BEATS -

the periodic variations in intensity at a given point due to the superposition of two sound waves of equal amplitude and of slightly different frequencies travelling in the same direction are called beats. Beats is an interesting phenomenon arising from the interference of waves.

When two harmonic sound waves of close frequencies are heard at the same time, we hear a sound of an average of two close frequencies with audibly distinct waxing and waving of the intensity of the sound, with a frequency equal to the difference in the two close frequencies.

Let us consider two harmonic sound waves of nearly equal angular frequency $\dpi{120} \fn_cm \omega_1\;and\; \omega_2\;(\omega_1 > \omega_2)$ travelling in +ve x axis.

$\dpi{120} \fn_cm s_1=A sin(\omega_1 t -k_1 x)$

and $\dpi{120} \fn_cm s_2=A sin(\omega_2 t-k_2 x)$

Let choose a p[article at x=0

i.e $\dpi{120} \fn_cm s_1=A sin(\omega_1 t)\;and\;s_2=Asin(\omega_2 t)$

The resultant displacement is, by the principle of superposition is

$\dpi{120} \fn_cm s=s_1 +s_2$

$\dpi{120} \fn_cm =A sin\omega_1t+A sin \omega_2 t$

$\dpi{120} \fn_cm =A [sin\omega_1t+ sin \omega_2 t]$

$\dpi{120} \fn_cm =A\; 2\sin(\frac{\omega_1 t+\omega_2 t}{2}).\cos(\frac{\omega_1 t-\omega_2 t}{2})$

$\dpi{120} \fn_cm \Rightarrow s=\left [ 2 \;A \cos(\frac{\omega_1-\omega_2}{2})t \right ] \sin (\frac{\omega_1+\omega_2}{2})t$

$\dpi{120} \fn_cm \large \Rightarrow s=\left [ 2 \;A \cos 2\pi(\frac{f_1-f_2}{2})t \right ] \sin 2\pi(\frac{f_1+f_2}{2})t$

The following points may be noted from above equation

a. The resultant wave has an effective frequency equal to the average frequency of the two sources i.e $\dpi{120} \fn_cm \left ( \frac{f_1 +f_2}{2} \right )$

b. The resultant wave has an amplitude given by $\dpi{120} \fn_cm A_r= 2 A cos 2\pi\left ( \frac{f_1-f_2}{2} \right ) t$

Here amplitude varies in time with a frequency given by $\dpi{120} \fn_cm \left ( \frac{f_1 -f_2}{2} \right )$ where $\dpi{120} \fn_cm f_1$ is close to $\dpi{120} \fn_cm f_2$

NOTE

Here cosine term contain $\dpi{120} \fn_cm \left ( \frac{\omega_1 -\omega_2}{2} \right )$ i.e cosine term is a slowly varying term and it is this which makes it possible for it to be regarded as amplitude.

The following graph shows the plots of

$\dpi{120} \fn_cm A_r= 2 A cos 2\pi\left ( \frac{f_1-f_2}{2} \right ) t\;\;\;\;\;(frequency \;is\; \frac{f_1-f_2}{2})$

$\dpi{120} \fn_cm B=\sin 2\pi \left ( \frac{f_1+f_2}{2} \right )t\;\;\;\;\;(frequency\;is\; \frac{f_1 +f_2}{2})$

and their product

$\dpi{120} \fn_cm s=\left [ 2 \;A \cos 2\pi(\frac{f_1-f_2}{2})t \right ] \sin 2\pi(\frac{f_1+f_2}{2})t$

as a function of time for a fixed x=0

BEAT FREQUENCY

It is the number of beat/ loud sounds per second. Note that loud sound will be detected when

$\dpi{120} \fn_cm \cos 2\pi \left ( \frac{f_1 -f_2}{2} \right )t=\pm 1$

$\dpi{120} \fn_cm i.e\;\; 2\pi \left ( \frac{f_1 -f_2}{2} \right )t=0,\pi,2\pi, 3\pi............$

$\dpi{120} \fn_cm \pi(f_1-f_2)t=0,\pi,2\pi,3\pi.........$

$\dpi{120} \fn_cm i.e \;\;at\;\;t_1=0\;\;\;\;\;(detect\;loud\;sound)$

$\dpi{120} \fn_cm at\;\;t_2=\frac{1}{f_1-f_2}\;\;\;\;\;(detect\;next\;loud\;sound)$

$\dpi{120} \fn_cm at\;\;t_3=\frac{2}{f_1-f_2}\;\;\;\;\;(detect\;next\;loud\;sound)$

and so on

so the time period of beat/ loud sound is $\dpi{120} \fn_cm =t_2-t_1=\frac{1}{f_1-f_2}$

$\dpi{120} \fn_cm \large \therefore \left [ Beat\;frequency=f_1-f_2 \right ]$

For example, if two tuning forks vibrate individually at frequencies of 442 Hz and 438 Hz, the resultant sound wave of two combinations would have a frequency= $\dpi{120} \fn_cm =\frac{f_1+f_2}{2}=440 Hz$ and beat frequency $\dpi{120} \fn_cm (f_1-f_2)=4Hz$ that is the listener would hear the 440 Hz sound wave go through an intensity maximum of four times every second.