Sound waves are produced by a vibrating body. The vibrating object moves in one direction and compresses the air directly in front of it. As the vibrating object moves in the opposite direction, the pressure on the air is lessened so that an expansion, or rarefaction, of air molecules occurs. One compression and one rarefaction make up one longitudinal wave. The vibrating air molecules move back and forth parallel to the direction of motion of the wave, receiving energy from adjacent molecules nearer the source and passing the energy to adjacent molecules farther from the source.

Intensity and pitch

The pitch of a sound depends on the frequency of the tone that the ear receives. High notes are produced by an object that is vibrating a greater number of times per second than for a low note.

The intensity of a sound is the amount of energy crossing a unit area in unit time or the power flowing through the unit area. The SI unit is watts per square meter. The loudness of the sound depends upon the subjective effect of intensity of sound waves on the human ear. In general, a more intense sound is also louder, but the ear does not respond similarly at all frequencies so that two tones of the same intensity but with different pitches may appear to have different loudness. The intensity of the threshold of hearing ( I o), which is the intensity that can be barely heard by a normal person, is about 0 −12 (watt/m 2) when measured by acoustical devices. The relation between loudness and intensity is nearly logarithmic. The intensity level of sound is measured in decibels and is given by the equation β = 10 log I/ I o, β (the Greek letter beta) is the intensity in decibels, I is the sound intensity, and I o is the intensity of the threshold of hearing. For example, normal conversation is about 60 decibels, and a power saw is about 110 decibels.

Doppler effect

When a siren approaches, the pitch is high, and when it passes, the pitch drops. As a moving sound source approaches a listener, the sound waves are closer together, as shown in Figure 1, causing an increase in the frequency of the sound heard. As the source passes the listener, the waves spread out, and the observed frequency lowers.

Figure 1

The Doppler effect.

This change in observed frequency due to relative motion is called the Doppler effect. The equation for a stationary observer and moving source is where f′ is the frequency heard by the observer, f is the source frequency, v is the speed of sound in air, and v, is the speed of the source.

Forced vibrations and resonance

The tuning fork is a useful instrument for investigating sound because it vibrates at only one frequency, in contrast to most musical instruments that produce several different frequencies simultaneously. A struck tuning fork vibrates at a natural frequency that depends upon the fork's manufacture—the dimensions and the material from which it is made. If the stem of a vibrating tuning fork is set on a table top, the tone becomes louder because the fork forces the table top to vibrate. Because the table top has a larger vibrating area, the sound is more intense. This principle of forced vibrations is applied in most musical instruments by using a part of the instrument, such as the piano sounding board, to intensify the sound.

Imagine two matched tuning forks with the same frequency mounted on sounding boxes. As shown in Figure 2, the vibrating air column set up by one tuning fork will cause the other tuning fork to vibrate weakly. This action is called resonance or sympathetic vibration. Resonance occurs when the natural vibration rates of two objects are the same or when one has a natural vibration rate that is a multiple of the other. The requirement that the two objects have the same natural frequency (or multiple thereof) can be demonstrated by a violin and a tuning fork that vibrates at the pitch of one of the violin strings. First, set small pieces of paper on each violin string. Then hold the tuning fork very near the violin strings. The small paper will fall off the string that has the same natural frequency as the tuning fork because the string experiences a weak sympathetic vibration.

Figure 2

Sympathetic vibration between two tuning forks.


The discussion of standing waves analyzed the superposition of waves with the same frequency. A different interference effect occurs when two waves with slightly different frequencies are heard at the same time. The top graph in Figure represents the individual waves of two slightly different frequencies. The bottom graph shows the resultant wave. At time t a, the two waves destructively interfere (cancel each other out). At a later time ( t b), the waves constructively interfere because the amplitudes are both in the same direction. A listener will hear the alternating loudness, known as beats. The number of beats per second, called the beat frequency, equals the difference between the frequencies of the two individual waves. To tune an instrument accurately, a musician listens carefully and adjusts her instrument to eliminate beats between the instrument and a given pitch.

Figure 3

Beats are created by the interference of two waves with different frequencies.