A sound source S is moving along a straight track with speed $$v$$, and is emitting sound of frequency $$v_0$$. An observer is standing at a finite distance, at the point O, from the track. The time variation of frequency heard by the observer is best represented by: ($$t_0$$ represents the instant when the distance between the source and observer is minimum)

Apparent frequency formula:   $$\nu = \nu_0 \left( \frac{v_s}{v_s - v \cos\theta} \right)$$

As source approaches from infinity ($$t \to -\infty$$):

$$\theta \to 0^\circ \implies \cos\theta \to 1 \implies \nu_{\text{max}} = \nu_0 \left( \frac{v_s}{v_s - v} \right) > \nu_0$$

At the closest approach ($$t = t_0$$):

$$\theta = 90^\circ \implies \cos\theta = 0 \implies \nu = \nu_0$$

As source recedes to infinity ($$t \to \infty$$):

$$\theta \to 180^\circ \implies \cos\theta \to -1 \implies \nu_{\text{min}} = \nu_0 \left( \frac{v_s}{v_s + v} \right) < \nu_0$$

Since $$\theta$$ changes continuously with time:

$$\nu \text{ decreases smoothly and continuously from } \nu_{\text{max}} \text{ to } \nu_{\text{min}}\text{, passing through } \nu_0 \text{ at } t_0.$$