Standing Waves

Standing Waves
Fields: Dynamic Systems and Other
Image Created By: Tyler Sammann

Standing Waves

This image depicts a steel string acoustic guitar fret board. This is an instrument which uses standing waves in the strings to produce sounds.

Basic Description

A standing wave is a wave which does not travel from one place to another like most sound waves, light waves, etc. Though standing waves typically are not those which reach our ears, most acoustic musical instruments utilize them. A standing wave in a musical instrument allows for consistency and predictability of the wave in order to produce a tone.

In wind instruments (clarinet, saxophone, trumpet, etc.), a standing wave is “trapped” by resonance in a resonating chamber, and in string instruments (guitar, violin, cello, piano, etc.) the standing wave is continuously reflected between the string's two anchor points. Standing waves are rarely pure, meaning that they vibrate at many frequencies simultaneously, and the series of harmonic partial frequencies are musically utilized in various ways.

Why is it Interesting?

Standing waves are important to sound, and extremely important to music. Nearly all acoustic instruments including wind, brass, string, and some percussion instruments would not function without the predictability and amplification effects of the standing wave phenomenon. Speech and singing are possible because of standing waves in human vocal cords.

Much of what we hear in our everyday lives comes from standing waves, because of their unique and incredibly useful properties. Standing waves are also very important to engineering fields, and many other sciences (Civil engineering in particular). When designing and building structures, understanding the properties of standing waves and resonance can make huge differences in structural integrity. By understanding the properties of standing waves, some properly built structures can survive earthquakes. By misunderstanding or neglecting these properties, on the contrary, a structure could be improperly built and be in danger of collapsing in the presence of strong wind or other minor vibrations.

Harmonics

In a standing wave system like an instrument, the fundamental frequency is the frequency that has the longest wavelength possible. The fundamental frequency in such a system is therefore the lowest frequency possible. In a vibrating string system, such as that of a guitar or piano, the wavelength of the wave which produces the fundamental frequency equals the length of the entire string from anchor to anchor. The fundamental frequency is what defines the tone produced by a standing wave system, because the pitch of this frequency is the loudest and most identifiable.

In such a system, there are many other frequencies besides the fundamental. These are known as partials, which can be defined as either harmonic or inharmonic. Harmonic partial frequencies, as opposed to inharmonic frequencies, are defined to be integer multiples higher than the fundamental frequency (in Hz). This also means that the wavelengths of harmonics are even fractions of the fundamental frequency's wavelength, of the form $1/n$ where n is a positive integer greater than 1. Inharmonic frequencies do not have the same easily defined stationary nodes that harmonic frequencies do, and their wave shapes are much more complicated, leading to more complex fraction divisions. By nature, humans are sensitive to these kinds of proportional relationships involving pitch. This sensitivity allows us to distinguish between dissonance and consonance, and allow us to easily recognize octaves (a pitch relationship of 1:2). Relationships with less complicated proportional fractions tend to sound more "pleasing" to us.

Harmonic frequencies are usually much louder than inharmonic frequencies. Harmonic frequencies also have more sustain than inharmonic frequencies. Sustain is a term used in music to describe the length of time before the string completely stops vibrating, and it becomes silent. More sustain refers to vibrations which last for longer periods of time.

[Click to use the Flash App]

String Harmonics App

[Click to hide the Flash App]

Click on the finger to begin using this flash app!
To discover different string harmonics, press the finger (click and hold down the mouse) on different parts of the string.

In this application, the aim is to lightly press down on an already vibrating guitar string. In this case, it is a steel low E string for an electric guitar. This simulates how one might play a harmonic on a guitar string. After lifting the finger off the string (mouse up), the fundamental frequency is played again for comparison and demonstration purposes. On a real string however, the harmonic frequency would continue to sound even after the finger had lifted off the string. When the finger is pressed, the colored dots on the string represent nodes in the wave other than the node underneath the finger.

Source Code for this app is available here http://mathforum.org/mathimages/index.php/GoogleCode
1/5 A♭-2nd octave + Major 3rd
1/4 E -2nd octave
1/3 B -octave + 5th
1/2 E -octave above
1/1 E -fundamental

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Harmonic waves travelling in opposite directions can be represented by the equations below:

(The [...]

[Click to hide A More Mathematical Explanation]

Harmonic waves travelling in opposite directions can be represented by the equations below:
(The colored diagram below serves as a legend for the variables in these equations)

$y_1\; =\; y_0\, \sin(kx - \omega t)\,$

and

$y_2\; =\; y_0\, \sin(kx + \omega t)\,$

where:

• y₀ is the amplitude of the wave,
• ω (called angular frequency, measured in radians per second) is 2π times the frequency (in hertz),
• k (called the wave number and measured in radians per meter) is 2π divided by the wavelength λ (in meters),
• x is the longitudinal position, and
• t is time in seconds.

So the resultant wave y equation will be the sum of y₁ and y₂:

$y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,$

Using the trigonometric sum-to-product identity for 'sin(u) + sin(v)' to simplify:

$y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,$

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(kx). At locations x = 0, λ/2, λ, 3λ/2, ... called the nodes, the amplitude is always zero, whereas at locations x = λ/4, 3λ/4, 5λ/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is λ/2.

The Harmonic Series In Practice

String Instruments

The list of harmonic partials for a particular wave is sometimes referred to as a scale of harmonics, or harmonic series. In the case of some string instruments, including guitar and violin, a player is able to dampen, or mute the fundamental frequency, along with other lower frequencies of the harmonic series, giving the listener the ability to better hear the more quiet, high partials.

If the player lightly touches the string halfway between the two anchor points while it is vibrating, the fundamental frequency, along with some other frequencies, will be dampened, and the listener will be better able to hear the other harmonic partials. They will be best able to hear the 'octave' harmonic, with half the wavelength, and twice the frequency in Hertz (Hz) as the fundamental frequency. The listener can still hear these harmonic frequencies because the pressed finger lies on a node between two wavelengths, and therefore does not interfere with the wave.

Wind Instruments

Standing waves in wind instruments work much like they do in string instruments, but the vibrating component is not a string. Vibration in string instruments comes from reeds in most woodwind instruments, and from the players lips in brass instruments. Both types of wind instruments utilize these vibrations to produce a standing wave with the column of air trapped inside the instrument (known as the resonating chamber). The wave's shape can then be disrupted to change the tone produced in numbers of different ways. Strategically covering and uncovering holes in the resonating chamber (i.e. a Clarinet), changing the resonating chambers shape/length (i.e. a Trombone), changing the vibration of the lips by altering embrasure, or "mouth shape" (i.e. a Bugle), etc. are all ways to alter the waveform and thereby alter the tone and pitch.

Teaching Materials

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About the Creator of this Image

Tyler Sammann User: Sammat

References

• http://en.wikipedia.org/wiki/Standing_wave[1]
• http://fiddlerman.com[2]

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