Doppler geometry and reflected shifts

The head-on formula is a special case of one question: how fast is the propagation path lengthening or shortening? For a source moving through a still medium, with r the source-to-observer distance, f_obs = f_src / (1 + (1/v) dr/dt), and for a source moving at speed u at angle θ to the line of sight, dr/dt = −u cos θ. The rest of the velocity is perpendicular to the path and changes r only at second order. The same projection governs a moving observer, but there the component enters the numerator as (v + u_O cos θ_O) rather than the denominator, so dr/dt on its own never fixes the answer in a medium: the two motions are not interchangeable. A patrol car at 32 m s⁻¹ passing a pedestrian 25 m from the road emits, while still 60 m short of closest approach, along a line of sight of √(60² + 25²) = 65 m. Then cos θ = 60/65 = 0.9231 and the closing speed is 29.54 m s⁻¹, so a 1500 Hz siren arrives at 1500 × 343/(343 − 29.54) = 1641 Hz. The head-on limit is 1654 Hz and the far-behind limit 1372 Hz, and 1641 Hz correctly sits inside the approaching end.

A wave carries information about where the source was, not where it is. The angle in u cos θ is therefore the angle at the moment of emission, and the shift it produces arrives r/v later. For the patrol car above, the sound emitted exactly at closest approach carries no shift at all, since cos θ = 0 there, but it reaches the pedestrian 25/343 = 0.073 s after the car has gone past, by which time the car is 2.3 m further down the road. Use the angle you can see at the instant you hear the note and you place the unshifted moment too early. The pitch drop occupies emission times of order d/u = 25/32 = 0.78 s either side of closest approach, which is why a fast car on a near road drops pitch abruptly while a distant aircraft glides down over several seconds.

Air that is itself moving is handled by changing frame, not by inventing a new formula. In the air's own frame the medium result holds unchanged; a wind of component w along the source-to-observer path simply adds w to the wave speed measured over the ground, giving f_obs = f_src (v + w + u_O)/(v + w − u_S) with u_O and u_S now ground-frame speeds. Two consequences follow. If source and listener are both at rest over the ground, the same v + w appears above and below and cancels exactly: a steady wind produces no shift whatever, however strong. And when something is moving, the wind changes the size of the shift because it changes the denominator. A 1000 Hz siren approaching at 30 m s⁻¹ gives 1096 Hz in still air, 1092 Hz with a 12 m s⁻¹ tailwind, and 1100 Hz into a 12 m s⁻¹ headwind. What a wind does change with nothing moving is the wavelength over the ground, λ = (v + w)/f, because the crests are being carried along.

A target that scatters a wave back plays two roles in turn. Approaching at speed u along the beam, it first acts as a moving observer and meets crests at f₁ = f₀ (v + u)/v. It then re-radiates that very frequency while acting as a moving source, so the transducer receives f₂ = f₁ v/(v − u) = f₀ (v + u)/(v − u). The two factors multiply, and subtracting f₀ leaves Δf = 2 f₀ u/(v − u), which for u ≪ v is 2 f₀ u/v, twice what a single shift would give. With the beam at angle θ to the target's velocity, u becomes u cos θ throughout. A 38 kHz echo sounder in seawater at 1500 m s⁻¹ watching a shoal rise at 0.85 m s⁻¹ toward the hull gets Δf = 2 × 38000 × 0.85/1500 = 43.07 Hz, while the exact expression, dividing by 1499.15 instead of 1500, gives 43.09 Hz. The approximation costs 0.06% here.

Turn the reflector result around and you have the working equation of a flowmeter, u = v Δf / (2 f₀ cos θ). Every quantity on the right is known precisely except θ, which the operator estimates from the image. Differentiating, the fractional error in u is tan θ times the angular error in radians: at θ = 20° one degree of misjudgement costs 0.6%, at 60° it costs 3.0%, and at 80° it costs 9.9%. That is why sonographers steer the beam to keep θ well below 70°, and why a vessel crossing the beam at right angles is invisible to Doppler however fast the blood inside it moves. Note also the scale of Δf: for 0.6 m s⁻¹ blood read at 60° with a 5.0 MHz probe it is about 1.9 kHz, four parts in ten thousand of the carrier. The instrument never measures f₀ and fᵣₑₜ separately; it mixes them and listens to the beat, which lands in the audible range.

Every formula above needed a medium with a rest frame. Light has none, and the search for one failed: no experiment has found an aether wind, so there is no w to add and no way to say which party is really moving. The relativistic one-way factor for recession is √((1 − β)/(1 + β)), a function of relative velocity alone. Expanded it is 1 − β + β²/2 − …, while the classical medium answers are 1 − β exactly for a receding observer and 1/(1 + β) = 1 − β + β² − … for a receding source. The relativistic factor is exactly the geometric mean of those two, which is how relativity declines to distinguish them while agreeing with both to first order in β — the correspondence check. Two predictions then differ. A source passing at closest approach, where cos θ = 0, is classically unshifted but relativistically redshifted by 1/γ = √(1 − β²), a transverse shift of order β²/2 that exists only because of time dilation. And the two-way radar ratio is exactly (1 + β)/(1 − β), the same algebra as the sonar result with v replaced by c, though reached by squaring the geometric mean rather than by any medium argument.