Doppler Effect in Sound

What the Doppler effect is

On a railway platform the pitch of a whistle is higher as the engine approaches and lower as it moves away; the horn of a bus climbing a hill changes note continuously. The Doppler effect is the apparent change of frequency observed because of relative motion between the observer and the source. The frequency the source actually emits does not change — only the frequency that reaches the listener does.

A point that students lose marks on at the outset: a moving source does not change the speed of sound. The speed $v$ is a property of the medium alone; the wave forgets its source the moment it leaves. What motion alters is either the wavelength packed into the medium (when the source moves) or the rate at which crests are intercepted (when the observer moves). Keeping these two mechanisms separate is the whole secret of the topic.

A moving source crowds the waves

Suppose the source, the observer, and the sound all travel from left to right, and consider first only the source moving with speed $v_s$. A particular note that leaves the source arrives one second later at a point A a distance $v$ away, since $SA = v$. In that same second the source has advanced a distance $v_s$. All the $n$ waves emitted in that second are therefore squeezed into a length $v - v_s$ instead of $v$. The wavelength in the medium shrinks to

$$\lambda' = \frac{v - v_s}{n}$$

A shorter wavelength carried at the same speed $v$ means a higher frequency reaches a stationary observer ahead of the source. Behind the source the length available is $v + v_s$, the wavelength stretches, and the pitch falls. This is the mechanism shown in Figure 1: the source motion physically reshapes the wavelength of the wave in the medium.

A moving observer sweeps the waves

Now hold the source still and let the observer move with speed $v_o$. A crest reaching the observer's position would, one second later, lie at a point B with $OB = v$. But in that second the observer has moved from O to O′, so only the crests contained in the shortened gap O′B actually cross him. The number of waves passing the observer per second becomes

$$n' = \frac{v - v_o}{\lambda'}$$

Here the wavelength $\lambda'$ in the medium is untouched — the observer does not compress anything. He simply runs into, or away from, crests faster. Substituting the moving-source wavelength gives the combined result derived next.

The general formula and sign convention

Combining the two mechanisms — wavelength set by the moving source, interception rate set by the moving observer — gives the apparent frequency $f'$ when both move along the source-to-observer line:

$$f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$$

where $f$ is the true emitted frequency, $v$ is the speed of sound in the medium, $v_o$ the observer's speed and $v_s$ the source's speed. The signs are not chosen arbitrarily. The convention used here, following the derivation, is to take the positive direction as the direction running from the source to the observer; the speed of sound $v$ is positive along this direction, and $v_o$ and $v_s$ are positive when their velocities point the same way as $v$ and negative otherwise.

In practice it is faster, and far safer, to fix the upper/lower signs by physical reasoning: approach must raise the frequency, recession must lower it. So when the observer moves toward the source use $v + v_o$ (numerator grows); when the source moves toward the observer use $v - v_s$ (denominator shrinks). Both effects then enlarge $f'$, exactly as an approach should.

The four cases — master table

Each of the four single-motion situations follows from the general formula by switching off whichever speed is zero. Reading off the sign of each term and the direction of the pitch shift is the single most efficient way to revise this topic.

Why source motion ≠ observer motion

A defining feature of the sound Doppler effect is that a moving source and a moving observer, even at identical speeds, do not produce identical frequency shifts. The reason is structural: the source speed sits in the denominator, the observer speed in the numerator. These are different operations on $f$, so $\dfrac{v}{v - u} \neq \dfrac{v + u}{v}$ for any non-zero $u$.

Physically, a moving source genuinely alters the wavelength laid down in the medium, because successive crests originate from advancing positions. A moving observer leaves the wavelength untouched and merely changes how quickly he meets the existing crests. Because the medium provides a fixed reference frame for sound, these two motions are not interchangeable.

Effect of wind and the medium

The formula assumes a still medium. When wind blows, it shifts the entire medium, so the effective speed of sound becomes $v + w$ downwind and $v - w$ upwind, where $w$ is the wind speed component along the line of propagation. This modified speed is then used in place of $v$ for both numerator and denominator.

A subtle but examinable point: steady wind by itself, with both source and observer stationary, produces no change in observed frequency. Wind alters the wavelength and the speed together, but the number of crests leaving the source per second equals the number arriving per second, so frequency is preserved. A pitch change requires genuine relative motion of source or observer through the medium.

Applications: siren, radar, SONAR

The everyday signature of the effect is the siren drop: an ambulance approaching sounds high-pitched, and the note falls audibly the moment it passes and begins to recede. Crucially, the change is a sudden step at the moment of passing, not a gradual slide, because the radial velocity reverses sign as the vehicle goes by.

The utility of Doppler's effect arises because it applies to light as well as sound. SONAR sends ultrasonic pulses through water and reads the Doppler shift of the echo to find the speed of submarines or shoals of fish. Radar does the same with electromagnetic waves to measure vehicle and aircraft speeds. In each case the target acts first as a moving observer receiving the wave, then as a moving source re-radiating it, so the echo carries a double shift from which the target's speed is extracted.