6-19-98
When two waves which are of slightly different frequency interfere, the interference cycles from constructive to destructive and back again. This is known as beats; two sound waves producing beats will generate a sound with an intensity that continually cycles from loud to soft and back again. The frequency of the sound you hear will be the average of the frequency of the two waves; the intensity will vary with a frequency (known as the beat frequency) that is the difference between the frequencies of the two waves.
When two (or more) waves of the same frequency interfere, a variety of different results can be obtained. Consider first the special case of two sources separated by a small distance d, sending out waves of the same frequency. The sources are in phase with each other. For a particular separation and wavelength, the pattern is as shown in the diagram, with constructive interference taking place at certain angles and destructive interference taking place at other angles.
When the sources send out waves in phase, constructive interference will occur at a particular point if the path lengths from the two sources to that point differ by an integral number of wavelengths.
Destructive interference occurs at a particular point if the path lengths from the two sources to that point differ by an integral number of wavelengths + 1/2 a wavelength.
Diffraction is the bending of waves that takes place when the wave encounters openings or obstacles. The most interesting cases (i.e., the ones with interesting patterns of maxima and minima) are those in which the size of the openings or obstacles is about the same as the wavelength of the wave.
Interference, both constructive and destructive, is important to understanding why diffraction occurs. If the opening is divided into many small pieces, each piece can be thought of as an emitter of the wave. The waves from each piece of the opening are sent out in phase with each other; at some places they will interfere constructively, and at others they will interfere destructively.
Consider a point that is half a wavelength further from the center of the opening than from one side of the opening. This is the condition for destructive interference: the wave from the side of the opening will interfere destructively with the wave from the center of the opening. Similarly, the wave from the part of the opening next to the side will interfer destructively with the part of the opening next to the center, and so on - the waves from one half of the opening completely cancel the waves from the other half. It turns out that the points where this destructive interference occurs are all along one line, at an angle (measured from a line perpendicular to the opening) given by:
If D, the width of the opening, is less than the wavelength than there is no place where the interference is completely destructive. If D is greater than the wavelength there is at least one angle where destructive interference occurs; the diffraction patterns in such cases are similar to the interference patterns produces by two sources close together.
Moving on towards musical instruments, consider a wave travelling along a string that is fixed at one end. When the wave reaches the end, it will be reflected back, and because the end was fixed the reflection will be reversed from the original wave (also known as a 180° phase change). The reflected wave will interfere with the part of the wave still moving towards the fixed end. Typically, the interference will be neither completely constructive nor completely destructive, and nothing much useful occurs. In special cases, however, when the wavelength is matched to the length of the string, the result can be very useful indeed.
Consider one of these special cases, when the length of the string is equal to half the wavelength of the wave.
time to produce half a wavelength is t = T / 2 = 1 / 2f
in this time the wave travels at a speed v a distance L, so t = L / v
combining these gives L / v = 1 / 2f, so f = v / 2L
This frequency is known as the first harmonic, or the fundamental frequency, of the string. The second harmonic will be twice this frequency, the third three times the frequency, etc. The different harmonics are those that will occur, with various amplitudes, in stringed instruments.
In general, the special cases (the frequencies at which standing waves occur) are given by:
The first three harmonics are shown in the following diagram:
When you pluck a guitar string, for example, waves at all sorts of frequencies will bounce back and forth along the string. However, the waves that are NOT at the harmonic frequencies will have reflections that do NOT constructively interfere, so you won't hear those frequencies. On the other hand, waves at the harmonic frequencies will constructively interfere, and the musical tone generated by plucking the string will be a combination of the different harmonics.
Example - a particular string has a length of 63.0 cm, a mass of 30 g, and has a tension of 87.0 N. What is the fundamental frequency of this string? What is the frequency of the fifth harmonic?
The first step is to calculate the speed of the wave (F is the tension):
The fundamental frequency is then found from the equation:
So the fundamental frequency is 42.74 / (2 x 0.63) = 33.9 Hz.
The second harmonic is double that frequency, and so on, so the fifth harmonic is at a frequency of 5 x 33.9 = 169.5 Hz.
A typical guitar has six strings. These are all of the same length, and all under about the same tension, so why do they put out sound of different frequency? If you look at the different strings, they're of different sizes, so the mass/length of all the strings is different. The one at the bottom has the smallest mass/length, so it has the highest frequency. The strings increase in mass/length as you move up, so the top string, the heaviest, has the lowest frequency.
Tuning a guitar simply means setting the fundamental frequency of each string to the correct value. This is done by adjusting the tension in each string. If the tension is increased, the fundamental frequency increases; if the tension is reduced the frequency will decrease.
To obtain different notes (i.e., different frequencies) from a string, the string's length is changed by pressing the string down until it touches a fret. This shortens a string, and the frequency will be increased.
Pipes work in a similar way as strings, so we can analyze everything from organ pipes to flutes to trumpets. The big difference between pipes and strings is that while we consider strings to be fixed at both ends, the tube is either free at both ends (if it is open at both ends) or is free at one end and fixed at the other (if the tube is closed at one end). In these cases the harmonic frequencies are given by:
A pipe organ has an array of different pipes of varying lengths, some open-ended and some closed at one end. Each pipe corresponds to a different fundamental frequency. For an instrument like a flute, on the other hand, there is only a single pipe. Holes can be opened along the flute to reduce the effective length, thereby increasing the frequency. In a trumpet, valves are used to make the air travel through different sections of the trumpet, changing its effective length; with a trombone, the change in length is a little more obvious.
Example : A tube open at one end has a length of 25.0 cm. The temperature is 20 °C. What is the fundamental frequency of this tube? What is the frequency of the fifth harmonic?
If we blow through the tube, it will make a musical tone, and that's what we're talking about here. The velocity involved in the frequency equation is therefore the speed of sound, which is 343 m/s at 20 °C. The fundamental frequency is then:
So the fundamental is 343 / ( 4 x 0.25) = 343 Hz.
A tube like this, closed at one end, only has odd harmonics (n = 1, 3, 5, etc.). The fifth harmonic is five times the fundamental, and it's also given by:
So the fifth harmonic is 1715 Hz.