An engine whistling at a constant frequency $n_0$ and moving with a constant velocity goes past a stationary observer. As the engine crosses him,the frequency of the sound heard by him changes by a factor $f$. The actual difference in the frequencies of the sound heard by him before and after the engine crosses him is

• A
  $\frac{1}{2}n_0 (1 - f^2)$
• B
  $\frac{1}{2} n_0 \left( \frac{1 - f^2}{f} \right)$
• C
  $n_0 \left( \frac{1 - f}{1 + f} \right)$
• D
  $\frac{1}{2} n_0 \left( \frac{1 - f}{1 + f} \right)$