Question

When a train approaches a stationary observer, the apparent frequency of the whistle is n' and when the same train recedes away from the observer, the apparent frequency is n", Then the apparent frequency n when the observer sitting in the train is

• A. $n=\frac{n^{\prime }+n"}{2}$
• B. $n=\sqrt{n^{\prime }n"}$
• C. $n=\frac{2n^{\prime }n"}{n^{\prime }+n"}$
• D. $n=\frac{2n^{\prime }n"}{n^{\prime }-n"}$

Answer 1

The correct option is C $n=\frac{2n^{\prime }n"}{n^{\prime }+n"}$

When train approaches stationary observer, the apparent frequency $n^{{}^{\prime }}=\frac{vn}{v-v_s}$ and when the train recedes away train the observer, the apparent frequency is

$n^{\prime \prime }=\frac{vn}{v+v_s}$

or

$\frac{n}{n^{{}^{\prime }}}=1-\frac{v_s}{v}$ ........(1)

and

$\frac{n}{n"}=1+\frac{v_s}{v}$ ........(2)

On adding equation (1) and (2) , we get

$\frac{n}{n{}^{\prime }}+\frac{n}{n^{\prime \prime }}=2$

That is,

$n=\frac{2n^{{}^{\prime }}{n}^{\prime \prime }}{n^{{}^{\prime }}+n^{\prime \prime }}$