Concept:

When a source is moving towards a stationary observer, observed frequency is given by

\({f_{observed}} = f\left( {\frac{v}{{v + {v_s}}}} \right)\)

Where, f = frequency of sound from the source,

v = speed of sound and

v_s = speed of source.

Calculation:

Given,

Speed of the train = 34 m/s

Frequency when the whistle blows is f₁

Speed of the train is reduced to 17 m/s after sometime

Speed of the sound = 340 m/s

Frequency registered after = f₂

Now applying above formula to two different conditions given in problem, we get

f₁ = Observed frequency initially

\(\Rightarrow {f_1} = f\left( {\frac{{340}}{{340 - 34}}} \right)\)

\(\therefore {f_1} = f\left( {\frac{{340}}{{306}}} \right)\)

f₂ = Observed frequency when speed of source is reduced 

\(\Rightarrow {f_2} = f\left( {\frac{{340}}{{340 - 17}}} \right)\)

\(\therefore {f_2} = f\left( {\frac{{340}}{{323}}} \right)\)

So, the ratio f₁ : f₂ is

\(\Rightarrow \frac{{{f_1}}}{{{f_2}}} = \frac{{f\left( {\frac{{340}}{{306}}} \right)}}{{f\left( {\frac{{340}}{{323}}} \right)}} = \frac{{340}}{{306}} \times \frac{{323}}{{340}}\)

\(\Rightarrow \frac{{{f_1}}}{{{f_2}}} = \frac{{323}}{{306}} = \frac{{19}}{{18}}\)

\(\therefore \frac{{{f_1}}}{{{f_2}}} = \frac{{19}}{{18}}\)