Two major considerations will help us develop a clear understanding of an acoustic phenomena called ” sound beats “. First, consider that a longitudinal sound wave is a pressure wave moving through a ” springy ” medium. The Bulk Modulus and temperature of the medium influence the velocity of sound waves. Of interest here, however, are the alternating regions of high and low pressure created by sound waves and the ” back and forth ” motion these sound waves create within the medium through which they travel.
Imagine a sound’s compression wave causing a single air molecule to revolve back and forth between two neighboring air molecules each second. This back and forth motion can be modeled with the periodic motion of a swing. Assume that a swing is put into motion by sound waves that carry it in the ( -x ) direction. As the sound wave moves the swing through a ( -x ) displacement past some midpoint ( O ), let’s assume that the sound wave will continue moving in the ( -x ) direction until it reaches a peak amplitude ( displacement ) after ½ second. After the swing reaches its amplitude in the ( -x ) direction, it will swing back in the ( + x ) direction passing the midpoint ( O ) until it returns to its ( + x ) origin. Thus, the entire round trip ( cycle ) took a period ( T ) of 1 second to complete. Soon afterward, another sound wave will carry the swing through ½ revolution. The period ( T ) is defined as the time ( t ) needed for a complete revolution of some system to occur in seconds per cycle. Conversely, the cycles per second is the inverse of the period ( T ), and cycles per second is defined as the frequency ( f ) of a system, where f = ( 1 / T ). Frequency ( f ) is quantified in units of Hertz ( Hz ). In the above scenario, f = 1Hz.
As long as the point source of sound vibrates with a frequency of 1 Hz, the hypothetical molecule will sway back and forth in synchronicity with the source. If another point source begins emitting 1 Hz sound waves that are in phase ( in step ) with the first source of sound, both waves will harmoniously push the swing in the ( -x ) direction with a resultant change in amplitude ( A ) that increases proportionally. MOVEMENT IN THE ( -x ) DIRECTION IS ASSOCIATED WITH THE PRESSURE REGIONS OF THE SOUND WAVE, AND MOTION IN THE ( +x ) DIRECTION IS ASSOCIATED WITH REGIONS OF LOW PRESSURE ( RAREFACTIONS ) BETWEEN EACH WAVEFRONT.
Let’s now imagine that the sources of sound vibrate 180 degrees out of phase with one another with the same frequency. The physical consequence is such that when the swing is in the maximum ( -x ) displacement, which is ” pi ” revolutions of the period ( T ), before it has a chance to fall back in the ( +x ) direction, the second wave sweeps in from the right and keeps the swing suspended in space. As the second wave moves past the suspended swing further into the ( -x ) direction, another wave from the first source moves into place from the right and ” cancels ” the ( +x ) motion of the swing. In essence, the swing is now MOTIONLESS due to the sound waves being out of phase with one another. WITHOUT ANY MOTION, FOR ALL PRACTICAL PURPOSES, SYSTEM NOW HAS AN AMPLITUDE OF ZERO! This analogy is by no means perfect, but it conveys the point that waves out of phase with one another can negate or alter the motion of molecules in a way that neither wave would acting alone.
We are now ready to examine the phenomena referred to as ” beats. ” When two sources of sound emit different frequencies of sound and are slightly out of phase with one another, there will be points in space where the waves are completely in cooperation with one another, other regions where the waves are in complete interference with one another, and other regions where the cumulative physical effect of the two wave’s presence is somewhere between these two extremes. The combined waves create a moving fluctuation that is interpreted by the ear and other receptacles as a ” humming ” sound that harmonically increases and fades with the ” beat frequency ” f = ( f1 – f2 ). Since beats are most noticeable when the frequencies of point sources differ slightly, the difference between f1 and f2 is small, but the period ( T ) = ( 1 / f ) is large enough a period of time to allow for observation of the beat.
Consider a scenario where a sprinter wearing a red uniform represents a sound wavefront #1. It takes sprinter #1 a minute to run around a track. Let’s assume that equally fast sprinters in blue uniforms begin running the race every 50 seconds. As the race begins, sprinter #1 has a runner begin to trail them after 10 seconds. Runner #1 is now 50 seconds away from passing the starting point. As runner #1 passes the starting point, another runner in blue begins running beside them. When 50 seconds pass by, sprinter #1 will be approaching the starting point, but this time, a runner in blue will begin running 10 seconds before runner #1 passes the starting point.
An observer on the sidelines at the starting point would observe sprinter #1 passing the starting point being trailed by a runner during the first trip around the track. The second time around the track, sprinter #1 would pass the starting line paired up with another runner. The third time around, sprinter #1 would pass the starting line trailing another runner in blue. This scenario is akin to what happens when two waves with slightly different frequencies combine to form wave beats. An observer of such a wave will hear a rising and falling humming sound that consists of rising and falling cooperative effects of wave components going into and out of phase with one another.
For example, let’s assume that a tuning fork is observed to produce a quavering tone ( beats ) in the presence of a siren that emits a sound wave of f1 = 440 Hz. The frequency ( f2 ) of the tuning fork is determined by measuring the period ( T ) that passes before the tone repeats itself. Let’s assume that T = ( 1 / 4 ) of a second. Since the frequency of the system is f = ( 1 / T ), the frequency of the tuning fork can be determined in the following manner: ( 1 / T ) = ( f1 – f2 ). Therefore, ( 1 / ( 0.25 s ) ) = 4Hz = ( 440 Hz – f2 ). Finally, f2 = ( 440 Hz – 4 Hz ) = 436 Hz.
By lowering the frequency of the siren until beats were observed, we were able to determine that the siren frequency had the higher of the two frequencies, and determining f2 was simplified as a consequence.
RENAISSANCE PHYSICS 