Introduction To Beats Frequency Philosophy Essay

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The sound of a beat frequency or beat wave is a fluctuating volume caused when you add two sound waves of slightly different frequencies together. If the frequencies of the sound waves are close enough together, you can hear a relatively slow variation in the volume of the sound. A good example of this can be heard using two tuning forks that are a few frequencies apart.

A sound wave can be represented as a sine waves, and you can add sine waves of different frequencies to get a graphical representation of the waveform. When the frequencies are close together, they are enclosed in a beat envelope that modulates the amplitude or loudness of the sound. The frequency of this beat is the absolute difference of the two original frequencies

Examples and applications of beat frequencies:-

A good demonstration of beat frequencies can be heard in the animation below. A pure sound of 330 Hz is combined with 331 Hz to give a rather slow beat frequency of 1 Hz or 1 fluctuation in amplitude per second. When the 330 Hz sound is combined with a 340 Hz sound, you can hear the more rapid fluctuation at 10 Hz.

Another example of beats:-

When you fly in a passenger plane, you may often hear a fluctuating droning sound. That is a beat frequency caused by engine vibrations at two close frequencies.

Application of beats:-

A piano tuner will strike a key and then compare the note with a tuning fork. If the piano is slightly out of tune, he will be able to hear the beat frequency and then adjust the piano wire until it is at the same frequency as the tuning fork. If the piano is severely out of tune, it makes the job more difficult, because the beat frequency may be too fast to readily hear.

Adding sine waves :-

Although sound is a compression wave that travels through matter, it is more convenient to illustrate the sound wave as a transverse wave, similar to how a guitar string vibrates or how a water wave appears. The shape of such a wave for a single frequency is called a sine wave. Its fig is……

in fig:-

Here Sine wave represents a single frequency of sound with constant amplitude

When we add sound waves traveling in the same direction together, elements of the sine wave add or subtract, according to where they are in the waveform. we add the amplitude of each wave, point by point. Making a graphical representation of the sum of two waves can be done by hand, but that can be be tedious.

Beat envelope:-

If we add two waves of slightly different frequencies, the resulting amplitude will vary or oscillate at a rate that is the difference between the frequencies. That beat frequency will create a beat envelope around the original sine wave.

In this figure beat envelope modulates the amplitude of the sound

Since the frequencies of the two sounds are so close and we would hear a sound that is an average of the two. But we would also hear the modulation of the amplitude as a beat frequency, which is the difference between the initial frequencies.

fb = | f1 − f2 |


fb is the beat frequency .

f1 and f2 are the two sound frequency.

| f1 − f2 | is the absolute value or positive (+) value of the difference .


For example, if we add a wave oscillating at 445 Hz with one that is at 450 Hz, the resulting frequency will be an average of the sum of the two waves. (445 Hz + 450 Hz)/2 = 447.5 Hz. This waveform is close to a sine wave, since the frequency are almost the same.

The amplitude of volume of this combination will oscillate at the beat frequency of the difference between the two: (450 Hz - 445 Hz) = 5 Hz.

Now, if we add 440 Hz and 500 Hz notes, the resulting waveform will be a complex version of a sine wave and will sound like a fuzzy average of the two tones. The average frequency of this complex wave will be (440 Hz + 500 Hz)/2 = 470 Hz.

Also, its beat frequency will be 60 Hz, which would sound like a very low-pitched hum instead of a fluctuating volume.

When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference causes the sound to be alternatively soft and loud - a phenomenon which is called "beating"or producing beats. The beat frequency is equal to the absolute value of the difference in frequency of the two waves.


-:Applications of Beats:-


-:Envelope of Beat Production:-

Beats are caused by the interference of two waves at the same point in space. This plot of the variation of resultant amplitude with time shows the periodic increase and decrease for two sine waves.

The image below is the beat pattern produced by a London police whistle, which uses two short pipes to produce a unique three-note sound.

