5.0 Beats

When two waves of nearly equal (but not exactly same) frequencies traveling with the same speed in the same direction superpose on each other, they give rise to beats.

Let the displacement produced at a point by one wave be, $${y_1} = A\sin ({\omega _1}t - kx)$$ and the displacement produced at the point by other wave of equal amplitude be, $${y_2} = A\sin ({\omega _2}t - kx)$$

From the principle of superposition, the resultant displacement is, $$\begin{equation} \begin{aligned} y = {y_1} + {y_2} \\ y = A\sin ({\omega _1}t - kx) + A\sin ({\omega _2}t - kx) \\\end{aligned} \end{equation} $$Assume that the detector is at origin. So, $x=0$, $$\begin{equation} \begin{aligned} y = A\sin ({\omega _1}t) + A\sin ({\omega _2}t) \\ y = 2A\cos \left[ {\left( {\frac{{{\omega _2} - {\omega _1}}}{2}} \right)t} \right]\sin \left[ {\left( {\frac{{{\omega _2} + {\omega _1}}}{2}} \right)t} \right] \\\end{aligned} \end{equation} $$

First maxima would occur when,$$\begin{equation} \begin{aligned} \cos \left[ {\left( {\frac{{{\omega _2} - {\omega _1}}}{2}} \right)t} \right] = 1 \\ \left( {\frac{{{\omega _2} - {\omega _1}}}{2}} \right)t = 0 \\ t = 0\quad ...(i) \\\end{aligned} \end{equation} $$

Second maxima would occur when, $$\begin{equation} \begin{aligned} \cos \left[ {\left( {\frac{{{\omega _2} - {\omega _1}}}{2}} \right)t} \right] = - 1 \\ \left( {\frac{{{\omega _2} - {\omega _1}}}{2}} \right)t = \pi \\ t = \frac{{2\pi }}{{{\omega _2} - {\omega _1}}}\quad ...(ii) \\\end{aligned} \end{equation} $$

Time for one beat is $\Delta t$, $$\begin{equation} \begin{aligned} \Delta t = {t_2} - {t_1} \\ \Delta t = \left( {\frac{{2\pi }}{{{\omega _2} - {\omega _1}}}} \right) \\\end{aligned} \end{equation} $$ or $$\Delta t = \frac{1}{{{f_2} - {f_1}}}$$

Similarly, the time between two consecutive minima is, $$\Delta t = \frac{1}{{{f_2} - {f_1}}}$$

Beat frequency: It is defined as the number of beats heard per second.

Mathematically, it is given by, $$\begin{equation} \begin{aligned} f = \frac{1}{{\Delta t}} \\ f = {f_2} - {f_1} \\\end{aligned} \end{equation} $$

Application of the phenomenon of beats