Theory

Waves transmit energy from one point in space to another without the transfer of matter. Sound waves are longitudinal waves traveling through an elastic medium. The source of sound waves is a vibrating object such as a loudspeaker or a tuning fork. The air (or any other elastic medium) in the path of the sound wave becomes alternately denser and rarer. When the molecules are forced closer together than normal, the region of higher density and air pressure is called a compression or condensation. (See the figure below). When the molecules are spread farther apart than normal, the region of lower density is called a rarefaction.

Compressions and rarefactions in a longitudinal wave correspond to the crests and troughs, respectively, in a transverse wave.
The changes in air pressure caused by the vibrating sound source cause our eardrums to vibrate with the same frequency, producing the physiological sensation of sound. The average human ear is responsive only to sound waves having frequencies in the range of 20 Hz to 20,000 Hz.

If a vibrating tuning fork is held over the open end of a tube that has one end closed, air waves will be sent inward along the column of air in the tube. These waves will be reflected when they strike a boundary, such as the disk attached at the end of a wooden rod (the disk serves as the closed end of the tube). The reflected waves will then travel back outward along the tube. The waves sent out by the tuning fork and the reflected waves will superimpose on each other and form complex patterns called standing waves. If the length of the air column in the tube is such that the reflected waves arrive at the tuning fork are in phase with waves sent out by the tuning fork, the two waves will constructively interfere with each other. Since the two waves reinforce each other, a louder sound will then be produced. This phenomenon is called resonance.

Resonance will occur when the standing wave pattern in the vibrating air column has an antinode (largest displacement) at the open end of the tube and a node (smallest displacement, which is zero) at the closed end of the tube (see the figure below).

(Keep in mind that the waves in the air column are longitudinal--particle displacement is in the direction of wave propagation. The transverse nature of the drawings is for illustration purpose only).
The length of the vibrating air column is the distance between the open end of the resonance tube and the closed end. The length of the air column can be changed by sliding the wooden rod inward or outward.

As can be seen in the figure above, the first resonance length is the shortest air column length L1, which corresponds to a standing wave pattern of one quarter of a wavelength. Therefore L1 = /4. The air column length L2 at the second resonance length corresponds to a standing wave pattern of three quarters of a wavelength, that is L2 = 3 /4. The difference between the second and the first resonance lengths, therefore, corresponds to half of a wavelength, that is



The difference between the third and the second resonance lengths also corresponds to half of a wavelength, that is


Thus the wavelength of the sound wave is equal to twice the difference between the two successive resonance lengths, that is


If the frequency f of the vibrating tuning fork is known, the speed of sound in air v can be determined as


The speed of sound in air is temperature-dependent and is given to a good approximation by the emperical relationship
v = (331.45 + 0.61 TC) m/s,

where TC is the air temperature in Celsius degree.
The equation shows that the speed of sound at 0oC is approximately 331.45 m/s and increases by 0.61 m/s for each degree of temperature increase. For example, if the air temperature is TC = 20oC which is about the room temperature (68 oF), then v = 331.45 + (0.61)(20) = 343.7 m/s.