Sum and difference frequencies

Interference and Beats:-

Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. If two upward displaced pulses having the same shape meet up with one another while traveling in opposite directions along a medium, the medium will take on the shape of an upward displaced pulse with twice the amplitude of the two interfering pulses. This type of interference is known as constructive interference. If an upward displaced pulse and a downward displaced pulse having the same shape meet up with one another while traveling in opposite directions along a medium, the two pulses will cancel each other's effect upon the displacement of the medium and the medium will assume the equilibrium position. This type of interference is known as destructive interference. The diagrams below show two waves - one is blue and the other is red - interfering in such a way to produce a resultant shape in a medium; the resultant is shown in green. In two cases (on the left and in the middle), constructive interference occurs and in the third case (on the far right, destructive interference occurs.

But how can sound waves that do not possess upward and downward displacements interfere constructively and destructively? Sound is a pressure wave that consists of compressions and rarefactions. As a compression passes through a section of a medium, it tends to pull particles together into a small region of space, thus creating a high-pressure region. And as a rarefaction passes through a section of a medium, it tends to push particles apart, thus creating a low-pressure region. The interference of sound waves causes the particles of the medium to behave in a manner that reflects the net effect of the two individual waves upon the particles. For example, if a compression (high pressure) of one wave meets up with a compression (high pressure) of a second wave at the same location in the medium, then the net effect is that that particular location will experience an even greater pressure. This is a form of constructive interference. If two rarefactions (two low-pressure disturbances) from two different sound waves meet up at the same location, then the net effect is that that particular location will experience an even lower pressure. This is also an example of constructive interference. Now if a particular location along the medium repeatedly experiences the interference of two compressions followed up by the interference of two rarefactions, then the two sound waves will continually reinforce each other and produce a very loud sound. The loudness of the sound is the result of the particles at that location of the medium undergoing oscillations from very high to very low pressures. As mentioned in a previous unit, locations along the medium where constructive interference continually occurs are known as anti-nodes. The animation below shows two sound waves interfering constructively in order to produce very large oscillations in pressure at a variety of anti-nodal locations. Note that compressions are labeled with a C and rarefactions are labeled with an R.

Now if two sound waves interfere at a given location in such a way that the compression of one wave meets up with the rarefaction of a second wave, destructive interference results. The net effect of a compression (which pushes particles together) and a rarefaction (which pulls particles apart) upon the particles in a given region of the medium is to not even cause a displacement of the particles. The tendency of the compression to push particles together is canceled by the tendency of the rarefactions to pull particles apart; the particles would remain at their rest position as though there wasn't even a disturbance passing through them. This is a form of destructive interference. Now if a particular location along the medium repeatedly experiences the interference of a compression and rarefaction followed up by the interference of a rarefaction and a compression, then the two sound waves will continually each other and no sound is heard. The absence of sound is the result of the particles remaining at rest and behaving as though there were no disturbance passing through it. Amazingly, in a situation such as this, two sound waves would combine to produce no sound. location along the medium where destructive interference continually occurs are known as nodes.


Two Source Sound Interference:-

A popular Physics demonstration involves the interference of two sound waves from two speakers. The speakers are set approximately 1-meter apart and produced identical tones. The two sound waves traveled through the air in front of the speakers, spreading our through the room in spherical fashion. A snapshot in time of the appearance of these waves is shown in the diagram below. In the diagram, the compressions of a wavefront are represented by a thick line and the rarefactions are represented by thin lines. These two waves interfere in such a manner as to produce locations of some loud sounds and other locations of no sound. Of course the loud sounds are heard at locations where compressions meet compressions or rarefactions meet rarefactions and the "no sound" locations appear wherever the compressions of one of the waves meet the rarefactions of the other wave. If we were to plug one ear and turn the other ear towards the place of the speakers and then slowly walk across the room parallel to the plane of the speakers, then you would encounter an amazing phenomenon. we would alternatively hear loud sounds as you approached anti-nodal locations and virtually no sound as you approached nodal locations. (As would commonly be observed, the nodal locations are not true nodal locations due to reflections of sound waves off the walls. These reflections tend to fill the entire room with reflected sound. Even though the sound waves that reach the nodal locations directly from the speakers destructively interfere, other waves reflecting off the walls tend to reach that same location to produce a pressure disturbance.)

