An Idea On Beat Frequency

To understand beat frequency, consider two frequencies of 440 Hz and 441 Hz. While playing both of the frequencies simultaneously, both the frequencies interfere with each other and cause beat frequency. Beat frequency can be defined as the absolute value of the difference between two frequencies. In the above-given example, the difference between both frequencies is 1 Hz, which is one cycle per second. Similarly, if you keep on changing the frequencies involved in creating beat frequencies, you can get the different numbers of cycles per second. The frequencies with the number of beats when experimented with are distinctly audible.

Why do beat frequencies happen?

When two sinusoidal frequencies are close to one another in frequency, their two-wave patterns overlap with each other to produce beat waves. 

Here, P1 and P2 are two different frequency waves. It is seen that when they overlap, the two waves alternate between being in phase, and therefore, constructively interfere with one another to create a larger amplitude. Being out-phase, they destructively interfere with one another to create a smaller amplitude. When this happens, the amplitude of the two waves combines to result in an increase and decrease in the beat frequency. This is why we can hear a distinct number of beats during the experiment. The difference between the two frequencies is the time period taken between two wave peaks, which when sounded together, results in beat frequency. During constructive interference, the sound will be quite loud compared to destructive interference. This can also be seen when you try to count the number of beats in a beat frequency. One can hear consecutive louder and softer sounds when the beat frequency phenomenon happens. The beats are the distinctive wobble in the resultant confluence of two frequencies. The more wobbles you hear, the larger the difference that exists between the two frequencies combining to form the beat frequency. 

Beat Frequency Formula

Out of the two waves combined to result in a beat frequency, let’s name the frequency of wave one as f1 and that of wave two as f2. Thus, the beat frequency fb is written as,

The beat frequency value cannot be negative. So always subtract from the higher frequency to the lower frequency. The value of beat frequency is the number of wobbles it takes to cover from one constructive wave to the other constructive wave per second. 

Derivation of Beat Frequency Formula

To understand the beat frequency formula, we need to understand the beat period. The beat period is the number of seconds taken per wobble. In other words, we can say the time required for the process to begin from constructive to destructive and then again to a constructive is called a beat period. We know that

f =  1T 

Frequency is one over the Time period, so the beat period will help us derive the beat frequency.

Assuming that two individual waves started in the constructive pattern. For getting the beat period, we need to calculate the time till that wave reaches the next consecutive constructive wave. 

The above diagram has two, a red and a blue, waves of different frequencies overlap to form the beat frequency. We need to find the value between two consecutive ones as marked above for getting the beat period. 

Observe that every time the red wave takes a whole cycle, its consecutive peak stands a bit more displaced than the blue wave’s consecutive peak. Speaking in terms of time, the red wave’s peak happens slightly later than the blue wave. The period of the red wave is longer than the period of the blue wave. This amount of time difference can be written as

T =T1(red wave) – T2(blue wave)

With further cycles, the peaks have more time differences. Thus, it can be written as 

(T1 – T2) nR 

Where, nR is the number of cycles of the red wave and (T1 – T2) is the time difference. 

The above equation will give you information on how far apart the two crests are in time. As seen after some number of cycles of the red wave, the peaks of both the waves lie over each other. That is, the period taken by the red wave becomes equal to the period taken by the blue wave. Thus,

(T1 – T2) nR = T2

The blue wave has advanced in time through so many cycles that its next peak is overlapping with this red wave’s peak, thus, becoming constructive again. The variable component here is that the number of cycles taken by a particular wave might differ depending on the individual frequencies. Thus, to make it a general formula applicable on any frequency, we need to change nR to

nR = tT1 

Here, the ‘t’ can be taken as any time variable. T2 is the time period of the red wave to complete one complete cycle. thus,

(T1 – T2) tT1  = T2

t = T1T2T1 – T2

Here, the ‘t’ is the beat period as this equation (T1 – T2) nR = T2 was for the time period from one consecutive to the other consecutive. Thus, we can solve further,

Tb = T1T2T1 – T2

For obtaining f =  1T,

1Tb = T1 – T2T!T2=T1T!T2-T2T1T2

 1Tb= 1T2-1T1

As we know, f =  1T thus,

fb = |f1 – f2|

Conclusion

The oscillation of a sound wave from loud to soft and then back to loud again results in beats. In the lab, this can be experimented with using two tuning forks of different frequencies. The difference between the sound of the two frequencies gives rise to beat frequency. It has a very simple formula where we subtract the different values of the frequency to obtain the beat frequency. The consecutive constructive and destructive waves give rise to loud and soft sounds of the beat. Beat frequencies can help us understand the difference in frequencies. It also helps in tuning different musical instruments to sound properly.