According to the formula to calculate the apparent frequency heard by an observer, the frequency is independent of the distance of the source from the observer. This means however close the observer is from the source the frequency would be constant as far as the velocity is constant.
Let's consider a source moving with a velocity $\ v$ towards an observer.Let its distance from the observer at an instant before it crosses the observer be $\ ds$. The frequency heard by the observer is $\nu_1$. When the source is exactly on the observer the frequency is the original frequency with which the waves were emitted, i.e., $\nu_o$. When the source crosses the observer and it is a distance $\ ds$ on the opposite side of the observer and the frequency now becomes $\nu_2$.
My question is:
This seems a very abrupt/instantaneous process although in our daily life it doesn't happen so abruptly. The change in frequency is not so sudden. Why does that happen?