few times. If you listen carefully, you'll notice that in addition to the strings of the note you played, a number of other strings also vibrate (though at much lower amplitude). These extra strings share harmonics with the original note. Air vibrating around the strings at a rate that matches their characteristic frequencies pushes them back and forth. This phenomenon is known as resonance, or sympathetic vibration.
The tendency of an object to resonate at characteristic frequencies explains why natural sounds are nearly always combinations of different frequencies. When you pluck a string on your guitar, initially that pluck creates chaos in the string. But within an instant, most of the chaotic vibrations traveling between the ends have cancelled each other out. All that “survives” is a combination of the string's characteristic frequencies, although the predominant pitch you hear is the string's fundamental frequency. The string's vibrations are passed on to the body of the instrument, which also vibrates in sympathy, causing a greater displacement of air around and within the instrument. The body, with its resonances, determines the timbre (or tone color) of the instrument's sound.
The frequencies that an instrument produces are called partials (each is part of the composite vibration). Partials are generally some integer multiple of the fundamental, and therefore are known as harmonic partials. (The fundamental itself is the first partial.) But some instruments, notably bells and pitched percussion, have partials that are not integer multiples of the fundamental. Instead, they are at strange multiples such as 1.4 or 2.73. Those frequencies are inharmonic partials. The term overtones is used as the name for all of the partials above the fundamental.
FIG. 2: Rooms, like strings, are able to support standing waves at harmonic frequencies of the fundamental. As with strings, the fundamental frequency has a wavelength that is twice the length of the room.
ROOM WITH A REFLECTION
Recurring (or periodic) changes in air pressure that occur in enclosed spaces are also standing waves. Showers can be a great place to test that because they are typically small and have parallel walls and highly reflective tiled surfaces, which are perfect conditions for standing waves. Try this experiment: imitate your favorite singer and create a low tone. Slide your pitch slowly upward, and notice that some notes “ring” out, while others fade. You can also try to play a 1 kHz sine wave with your software synth, and put the speaker on the floor facing a wall. Move the speaker about six inches from the wall, get the spot just right, and listen to the level suddenly drop. What's going on here?
Air molecules that vibrate within a room bounce off the walls. When they do, there is a moment of high pressure (high density) as they bunch together before they ricochet away, momentarily spread out, and create lower pressure (low density). Because of these oscillations in pressure at the walls, the room has a set of characteristic (harmonic) frequencies just as a guitar string does (see Fig. 2). The fundamental is the frequency that has a wavelength of twice the room's length, with pressure antinodes at the walls (where the pressure is alternately very high and then very low, with the levels at the two walls always exactly opposite) and a node in the center (where the pressure is most consistent, at near-zero change). Integer multiples of that frequency also resonate. Acousticians call those characteristic frequencies room modes. (Characteristic frequencies are also what happen in your microwave, but those are a different type of wave.)
If the room is a perfect cube, the same resonances may be amplified in all three directions — floor to ceiling, side to side, and front to back. Similar resonance sharing can occur if the room dimensions are integer multiples of each other. If a room is twice as long as it is high, the floor — ceiling harmonics can also resonate on the front-to-back axis of the room.
A room naturally resonates at some frequencies. Is that a good thing? The equivocal answer is maybe. Room acoustics is a complicated business, even for the pros. You want your recording or performance space to be neutral, allowing all frequencies to sound equally and clearly. Ideally, your recording space has no parallel walls and is built of materials that absorb enough of the sound so there isn't excessive reverberation. The materials should also reflect enough sound so that the room doesn't sound dead. That is an ideal situation, but in reality, people usually inherit a recording space — such as a former garage, attic, practice room, warehouse, or barn — that was never intended for recording. Chances are you'll need to take some measures to improve its sound. Entire books have been written on how to do that; however, I'll discuss some initial steps that you can take. For small spaces, ensure that you don't get coloration from standing waves, which can show up in two guises: flutter echo and comb filtering.
