Beats and the Doppler effect

Sound two equal-amplitude tones at one point and superposition gives y = A cos(2πf₁t) + A cos(2πf₂t). The identity cos a + cos b = 2 cos ½(a − b) cos ½(a + b) turns the sum into a product: a carrier at the mean frequency ½(f₁ + f₂), multiplied by a slow envelope 2A cos(π(f₁ − f₂)t). With 440 Hz and 444 Hz you hear 442 Hz, and the envelope has frequency 2 Hz. The ear responds to intensity, which goes as the envelope squared, and cos² peaks twice per cycle, so the loudness swells 4 times a second: f_beat = |f₁ − f₂|, not half of it. Physically the tones slide from in step to out of step and back once every 1/|f₁ − f₂| = 0.25 s. If the amplitudes differ, the minima are |A₁ − A₂| rather than zero, and the beat is shallower but no slower.

Beats report |f₁ − f₂|, and the modulus throws away the sign. A string that beats at 3 Hz against a 440 Hz fork is at 437 Hz or at 443 Hz, and no amount of careful counting separates the two. What separates them is a small change in a known direction. Tighten the string a little, which raises its frequency: if the beats slow, the string was below the fork and is closing on it; if they quicken, it was already above. Loading a fork's prong with a little wax lowers its frequency and serves the same purpose. The count itself is good to about one beat over the counting time, so 30 beats in 10 s gives 3.0 ± 0.1 Hz: a 0.02% statement about a 440 Hz tone, made with no frequency meter. Above roughly 15 Hz the swells blur into roughness and the method stops.

Let a source of frequency f move at vₛ straight toward a listener at rest in still air. It emits one front, and one period T = 1/f later it emits the next, but by then the first front has travelled vT and the source has followed it by vₛ T. The gap between fronts ahead is therefore λ′ = (v − vₛ)T = (v − vₛ)/f. Once emitted, the fronts still cross the air at v, since the air sets that speed and the source does not, so the listener receives f′ = v/λ′ = f v/(v − vₛ). For a 500 Hz siren at 30.0 m s⁻¹ with v = 343 m s⁻¹, the spacing ahead drops from 0.686 m to 0.626 m and the listener hears 548 Hz; behind, the spacing is (v + vₛ)/f = 0.746 m and the pitch is 460 Hz. The rise and the fall are not equal: +48 Hz against −40 Hz.

Now park the source and drive the listener toward it at vₒ. Nothing about the wave in the air has changed: the fronts are λ = v/f apart and cross the air at v. But the listener runs into them, so relative to the listener they close at v + vₒ, and the number met per second is f′ = (v + vₒ)/λ = f (v + vₒ)/v. Drive at 30.0 m s⁻¹ toward the same 500 Hz siren and you hear 544 Hz, not the 548 Hz of the moving-source case. The difference is real. Air carries the fronts, so it picks out a frame, and a moving source and a moving observer are different experiments in that frame. Write β = 30.0/343 = 0.0875: the two factors are 1/(1 − β) and 1 + β, which agree to first order in β and differ by about β² = 0.8%, the 4 Hz seen here.

Both effects act at once and independently, one on the spacing and one on the closing speed: f′ = f (v + vₒ)/(v − vₛ). The sign rule that never fails: draw the line from source to listener, and count a velocity as positive when it points along that line toward the other party. Then test the result. First, approach in either role must raise the pitch and retreat must lower it. Second, let source and listener share one velocity u, the listener fleeing (vₒ = −u) while the source chases (vₛ = +u): the formula gives f (v − u)/(v − u) = f, as it must when the gap between them never changes. Third, push to the limits: vₒ = −v gives f′ = 0, because no front ever overtakes the listener, and vₛ → v sends f′ to infinity as the spacing ahead closes to zero. That divergence marks the end of the model, not an infinite pitch.

Three conditions sit under the formula. Velocities are relative to the air: in a steady wind, subtract the wind velocity from both source and listener first. A siren and a listener both fixed to the ground hear no shift in any steady wind, because in the air's frame they share one velocity. Only the component along the line of sight counts, vₛ cos θ for a source whose velocity makes angle θ with the line to the listener, so a car passing at a distance slides smoothly from high pitch to low, and the sound it emits when abreast arrives unshifted. And a moving reflector is handled in two steps: it receives as an observer, then re-emits as a source. A wall approaching at u returns f (v + u)/(v − u). Mix that echo with the outgoing tone and the beat frequency, about 2fu/v, measures u: 40.0 kHz ultrasound off a hand closing at 0.50 m s⁻¹ beats at 117 Hz.