Why a passing siren drops in pitch, drawn wave by wave. A source glides to the right, sending out a circular wavefront every beat; because it moves between beats, the fronts bunch up ahead of it into a shorter wavelength and higher pitch (tinted blue), and stretch out behind into a lower one (tinted red). Set the source speed, the observer speed, the emitted frequency and the medium, and read the shifted frequencies and wavelengths from f′ = f·(v ± vo)/(v ∓ vs). Push the source past the wave speed and the fronts collapse into a Mach cone — the makings of a sonic boom. It all runs on your device.
You are in the Physics lab.
What the rings, the colours and the cone are telling you.
Each ring is a wave crest, expanding at the wave speed from the exact spot the source sat when it left. Because the source keeps moving, every new ring starts a little further to the right, so the circles are no longer concentric — ahead of the source the moving centres crowd the rings into a shorter wavelength, and behind it they leave the rings stretched apart. A listener meets the crowded crests more often (higher pitch) and the stretched ones less often (lower pitch).
The half of every ring facing the way the source moves is drawn blue — that is the compressed, higher-pitch side, and the region ahead carries a faint blue wash. The trailing half is drawn red for the stretched, lower-pitch side. The observer's readout is tinted the same way: blue when the frequency it hears is above f₀, red when it drops below. Move the observer toward the source to push its reading further into the blue, or away to slide it toward red.
Raise the source speed toward the wave speed and the rings ahead squeeze together until, at Mach 1, the source rides the front of its own waves. Past that the crests can never get ahead, so they all touch a single V-shaped envelope trailing the source — the Mach cone, drawn in red with its half-angle set by sin θ = 1/M. The sudden pressure jump as that cone sweeps past a listener is heard as a sonic boom. Faster source, narrower cone.
Crests emitted from a moving point, packed ahead and stretched behind.
The source emits a wavefront at a steady rate, and each front expands outward at the wave speed from the exact spot the source occupied when it left. Because the source has moved on by the time the next front is emitted, the circles are no longer concentric — their centres march forward with the source.
Ahead of the source the moving centres crowd the fronts together, shortening the wavelength, so a listener there meets crests more often — a higher frequency. Behind the source the fronts are spread apart, lengthening the wavelength, so a listener there hears a lower frequency. The readouts apply the general Doppler relation f′ = f·(v ± vo)/(v ∓ vs) to each side: in the numerator take plus when the observer moves toward the source, minus when it moves away; in the denominator take minus when the source approaches, plus when it recedes. The wavelength readouts come straight from λ = (v ∓ vs)/f, and every value is scaled to the medium you pick — sound in air at 343 m/s or in water at 1480 m/s.
As the source speed approaches the wave speed the fronts ahead squeeze into almost the same place and the approaching frequency shoots up. At exactly the wave speed — Mach 1 — the source rides the crest of its own waves, which pile into a single shock front; beyond it they trail as a Mach cone, heard as a sonic boom. Slide the speed up to see it happen.
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