Question

A train running at 108 km h⁻¹ towards east whistles at a dominant frequency of 500 Hz. Speed of sound in air is 340 m/s. What frequency will a passenger sitting near the open window hear? (b) What frequency will a person standing near the track hear whom the train has just passed? (c) A wind starts blowing towards east at a speed of 36 km h⁻¹. Calculate the frequencies heard by the passenger in the train and by the person standing near the track.

Answer 1

Given:
Velocity of sound in air v = 340 m/s
Velocity of source v_s = 108 `\text{ kmh}^\(-)`¹ =\[\frac{108 \times 1000}{60 \times 60} = 30   {\text { ms }}^{- 1}\]

Frequency of the source \[n_0\]= 500 Hz
(a) Since the velocity of the passenger with respect to the train is zero, he will hear at a frequency of 500 Hz.

(b) Since the observer is moving away from the source while the source is at rest:
Velocity of observer \[v_o\]= 0 
 Frequency of sound heard by person standing near the track is given by:

\[n = \left( \frac{v}{v + v_s} \right) n_0 \]

 Substituting the values, we get:

\[n = \frac{340}{340 + 30} \times 500 = 459 \text{ Hz }\]

(c) When medium (wind) starts blowing towards the east:

Velocity of medium v_m = 36 `\text { kmh}^\(-)`¹ =\[36 \times \frac{5}{18}   =   10   {\text { ms }}^{- 1}\]

The frequency heard by the passenger is unaffected (= 500 Hz).

 However, frequency heard by person standing near the track is given by:

\[n = \frac{\left( v + v_m \right)}{\left( v + v_m \right) + v_s} \times  n_0 \] 

\[     = \frac{\left( 340 + 10 \right)}{\left( 340 + 10 \right) + 30} \times 500\] 

\[     = 458 \text{ Hz }\]