9.4 Observer, source, and medium — three motions

The Doppler formula in lesson 9.1 was derived for the case of a stationary medium. In reality, all three “actors” can move: the source, the observer, and the air itself. The full formula accounting for all three is straightforward, but the bookkeeping is worth being explicit about because the result is not symmetric.

The general formula

Let $\mathbf{c}_s$, $\mathbf{v}_o$, $\mathbf{v}_s$, $\mathbf{U}$ be unit vectors and velocity magnitudes for:

• $\hat{\mathbf{n}}$: the unit vector from source to observer.
• $v_o$: component of observer velocity along $\hat{\mathbf{n}}$, positive if moving toward the source.
• $v_s$: component of source velocity along $\hat{\mathbf{n}}$, positive if moving toward the observer.
• $U$: component of medium velocity along $\hat{\mathbf{n}}$, positive if moving toward the observer.

Then the observed frequency is

If we set $U = 0$ this is the standard textbook formula. If we set $U = -v_s = v_o$ (source moves with the air, observer moves with the air, all in the same direction) we get $f_o = f_s$ — no shift, as expected: the source and observer have no relative motion with respect to the medium.

Why the asymmetry

The “medium” plays the role of a preferred frame in acoustics, the way the ether was (incorrectly) supposed to play in 19th-century optics. The wave equation is Galilean covariant — its form is the same in any frame moving uniformly with respect to the medium — but it has a preferred velocity, the sound speed $c$, defined relative to the medium.

This is fundamentally different from electromagnetism, where the wave equation is Lorentz invariant and there is no preferred medium. Light’s Doppler shift depends only on the relative velocity of source and observer; sound’s Doppler shift depends on all three.

A practical example: wind

A musician plays a tuning fork at $f_s = 440$ Hz, standing still in still air. An observer 100 m away, also stationary, hears 440 Hz. No shift.

Now suppose a 30 m/s wind blows from source to observer. The medium velocity component toward the observer is $U = +30$ m/s. The formula gives

No shift! The wind carries the sound faster, but neither stretches nor compresses the wavelengths. (The wavelengths in the air are the same — the wind moves the fluid carrying the waves but doesn’t change the rate of wavecrests passing any fixed point.)

But the time of arrival is shifter: the sound takes $100/(343 + 30) = 268$ ms with the wind, versus $292$ ms without. Different from a Doppler shift but real.

If instead the source moves at 30 m/s toward the stationary observer (in still air), the formula gives

— a 41-Hz shift, easily heard.

When everything moves

A car (source) driving 30 m/s east through 10 m/s eastward wind, with an observer 100 m east of the car (stationary on the ground). Then

• $v_s = +30$ (source approaching observer along $\hat{\mathbf{n}}$, which points east).
• $v_o = 0$.
• $U = +10$ (wind toward observer).

A 9.3% upward shift. Without the wind, the same source velocity would give $343/(343 - 30) = 1.096$ — 3% larger shift. The tailwind from source to observer reduces the Doppler shift slightly, because the wave fronts are now travelling faster (in the lab) and the source is less able to compress them.

In medical ultrasound

Doppler ultrasound for blood flow measurement uses essentially this formula. A transducer (source and observer) emits ultrasound; it reflects from moving red blood cells (which serve as moving secondary sources); the reflected wave is detected back at the transducer with a Doppler-shifted frequency. The shift is proportional to the velocity component of the blood along the ultrasound beam direction, providing a non-invasive map of flow velocities in arteries and veins.

Looking ahead

Doppler is a kinematic phenomenon — the wave equation itself doesn’t change shape, just gets applied in different frames. The next lesson considers a more dynamical case: sound generated by a source moving through, or embedded in, a flow. This is where aerodynamic sound generation (jet noise, Aeolian tones, propeller noise) comes in.