Question
A source of sound $$S$$ emitting waves of frequency $$100\,Hz$$ and an observer $$O$$ are located at some distance from each other. The source is moving with a speed of $$19.4\,m{s^{ - 1}}$$ at an angle of $${60^ \circ }$$ with the source observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer is (velocity of sound in air $$330\,m{s^{ - 1}}$$ )
A source of sound $$S$$ emitting waves of frequency $$100\,Hz$$ and an observer $$O$$ are located at some distance from each other. The source is moving with a speed of $$19.4\,m{s^{ - 1}}$$ at an angle of $${60^ \circ }$$ with the source observer line as shown in the figure. The observer is at rest. The apparent frequency observed by the observer is (velocity of sound in air $$330\,m{s^{ - 1}}$$ )
A.
$$103\,Hz$$
B.
$$106\,Hz$$
C.
$$97\,Hz$$
D.
$$100\,Hz$$
Answer :
$$103\,Hz$$
Solution :
Here, original frequency of sound, $${f_0} = 100\,Hz$$
Speed of source $${V_s} = 19.4\cos {60^ \circ } = 9.7$$
From Doppler's formula
$$\eqalign{ & f = {f_0}\left( {\frac{{V - {V_0}}}{{V - {V_s}}}} \right) \cr & f = 100\left( {\frac{{V - 0}}{{V - \left( { + 9.7} \right)}}} \right) \cr & f = 100\frac{V}{{V\left( {1 - \frac{{9.7}}{V}} \right)}} = \frac{{100}}{{\left( {1 - \frac{{9.7}}{{330}}} \right)}} = 103\,Hz \cr} $$
Apparent frequency $$f = 103\,Hz$$
Here, original frequency of sound, $${f_0} = 100\,Hz$$
Speed of source $${V_s} = 19.4\cos {60^ \circ } = 9.7$$
From Doppler's formula
$$\eqalign{ & f = {f_0}\left( {\frac{{V - {V_0}}}{{V - {V_s}}}} \right) \cr & f = 100\left( {\frac{{V - 0}}{{V - \left( { + 9.7} \right)}}} \right) \cr & f = 100\frac{V}{{V\left( {1 - \frac{{9.7}}{V}} \right)}} = \frac{{100}}{{\left( {1 - \frac{{9.7}}{{330}}} \right)}} = 103\,Hz \cr} $$
Apparent frequency $$f = 103\,Hz$$