Wave Beats

Wave beats occur when two waves having the same amplitude but slightly different frequencies overlap in a region of space. Here we discuss sound wave beats.

In Figure 1 you can see two sinusoidal sound waves having the same amplitude but slightly different frequencies overlap in a rigion of space.

Note that the difference between two frequencies should not exceed $6$ or $7 \text{Hz}$. The two sound waves superpose and the displacement of any particle at a point in the medium is given by the vector sum of the individual displacements due to each wave.

Figure 1 Sound wave beats.

Both are sinusoidal waves so the first wave shown by red line is say $y_1 = A \cos (\omega _1t)$ and the other shown by green line is say $y_2 = A\cos (\omega _2t)$.

Note that both waves are travelling in positive x-direction in our illustration in Figure 1.

Now we are going to add these two waves to get the resultant wave, that is $y = y_1 + y_2$. But before adding the two wave functions, you should know the trigonometric identity $\cos a + \cos b = 2\cos [\frac{1}{2}(a - b)]\cos [\frac{1}{2}(a + b)]$ and you'll get the result:

\[y = 2A\cos [\frac{1}{2}({\omega _1} - {\omega _2})t]\cos [\frac{1}{2}({\omega _1} + {\omega _2})t] \tag{1} \label{1}\]

You know that $\omega _1 = 2 \pi f_1$ and $\omega _2 = 2 \pi f_2$ where $f_1$ and $f_2$ are the corresponding frequencies. Substituting the values of $\omega _1$ and $\omega _2$ in the above equation you'll get another form:

\[y = 2A\cos [\frac{1}{2}2\pi ({f_1} - {f_2})t]\cos [\frac{1}{2}2\pi ({f_1} + {f_2})t] \tag{2} \label{2}\]

Notice the first part of the right hand side of the above equation which is $2A\cos [\frac{1}{2}2\pi ({f_1} - {f_2})t]$.

In one complete cycle of $2A\cos [\frac{1}{2}2\pi ({f_1} - {f_2})t]$ there are two amplitude peaks hence the beat frequency is twice the part $\frac{1}{2} (f_1 - f_2)$ which is simply $f_1 - f_2$.

The slight difference in frequencies between two waves having the same amplitude causes the variation in loudness called beats.

The Figure 1 shows the displacement-time graph of the superposition of two sinusoidal waves.

You can easily find the beat frequency of the resultant wave by just looking at the graph which is simply $4 \rm Hz$ in which the amplitude goes through four maxima and four minima in one second.

The actual wave function of the wave shown by red curve is $y_1 = 5 \cos (2\pi 10 x)$ and the actual wave function of the wave shown by green curve is $y_2 = 5 \cos (2\pi 14 x)$, so $f_1 = 10 \rm Hz$ and $f_2 = 14 \rm Hz$.

Therefore, the beat frequency is the difference between these two frequencies which is $\left| {{f_1} - {f_2}} \right| = 4 \rm Hz$.

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