Beats - Practice Questions & MCQ

Beats -

When any two sound waves of slightly different frequencies, travelling along the same direction in a medium and superimpose on each other then the intensity of the resultant sound at a particular position rises and falls regularly with time. This phenomenon of regular variation in intensity of sound with time at a particular position is called beats. 

If we struck two tuning forks of slightly different frequencies, one hears a sound of periodically varying amplitude. This phenomenon is called beating.

Beat frequency, equals the difference in frequency between the two sources which we will see below.

Let us consider two sound waves travelling through a medium having equal amplitude with slightly different frequencies $f_1$ and $f_2$. We use equations similar to equation $y=A \sin (k x-\omega t)$ to represent the wave functions for these two waves at a point such that $k x=\pi / 2$.

$
\begin{aligned}
& y_1=A \sin \left(\frac{\pi}{2}-\omega_2 t\right)=A \cos \left(2 \pi f_1 t\right) \\
& y_2=A \sin \left(\frac{\pi}{2}-\omega_2 t\right)=A \cos \left(2 \pi f_2 t\right)
\end{aligned}
$

By using superposition principle -

$
y=y_1+y_2=A\left(\cos 2 \pi f_1+\cos 2 \pi f_2 t\right)
$

We can also write the above equation by using trigonometric identity as -

$
y=\left[2 A \cos 2 \pi\left(\frac{f_1-f_2}{2}\right) t\right] \cos 2 \pi\left(\frac{f_1+f_2}{2}\right) t
$
 

The graph is like this - 

Graphs of the individual waves and the resultant wave are shown in the figure. We can see that the resultant wave has effective frequency equal to average frequency $\frac{f_1+f_2}{2}$. From the figure, we can see that this wave is multiplied by the envelope whose equation is given as -

$
y_{\text {envelope }}=2 A \cos 2 \pi\left(\frac{f_1-f_2}{2}\right) t
$

A maximum in the amplitude of the resultant sound wave is detected whenever

$
\cos 2 \pi\left(\frac{f_1-f_2}{2}\right) t= \pm 1
$

Hence, there are two maxima in each period of the envelope wave. Because the amplitude varies with frequency as $\frac{\left(f_1-f_2\right)}{2}$ the beat frequency is two times of this value and given by -

$
f_{\text {beat }}=\left|f_1-f_2\right|
$