Destructive interference of sound waves becomes an important issue in the design of concert halls and auditoriums. The rooms must be designed in such as way as to reduce the amount of destructive interference. Interference can occur as the result of sound from two speakers meeting at the same location as well as the result of sound from a speaker meeting with sound reflected off the walls and ceilings. If the sound arrives at a given location such that compressions meet rarefactions, then destructive interference will occur resulting in a reduction in the loudness of the sound at that location. One means of reducing the severity of destructive interference is by the design of walls, ceilings, and baffles that serve to absorb sound rather than reflect it.

The destructive interference of sound waves can also be used advantageously in noise reduction systems. Earphones have been produced that can be used by factory and construction workers to reduce the noise levels on their jobs. Such earphones capture sound from the environment and use computer technology to produce a second sound wave that one-half cycle out of phase. The combination of these two sound waves within the headset will result in destructive interference and thus reduce a worker's exposure to loud noise.


Musical Beats and Intervals:-

Interference of sound waves has widespread applications in the world of music. Music seldom consists of sound waves of a single frequency played continuously. Few music enthusiasts would be impressed by an orchestra that played music consisting of the note with a pure tone played by all instruments in the orchestra. Hearing a sound wave of 256 Hz , would become rather monotonous (both literally and figuratively). Rather, instruments are known to produce overtones when played resulting in a sound that consists of a multiple of frequencies. Such instruments are described as being rich in tone color. And even the best choirs will earn their money when two singers sing two notes i.e., produce two sound waves that are an octave apart. Music is a mixture of sound waves that typically have whole number ratios between the frequencies associated with their notes. In fact, the major distinction between music and noise is that noise consists of a mixture of frequencies whose mathematical relationship to one another is not readily discernible. On the other hand, music consists of a mixture of frequencies that have a clear mathematical relationship between them. While it may be true that "one person's music is another person's noise" (e.g., your music might be thought of by your parents as being noise), a physical analysis of musical sounds reveals a mixture of sound waves that are mathematically related.

To demonstrate this nature of music, let's consider one of the simplest mixtures of two different sound waves - two sound waves with a 2:1 frequency ratio. This combination of waves is known as an octave. A simple sinusoidal plot of the wave pattern for two such waves is shown below. Note that the red wave has two times the frequency of the blue wave. Also observe that the interference of these two waves produces a resultant (in green) that has a periodic and repeating pattern. One might say that two sound waves that

have a clear whole number ratio between their frequencies interfere to produce a wave with a regular and repeating pattern. The result is music.

Another easy example of two sound waves with a clear mathematical relationship between frequencies is shown below. Note that the red wave has three-halves the frequency of the blue wave. In the music world, such waves are said to be a fifth apart and represent a popular musical interval. Observe once more that the interference of these two waves produces a resultant (in green) that has a periodic and repeating pattern. It should be said again: two sound waves that have a clear whole number ratio between their frequencies interfere to produce a wave with a regular and repeating pattern; the result is music.

Finally, the diagram below illustrates the wave pattern produced by two dissonant or displeasing sounds. The diagram shows two waves interfering, but this time there is no simple mathematical relationship between their frequencies (in computer terms, one has a wavelength of 37 and the other has a wavelength 20 pixels). We observe that the pattern of the resultant is neither periodic nor repeating (at least not in the short sample of time that is shown). It is clear: if two sound waves that have no simple mathematical relationship between their frequencies interfere to produce a wave, the result will be an irregular and non-repeating pattern. This tends to be displeasing to the ear.


