A commuter train blows its\(200\;{\rm{Hz}}\)horn as it approaches a crossing. The speed of sound is\(335\;{\rm{m/s}}\). (a) An observer waiting at the crossing receives a frequency of\(208\;{\rm{Hz}}\). What is the speed of the train? (b) What frequency does the observer receive as the train moves away?

The apparent frequency of a sound source increases when the source moves towards the listener and vise-versa.

The apparent frequency is\(f' = 208\;{\rm{Hz}}\).

The actual frequency is\(f = 200\;{\rm{Hz}}\).

The speed of sound is\(v = 335\;{\rm{m/s}}\).

(a)

The Doppler Effect tells the apparent frequency is,

\(f' = f\left( {\frac{{v - {v_o}}}{{v - {v_s}}}} \right)\)

Now, the apparent frequency is,

\(\begin{array}{c}208 = 200\left( {\frac{{335 - 0}}{{335 - {v_s}}}} \right)\\\frac{{208}}{{200}} = \frac{{335}}{{335 - {v_s}}}\\335 - {v_s} = 322.11\\{v_s} = 12.88\;{\rm{m/s}}\end{array}\)

(b)

Now, the apparent frequency of the passing away ambulance is,

\(\begin{array}{c}f'' = 200\left( {\frac{{335 - 0}}{{335 - \left( { - 12.88} \right)}}} \right)\\ = 192.6\;{\rm{Hz}}\end{array}\)