Answer to Question #164206 in Optics for Zaru

Question

(a.) Beats are heard when a tuning fork and a sonometer wire vibrate simultaneously with

frequencies f1 and f2 respectively, f2 being slightly different from f1.

(b.) In an arrangement such as that described in (a), a sonometer wire vibrates under a constant

tension in its fundamental mode and two beats per second are heard when f1 = 512Hz. When the

length of the wire is increased by 2mm, beats are no longer heard. Calculate, to three significant

figures,

(i.) The initial frequency of vibration.

(ii.) The initial length of the wire. 

Answer 1

a) Answer

i) according to the physics concept

Beats are generated when two sounds have similar frequencies or almost equal to each other. The effect is that the sound's amplitude or volume oscillates or beats. The beat phenomenon is caused by constructive and destructive interference. The picture below shows the beat generated by two tuning forks with very close frequencies. so beats are formed.

ii)we know that The formula for beat frequency is the difference in frequency of the two superimposed waves.

$f_b =|f2−f1|$

f₁ and f₂ are the frequency of two waves. So we can say the beat frequency is equal to the difference between fi and f2·

b) 2 beats per second are heard

$f = 512Hz \space\space\space\space\space\space\space\space\space f_1\pm 2 = f \to f_1 = f - 2 = 510Hz$

Initial frequency - $f_1 = 510Hz$

$fl = const$ $\Delta l = 2mm$

$fl = f_1l_1 \to fl = f_1(l+\Delta l) \to l = \large\frac{f_1\Delta l}{f-f_1}$ = 510 mm

Initial length - 510mm