A train travelling at 108 km h^ -1 towards east whistles with a...

Speed of the train, $v_s = 108 \text{ km h}^{-1} = 108 \times \dfrac{5}{18} \text{ m/s} = 30 \text{ m/s}$

Speed of sound, $v = 340 \text{ m/s}$

Actual frequency, $f_0 = 500 \text{ Hz}$

Since the train has just passed the person, it is moving away from the stationary observer.

Using the Doppler effect formula for a source moving away from a stationary observer:

f = f_0 \left(\dfrac{v}{v + v_s} \right)

Substituting the values:

$f = 500 \left( \dfrac{340}{340 + 30} \right)$

f = 500 \left(\dfrac{340}{370} \right)

$f =$\dfrac{17000}{37} \approx$459.46 \text{ Hz}$

The frequency heard by the person is approximately $459 \text{ Hz}.$