A final application of physics to the world of music pertains to the topic of beats. Beats are the periodic and repeating fluctuations heard in the intensity of a sound when two sound waves of very similar frequencies interfere with one another. The diagram below illustrates the wave interference pattern resulting from two waves (drawn in red and blue) with very similar frequencies. A beat pattern is characterized by a wave whose amplitude is changing at a regular rate. Observe that the beat pattern (drawn in green) repeatedly oscillates from zero amplitude to a large amplitude, back to zero amplitude throughout the pattern. Points of constructive interference (C.I.) and destructive interference (D.I.) are labeled on the diagram. When constructive interference occurs between two crests or two troughs, a loud sound is heard. This corresponds to a peak on the beat pattern (drawn in green). When destructive interference between a crest and a trough occurs, no sound is heard; this corresponds to a point of no displacement on the beat pattern. Since there is a clear relationship between the amplitude and the loudness, this beat pattern would be consistent with a wave that varies in volume at a regular rate.

The beat frequency refers to the rate at which the volume is heard to be oscillating from high to low volume. For ex…, if two complete cycles of high and low volumes are heard every second, the beat frequency is 2 Hz. The beat frequency is always equal to the difference in frequency of the two notes that interfere to produce the beats. So if two sound waves with frequencies of 256 Hz and 254 Hz are played simultaneously, a beat frequency of 2 Hz will be detected. A common physics demonstration involves producing beats using two tuning forks with very similar frequencies. If a tine on one of two identical tuning forks is wrapped with a rubber band, then that tuning forks frequency will be lowered. If both tuning forks are vibrated together, then they produce sounds with slightly different frequencies. These sounds will interfere to produce detectable beats. The human ear is capable of detecting beats with frequencies of 7 Hz and below.

A piano tuner frequently utilizes the phenomenon of beats to tune a piano string. She will pluck the string and tap a tuning fork at the same time. If the two sound sources - the piano string and the tuning fork - produce detectable beats then their frequencies are not identical. She will then adjust the tension of the piano string and repeat the process

the beats can no longer be heard. As the piano string becomes more in tune with the tuning fork, the beat frequency will be reduced and approach 0 Hz. When beats are no longer heard, the piano string is tuned to the tuning fork; that is, they play the same frequency. The process allows a piano tuner to match the strings' frequency to the frequency of a standardized set of tuning forks.


Important Note:- Many of the diagrams on this page represent a sound wave by a sine wave. Such a wave more closely resembles a transverse wave and may mislead people into thinking that sound is a transverse wave. Sound is not a transverse wave, but rather a longitudinal wave. Nonetheless, the variations in pressure with time take on the pattern of a sine wave and thus a sine wave is often used to represent the pressure-time features of a sound wave.

Whenever two wave motions pass through a single region of a medium simultaneously, the motion of the particles in the medium will be the result of the combined disturbance due to the two waves. This effect of superposition of waves, is also known as interference. The interference of two waves with respect to space of two waves traveling in the same direction, has been described in previous section. The interference can also occur with respect to time (temporal interference) due to two waves of slightly different frequencies, travelling in the same direction. An observer will note a regular swelling and fading or waxing and waning of the sound resulting in a throbbing effect of sound called 'beats'.

Number of beats heard per second

Qualitative treatment:-

Suppose two tuning forks having frequencies 256 and 257 per second respectively, are sounded together. If at the beginning of a given second, they vibrate in the same phase so that the compressions (or rarefactions) of the corresponding waves reach the ear together, the sound will be reinforced . Half a second later, when one makes 128 and the other 128*1/2 vibrations, they are in opposite phase, i.e., the compression of one wave combines with the rarefaction of the other and tends to produce silence. At the end of one second, they are again be in the same phase and the sound is reinforced. By this time, one fork is ahead of the other by one vibration.


in the resultant sound, the observer hears maximum sound at the interval of one second. Similarly, a minimum loudness is heard at an interval of one second. As we may consider a single beat to occupy the interval between two consecutive maxima or minima, the beat produced in one second in this case, is one in each second. If the two tuning forks had frequencies 256 and 258, a similar analysis would show that the number of beats will be two per second. Thus, in general, the number of beats heard per second will be equal to the difference in the frequencies of the two sound waves.

