A train whistling at constant frequency is moving towards a station at a constant speed $v$. The train goes…

A train whistling at constant frequency is moving towards a station at a constant speed $v$. The train goes past a stationary observer on the station. The frequency $n$ of the sound as heard by the observer is plotted as a function of time $t$ (figure). Identify the expected curve. [NCERT Exemplar] Four graphs showing frequency n versus time t: (a) curve concave downwards, (b) triangular curve rising then falling linearly, (c) constant higher frequency dropping vertically to constant lower frequency, (d) smooth bell-like curve peaking at time of passing.
  1. Graph (a)
  2. Graph (b)
  3. Graph (c)
  4. Graph (d)

Solution

Let the original frequency of the source is $n_o$. Let the speed of sound wave in the medium is $v$. As observer is stationary, [Diagram shows a train moving with speed $v$ towards a stationary observer.] When train is approaching, Apparent frequency, $n_a = \left( \frac{v}{v - v_s} \right) n_o \Rightarrow n_a > n_o$ When the train is going away from the observer, Apparent frequency, $n_a = \left( \frac{v}{v + v_s} \right) n_o \Rightarrow n_a < n_o$ Hence, the expected curve is (c).

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