A sound source S is moving along a straight track with speed v, and is emitting, sound of frequency v₀ (see figure). 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:
(t₀ represents the instant when the distance between the source and observer is minimum)

Solution

Physics sir · 3 steps

Only the velocity component along the line to you counts, and it shrinks, vanishes, then reverses.

1. Line-of-sight component. When the source moves at an angle to the line joining it to the observer, only the component of its velocity along that line, v cos θ, causes a Doppler shift. Here c is the speed of sound and θ is the angle between the track and the line from S to O.

   ν = ν₀ c/(c - v cos θ)

   Stage	Angle θ	cos θ	Frequency heard
   Far away, approaching	close to 0°	close to +1	highest, near ν₀c/(c-v)
   Closest point, t₀	90°	0	exactly ν₀
   Far away, receding	close to 180°	close to -1	lowest, near ν₀c/(c+v)

2. Shape of the curve. As the source moves, θ changes smoothly, so cos θ falls smoothly from +1 to -1. The frequency therefore falls smoothly from above ν₀ to below it. There is no sudden jump because the observer is off the track.
3. So. The graph must fall continuously and cross ν₀ at t₀. That is option (d).

The idea to keep. For a source passing beside you, use ν₀c/(c - v cos θ). Pitch falls smoothly and equals ν₀ when the motion is perpendicular to your line of sight.

Answer from the NTA final answer key. Solution by PrepWiser.