Question

When a train approaches a stationary observer, the apparent frequency of the whistle is $n^{\prime }$ and when the same train recedes away from the observer, the apparent frequency is $n^{\prime \prime }$. Then, the apparent frequency $n$ when the observer moves with the train is :

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

Answer 1

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

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

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

and $\frac{n}{n^{\prime }}=1+\frac{v_s}{v}...\left(ii\right)$

On adding Eqs. (i) and (ii), we get

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

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