SOUND

6. Beats.
   Interference in Time. There are two types of interference phenomena for sound: spatial interference and temporal interference. Standing waves in strings and sounds in pipes are examples of spatial interference. In standing waves in strings the interference phenomena involve the superposition of two or more waves with the same frequency traveling in opposite directions. In this case the resultant wave form depends upon the coordinates of the distrubed medium: the nodes and antinodes are located at points in space on the string. These standing waves in a string is an interference pattern in space. But there is another interference phenomena that results from the superposition of two waves with slightly different frequencies traveling in the same direction. In this case, when the two waves are observed at a given point, they are periodically in and out of phase. That is, there is an alternation in time between constructive and destructive interference. This phenomena is called temporal interference or interference in time. As an example of temporal interference are beats, that is, the periodic variation in the intensity of the sound heard when two tuning forks of slightly different frequencies are struck. Beats may be defined as the periodic variation in the intensity of the sound heard at a given point due to the superposition of two waves having slightly different frequencies. The number of beats per second heard at the given point is called the beat frequency and is equal to the difference in frequency between the sounds produced from two sources. The maximum beat frequency that the human ear can detect is about 20 beats per second. When the beat frequency exceeds this value, the beats blend indistinguishably into the combined sounds producing the beats. Beats are used to tune a string instrument, such as a piano, by beating a note of a string on the instrument against a reference tone of a known frequency, such as produced by a tuning fork. The string can then be adjusted to equal the frequency of the tuning fork by tightening or loosening the string until the beats cannot be heard.

   Let us derive a mathematical equation for this beat phenomena. Consider two waves with equal amplitudes traveling through a medium in the same direction, but of slightly different frequencies f₁ and f₂. The displacement of each of these waves produced at point can be expressed mathematically as
   y₁ = A₀ cos 2πf₁t and y₂ = A₀ cos 2πf₂t.
   Using the superposition principle, the displacement of the resultant wave is
   y = y₁ + y₂ = A₀(cos 2πf₁t + cos 2πf₂t).
   Using the following trigonometric identity,
   cos a + cos b = 2cos[(a - b)/2]cos[(a + b)/2],
   and letting a = cos 2πf₁t and b = cos 2πf₂t, we get
   y = 2A₀ cos {2π[(f₁ - f₂)/2]t} cos{[2π(f₁ + f₂)]t}.
   From this equation we see that the resultant wave at a point has an effective frequency equal to the average frequency of the two component waves, that is, (f₁ + f₂)/2, and amplitude A given by A = 2A₀ cos [2π(f₁ - f₂)/2]t. This amplitude varies in time with a frequency equal to (f₁ - f₂)/2. When f₁ is close to f₂, this amplitude variation is slow. For example, if two tuning forks vibrate individually at frequencies of 438 Hz and 442 Hz, then the resultant sound wave from the two forks sounded together will have a frequency of 440 Hz (the fundamental of the A note on a piano) and a beat frequency of 4 Hz. That is, the listener will hear the 440 Hz sound wave go through an intensity maximum four times every second.

7. Droppler Effect.
   When a car or truck with its horn blowing passes a unmoving listener, the frequency of the sound that he hears as the vehicle approaches him is higher and lower as it moves away from him. This is an example of the Dropper effect. This effect is named for Austrian physicist Christian Johann Droppler (1803-1853), who discovered this phenomena for light waves. In general the Dropper Effect is experienced whenever there is arelative motion between the source of the wave motion and its observer. When the source and the observer are moving toward each other, the frequency experienced by the observer is higher than the frequency of the source. When the source and the observer are moving away from each other, the frequency experience by the observer is lower than the source frequency. Although this phenomena is most often experienced with sound waves, it is a phenomena common to all harmonic waves. For example, a shift in frequencies of the light waves are observed when the source of light and its observer are in relative motion.

   To derive a mathematical equation for the Droppler effect, let us consider the case of sound waves emitted by an acoustical source. First, consider the case where the observer O is moving and the source S is stationary. For simplicity we will assume that the air is also stationary. The observer moves with respect to source with a speed of v_o toward the source (consider to be a point source), which is at rest (v_s = 0). "At rest" means at rest with respect to the medium, the air. Let the frequency of source be denoted by f, its wavelength by λ, and the speed of sound by v. Now if the observer is also at rest, then he or she would detect f wave fronts per second. (That is, when v_o = 0 and v_s = 0, then the observed frequency will be equal to the source frequency.) When the observer travels toward the source, the observer will move a distance v_ot in time t seconds and in this time detects an additional v_ot/λ wave fronts. And in addition, the speed of the sound waves relative to the observer is v + v_o. Since the additional number wavefronts detected per second is v_o/λ. The frequency f′ detected by the observer is increased and expressed by
   f′ = f + δf = f + v_o/λ.
   Using the relation that λ = v/f, we get v_o/λ = (v_o/v)f. Hence,
   f′ = f[(v + v₀)/v].
   If the observer is moving away from the source, the observer detects fewer wavefronts per second. In this case, the speed of wave relative to the observers is v - v_o. Thus it follows that the frequency detected by the observed is lower. Hence,
   f′ = f[(v - v_o)/v].
   In general, when an observer moves with a speed vo relative to a stationary source, the frequency detected by the observer is
   f′ = f[(v ± v_o)/v],
   where the positive sign is used when the observer is moving toward the source and the negative sign is used when the observer is moving away from the source.

   Now consider the case where the source is in motion and the observer is at rest. If the source moves directly toward the observer, the wavefronts seen by the observer are closer together as a result of the motion of the source in the direction of the outgoing wave. As a result, the wavelength λ′ measured by the observer is shorter than the wavelength &lambda of the source. During each vibration, which lasts for a time T (the period), the source moves a distance v_sT = v_s/f and the wavelength is shortened by this amount. Thus the observed wavelength is
   λ′ = λ - δλ = λ - (v_s/f).
   Since λ = v/f, the frequency detected by the observer is
   f′ = v/λ′ = f[v/(v - v_s)]
   That is, the observed frequency is increased when the source moves toward to the observer.

   In a similar manner, when the source moves away from the observer at rest, the observer measures a wavelength λ' that is greater than the wavelength λ of source. The observer detects a decreased in the frequency f'. Hence,
   f' = f[v/(v + v_s)].
   Combining equations (19) and (20) we get a general equation for the observed frequency when the source is moving and the source is at rest. Hence,
   f' = f[v/(v ± v_s)].
   Now if both the source and the observer are in motion, then the general relationship is f' = f[(v ± v_o)/(v -+ v_s)]. In this equation the upper signs (+v_o and -v_s) refer to motion of one toward the other, and the lower signs (-v_o and +v_s) refer to motion of one away from the other.

To continue, click here.