An ambulance siren holds one steady note as far as its own speaker is concerned — it isn't sliding its pitch up and down on purpose. Standing on the pavement as it approaches, you hear something that clearly is sliding: a higher note swelling as it nears, then a distinct drop right around the moment it passes, settling into a lower note as it recedes. The siren never changed. What changed is that you were listening to it from a position the sound source was moving toward, and then, in an instant, from a position it was moving away from.
Compression on the way in, stretching on the way out
Sound travels as a series of pressure waves, and pitch is just how frequently those waves arrive at your ear — more wave-crests per second, higher pitch. A stationary siren emits crests at a steady rate, evenly spaced through the air in every direction. A moving siren emits each new crest from a slightly different position than the last one, because it's travelled a little further between emissions. Heading toward you, each successive crest is emitted from closer than the one before it, so the crests arrive at your ear more bunched together than they were made — a shorter effective wavelength, which your ear reads as a higher pitch than the siren is actually producing. Moving away, the opposite happens: each crest is emitted from further off than the last, so they arrive stretched further apart, and the pitch you hear drops below the siren's true note.
This is the Doppler effect, and the reason the shift feels concentrated right at the moment of passing rather than spread smoothly across the whole approach is about geometry, not sound. What actually matters for the shift is the rate the siren is closing the distance to you — its speed along the line straight to your ear, not its raw speed down the road. Well before the ambulance reaches you, most of its motion is angled somewhat toward you and the shift builds gradually. In the instant it's directly alongside you, its entire velocity is suddenly perpendicular to that line — for that instant, it's neither closing the distance nor opening it — and just after, its whole velocity swings to point directly away. That swing from maximally-approaching to maximally-receding happens fastest exactly at the closest point, which is why the ear registers the shift as a comparatively sudden drop centred right there rather than a slow fade.
The same effect, done to light instead of sound
Light is also a wave, and the same logic applies to it with one substitution: instead of a pitch shift, a wavelength shift, which for visible light reads as a colour shift. A star or galaxy moving away from Earth has its light stretched toward longer, redder wavelengths — redshift — and one moving closer gets compressed toward shorter, bluer ones — blueshift. Edwin Hubble's observation that most distant galaxies show redshift, and that the redshift grows with distance, was direct evidence the universe itself is expanding, discovered by applying exactly the physics behind a receding ambulance siren to light instead of sound.
What we're still unsure about
The "instant drop" framing is intuitive but slightly overstates how sudden the shift actually is — the pitch change is continuous the entire time, following the smoothly changing angle between the siren's velocity and the line to the listener; there's no literal discontinuity at the moment of passing, just a point where the rate of change is at its steepest. Describing it as an "instant" switch is a useful simplification for how it's perceived, not a precise description of the underlying waveform, and it's worth being clear about which one a given explanation is actually claiming.
This sits inside The Doppler Effect, one of eight topics in Waves & Optics, one of five domains in Physics, one of seventeen subjects the app can quiz you on.