Flutter echo refers to echoes bouncing back and forth between surfaces. If you stood in a completely empty room (bare walls, bare floor, and bare ceiling) and you clapped your hands or snapped your fingers, you would hear a buzzing “reverberation tail.” Flutter echo can be produced by short, impulsive sounds, and it's probably not a feature you'd want on every percussion track.
Sustained tones or chords won't create flutter echo, but they may be colored by comb filtering. If the chord contains frequencies that are also room modes, they ring nicely in the room — perhaps too nicely. Other frequencies are reflected out of phase with themselves and are largely cancelled (for more on phase, see the Square One column “About Phase” in the May 2004 issue of EM). The room may have an overall frequency response with a regular series of peaks and nulls, as shown in Fig. A. Instructions for estimating your studio's room modes can be found in the sidebar “Room à la Modes.”
If you play and hold a chord in a room, it can actually sound different as you walk around, passing through various nodes and antinodes. For high frequencies (which have short wavelengths), you may be able to hear the differences simply by tilting your head back and forth. A cello being played in a corner can produce low-end resonances. Moving the cello player toward the center of the room and out of the antinode can yield a more even frequency response. Similarly, a microphone can pick up different-sounding results depending on whether it's placed in a nodal or antinodal region (different microphones respond differently in different spaces).
Depending on how your recording room sounds, you may want to absorb or diffuse standing-wave frequencies. If your room tends to ring or rumble, you may want to absorb some of that extra sound. Strategically placed materials, such as drapes or carpets, can soak up high frequencies, while surfaces such as plywood or glass are better for absorbing lower frequencies.
Keep in mind that absorbing too much can muffle everything. You may be better off trying to diffuse the sound — that is, scattering the reflections so they don't bounce right back on top of each other and form standing waves. Sometimes a low-end solution is as simple as putting bookcases against the wall or adding extra furniture to the room. But you have more flexibility with acoustic diffusers, which include foam that has regular slits cut into it to scatter high frequencies or bass traps that are placed in corners to break up the low tones. You can also purchase an AcoustiKit from Acoustics First (www.acousticsfirst.com/), which is a modular set of diffusers of various shapes and sizes.
THE SCOPE OF IT ALL
We've only been able to scratch the surface here, so for more on the behavior of standing wave behaviors, check out Zona's Standing Wave page http://id.mind.net/*zona/mstm/physics/waves/standingWaves/standingWaves.html or the University of Colorado at Boulder's Physics Dept. page: www.colorado.edu/physics/2000/microwaves/standing_wave1.html. You'll see that resonance is at the heart of so many things, and once you get the concepts, you'll be astounded at the things you'll notice resonating all around you.
is assistant professor of music technology at Penn State. Resonance is one of his favorite teaching topics.
FIG. A: Reflective surfaces in a room can lead to a regular series of waves that either reinforce or cancel each other.
ROOM À LA MODES
Wavelength and frequency are intimately connected: if you know one, you can figure out the other. Given a certain wavelength (l), the corresponding frequency (f) is the speed of sound (c 1,000 feet per second, with slight variations depending on altitude and temperature) divided by the wavelength:
f = c /__
Given a certain frequency, the wavelength is the speed of sound divided by frequency:
_ = c / f
In a room, the wavelength of the fundamental room mode is twice the room's length (L). The room will therefore have natural resonances at integer multiples (n) of the fundamental frequency:
[room modes] = nc/2L
For example, a room with walls that are 10 feet apart may support standing waves at harmonics of:
nc/2L = n1000/20 = n50 Hz
But frequencies that are halfway between the harmonics of 50 Hz (for example, 75 Hz, 125 Hz, and 175 Hz) are cancelled. In worst-case conditions, the room's frequency response has a regular series of peaks and dips. The comb shape of that response curve gives the effect the name comb filtering (see Fig. A).