Question

One train is approaching an observer at rest and another train is receeding from him with the same velocity $4\text{m/s}$. Both the trains blow whistles of same frequency of $243\text{Hz}$. The beat frequency in $\text{Hz}$ as heard by the observer is (speed of sound in air is $320\text{m/s}$).

Answer 1

When the train is approaching apparent frequency heard by the observer is
$n_1=\frac{v}{v-v_s}\times n=\frac{320}{320-4}\times 243=\frac{80}{79}\times 243$
When the train is receding apparent frequency heard by the observer is
$n_2=\frac{v}{v+v_s}\times n=\frac{320}{324}\times 243=\frac{80}{81}\times 243$
Beat frequency is
$n=n_1-n_2=80\times 243(\frac{1}{79}-\frac{1}{81})\approx 6\text{Hz}$