Analytical treatment:-

Consider two simple harmonic sound waves each of amplitude A, frequencies f1 and f2 respectively, travelling in the same direction. Let y1 and y2 represent the individual displacements of a particle in the medium, that these waves can produce. Then the resultant displacement of the particle, according to the principle of superposition will be given by


This equation represents a periodic vibration of amplitude R and  frequency . The amplitude and hence the intensity of the resultant wave, is a function of the time. The amplitude varies with a  frequency

Since intensity (amplitude)2, the intensity of the sound is maximum in all these cases. For  to assume the above values like 0, p, 2p, 3p, 4p,....

Thus, the time interval between two maxima or the period of beats =

When the difference in the frequency of the two waves is small, the variation in intensity is readily detected on listening to it. As the difference increases beyond 10 per second, it becomes increasingly difficult to distinguish them. If the difference in the frequencies reaches the audible range, an unpleasant note of low pitch called the beat note is produced. The ability to hear this beat note is largely due to the lack of linearity in the response of the ear.

Demonstration of beats:-

Let two tuning forks of the same frequency be fitted on suitable resonance boxes on a table, with the open ends of the boxes facing each other. Let the two tuning forks be struck with a wooden hammer. A continuous loud sound is heard. It does not rise or fall. Let a small quantity of wax be attached to a prong of one of the tuning forks.. This reduces the frequency of that tuning fork. When the two forks are sounded again beats will be heard.

Uses of beats:-

The phenomenon of beats is used for tuning a note to any particular frequency. The note of the desired frequency is sounded together with the note to be tuned. If there is a slight difference in frequencies, then beats are produced. When they are exactly in unison, i.e., have the same frequency, they do not produce any beats when sounded together, but produce the same number of beats with a third note of slightly different frequency. Stringed musical instruments are tuned this way. The central note of a piano is tuned to a standard value using this method.

The phenomenon of beats can be used to determine the frequency of a tuning fork. Let A and B be two tuning forks of frequencies fA (known) and fB (unknown). On sounding A and B, let the number of beats produced be n. Then one of the following equations must be true.

fA - fB = n ……………. (i)

or fB - fA = n ……………. (ii)

To find the correct equation, B is loaded with a little wax so that its frequency decreases. If the number of beats increases, then equation (i) is to be used. If the number of beats decreases, then equation (ii) is to be used. Thus, knowing the value of fA and the number of beats, fB can be calculated.

Sometimes, beats are deliberately caused in musical instruments in a section of the orchestra to create sound of a special tonal quality.

The phenomenon of beats is used in detecting dangerous gases in mines. The apparatus used for this purpose consists of two small and exactly similar pipes blown together, one by pure air from a reservoir and the other by the air in the mine. If the air in the mine contains methane, its density will be less than that of pure air. The two notes produced by the pipes will then differ in the pitch and produce beats. Thus, the presence of the dangerous gas can be detected.

The super heterodyne type of radio receiver makes use of the principle of beats. The incoming radio frequency signal is mixed with an internally generated signal from a local oscillator in the receiver. The output of the mixer has a carrier frequency equal to the difference between the transmitted carrier frequency and the locally generated frequency and is called the intermediate frequency. It is amplified and passed through a detector. This system enables the intermediate frequency signal to be amplified with less distortion, greater gain and easier elimination of noise


A beat frequency is the combination of two frequencies that are very close to each other. The sound you hear will fluctuate in volume according to the difference in their frequencies. You may often hear beat frequencies when objects vibrate. Beat frequencies can be graphically shown by adding two sine waves of different frequencies. The resulting waveform is a sine wave that has an envelope of modulating amplitude.