Question

Three sound waves of equal amplitudes have frequencies (v - 1), (v) and (v + 1). They superpose to give beats. The number of beats produced per second will be

• A. $v$
• B. $\frac{v}{2}$
• C. $2$
• D. $1$

Answer 1

The correct option is D

$1$

When the three waves superpose at a point, then from the superposition principle, the resultant particle displacement at that point is given by

$y=y_1+y_2+y_3$

$=asin\left\{2\pi \right(v-1\left)t\right\}+asin\left(2\pi vt\right)+asin\left\{2\pi \right(v+1\left)t\right\}$

Now $sin\left\{2\pi \right(v-1\left)t\right\}+sin\left\{2\pi \right(v+1\left)t\right\}=2cos2\pi tsin2\pi vt$

Therefore, $y=y_1+y_2+y_3$

$y=a(1+2cos2\pi t)sin2\pi vt$

or $y=Asin2\pi vt$

where $A=a(1+2cos2\pi t)$ is the resultant amplitude.

Now, the resultant intensity $\propto$ $A_2$. Now $A_2$ will be maximum when

$cos2\pi t=+1$

or $2\pi t=0,2\pi ,4\pi ,$ . . . . etc.

or t = 0, 1s, 2s,. . . . etc.

$\therefore$ Time period of beats = time interval between two consecutive maxima = 1s. Hence the beat frequency is 1 Hz, i.e., one beat is heard per second which is choice (d).