Sound

Sound consists of longitudinal waves and requires a medium to travel in. It cannot travel in a vacuum.

When we listen to a sound, we perceive its loudness (intensity), pitch (frequency), and timbre (harmonic composition).

Velocity of Sound in Different Media

Velocity of sound in a solid > velocity of sound in a liquid > velocity of sound in a gas

Velocity of sound in a gas or liquid, v = Ö (B/r)

where B is the bulk modulus and r is density.

Also, absolute temperature, T µ KE of molecules
And since KE = ½mv2

T µ ½mv²
or T µ ½(mw)v²
where mw is the molecular weight


\ v µ Ö (T/mw)

So when T increases so does v, and when mw is lower v is higher.

Velocity of sound in a solid = Ö (Y/r)
where B is the Young's modulus and r is density.

Velocity of waves in a taut string = Ö (tension/[mass/length]),
and this is proportional to the frequency of sound produced from it.
So if the tension is increased the frequency increases, and if the mass/length increases the frequency decreases.

Interference

When the paths of two or more waves intersect, the result is interference. At any given time and place where there is interference, the resulting disturbance is the sum of the disturbances of the individual waves at that time and place.

When two waves always meet crest-to-crest and trough-to-trough, they are exactly in phase and exhibit constructive interference.

Conversely, when two waves always meet crest-to-trough, they are exactly out of phase and exhibit destructive interference.

Constructive interference and destructive interference occur when the frequencies from two sound sources, placed a distance apart, are equal.

Beats

When there are two sound sources of slightly different frequencies, a different pattern of interference is established. A listener would hear beats, which are alternating periods of loudness and softness. The frequency of the beats is given by:

Beat frequency = difference between the two frequencies

Harmonics

A harmonic is a frequency that is a multiple of another frequency; e.g. 800 Hz and 1,200 Hz are harmonics of 400 Hz. Using this example, 400 Hz is called the fundamental frequency or first harmonic. Harmonics higher than the first harmonic are sometimes referred to as overtones. When the disturbances from each harmonic are summed, a complex sound wave is created. The relative amplitudes of the different harmonics gives sound its timbre.

Standing Waves

A standing wave is one that does not travel; that is, it stands still, unlike the waves that produce it. A standing wave is usually set up when a wave reflects off a surface and travels back along the same path in which it came. It interferes with itself creating the effect of a wave where at certain fixed points there is no vibration (nodes) and in between these there are fixed points of maximum vibration (antinodes). Since the vibration at the antinodes is greater than the amplitude of the travelling wave that caused it, a resonance effect is occurring.

Standing Wave in a Tube Open at Both Ends

Supposing a standing wave is produced in a tube of length l with both ends open. Sound waves enter one end of the tube and travel down it and are reflected back at the other end (even though it is open). At the end closest to the sound source is an antinode. At the opposite end is also an antinode because the air molecules there are free to move longitudinally. At the middle of the tube is a node. Thus,

l = l/2
l = 2l
f₁ = v/l = v/2l = 0.5v/l
where f₁ is the first harmonic, v is the velocity of sound, and l is the wavelength

Now, the frequency of sound can be increased to a level where another standing wave would be established. Such a standing wave would again have an antinode at one end of the tube and an antinode at the opposite end; however,

l = l
f₂ = v/l = v/l
f₂ is the second harmonic since it is twice f₁

And if the frequency is increased again until resonance occurs again,
l = 3l /2
l = 2l/3
f₃ = v/l = 3v/2l = 1.5v/l
f₃ is the third harmonic since it is three times f₁

And so on for f₄, f₅, etc.

Standing Wave in a Tube Closed at One End

Supposing a standing wave is produced in a tube of length l with one end closed and the other end open. At the closed end is a node because the air molecules there are not free to move longitudinally. At the open end is an antinode because the air molecules there are free to move longitudinally. Thus,

l = l/4
l = 4l
f₁ = v/l = v/4l = 0.25v/l
f₁ is the first harmonic

Now, the frequency of sound can be increased to a level where another standing wave would be established. Such a standing wave would again have a node at the closed end of the tube and an antinode at the open end; however,

l = 3l/4
l = 4l/3
f₃ = v/l = 3v/4l = 0.75v/l

f₃ is the third harmonic (since it is three times the frequency of f₁; the second harmonic, which should be 0.5v/l, is not feasible).

Similarly, the next higher frequency at which there would be resonance can be calculated as follows:

l = 5l/4
l = 4l/5
f₅ = v/l = 5v/4l = 1.25v/l

f₅ is the fifth harmonic (since it is five times the frequency of f₁; the fourth harmonic, which should be v/l, is not feasible).

Thus, a tube open only at one end can develop standing waves at only the odd harmonic frequencies (f₁, f₃, f₅, etc.), while a tube open at both ends can develop standing waves at all harmonic frequencies (f₁, f₂, f₃, f₄, f₅, etc.). This results in differences in the timbre of sounds produced from each type of tube. Furthermore, in order to produce the same fundamental frequency, a tube with both ends open must be twice as long as a tube with one end closed.

An example of where standing waves in tubes has applications is in the design of musical instruments such as flutes and trumpets. The relative amplitudes of the different harmonics give each instrument its unique timbre.