Question

Two sound sources S1 and S2 of frequencies 324 Hz and 320Hz are placed at certain separation .  An observer is moving away from S1 and towards S2 on line joining them. If he hears no beats then speed of observer is 

Answer 1

Given :

Frequency of First Source S1 = 324 Hz

Frequency of second source S2 = 320 Hz

Observer moving away from S1 and towards S2

Observer cant hear beats.

To find :

Speed of observer.

Solution :

When a sound source is producing sound and the observer is moving then there will be difference between the original frequency and the apparent frequency with respect to the observer .

So, when the observer is moving away from source S1,

The formula of apparent frequency :

$\bf \: f_1 = f_0 \frac{(v - v_0)}{v}$

(equation 1)

Original frequency of first source.

f0 = 324 Hz

Speed of sound v = 344 meter per second.

so,

putting these values in equation 1,

Apparent frequency of source S1 :

$\bf \: f_1 = 324 \times \frac{(344 - v_0)}{344}$

(equation 2)

similarly

When the observer is moving towards the source S2,

The formula of apparent frequency :

$\bf \: f_2 = f_0 \frac{(v + v_0)}{v}$

(equation 3)

Original frequency of second source.

f0 = 320 Hz

Speed of sound v = 344 meter per second.

so,

putting these values in equation 3,

$\bf \: f_2 = 320 \times \frac{(344 + v_0)}{344}$

(equation 4)

It is given that the observer hears no beat,

it means the difference between apparent frequencies of both the sources is equal to zero and

Apparent frequency of first source is equal to apparent frequency of second source.

$\bf \: f_1 - f_2 = 0 \\ </p><p> \bf \: f_1 = f_2$

so, by equation 2 and equation 4.

$\bf \: 324 \times \frac{(344 - v_0)}{344} = \bf \: 320 \times \frac{(344 + v_0)}{344} \:$

$\bf \: 324 \times (344 - v_0)= \bf \: 320 \times (344 + v_0) \:$

$\bf(320 + 324)v_0 = 344(324 - 320) \\ \\ \bf v_0 \: = \frac{344 \times 4}{644}$

on solving this we get,

Speed of observer v0 = 2.1 meter per second