The Doppler Effect

What Changes in the Doppler Effect?

The source frequency stays the same, but the observed frequency changes because relative motion alters how the wavefronts arrive at the observer.

If the source moves towards the observer, each successive wavefront is emitted from a position slightly closer to the observer than the one before. This compresses the wavefronts in front of the source, shortening the effective wavelength. Because the wave speed in the medium is unchanged, a shorter wavelength means a higher observed frequency. If the source moves away, the wavefronts are more spread out, the effective wavelength increases, and the observed frequency drops.

A moving observer experiences a similar shift for a different reason: the observer changes the rate at which they encounter wavefronts. Moving towards the source means passing through wavefronts faster, raising the observed frequency. Moving away means encountering them more slowly, lowering the frequency. In both cases, the wave speed in the medium depends only on the properties of the medium — it does not change because of the motion.

Approaching vs Receding

The direction of relative motion decides whether the observed frequency rises or falls.

Imagine a source moving through still air. In front of it, the wave crests are squeezed together like the coils of a compressed spring. The observer in front receives more crests per second, so $f' > f$. Behind the source, the wave crests are stretched out like a relaxed spring. The observer behind receives fewer crests per second, so $f' < f$. The faster the source moves, the more dramatic the compression and stretching, and the larger the frequency shift.

The exact moment the source passes the observer is the transition point: the observed frequency drops sharply from above $f$ to below $f$. This is the classic drop in pitch you hear when an ambulance passes by on the street.

Two Common Doppler Cases

The moving-source and moving-observer situations use related but different equations because the physics of the shift is different in each case.

For a moving source, the source motion changes the physical spacing of the wavefronts in the medium. The new wavelength is $\lambda' = (v \pm v_s)/f$, so the observed frequency becomes $f' = f \times v/(v \pm v_s)$. For a moving observer, the wavelength in the medium is unchanged, but the observer’s motion changes the relative speed of the wavefronts. The observer meets wavefronts at speed $v \pm v_o$, giving $f' = f \times (v \pm v_o)/v$.

The sign conventions are designed so that approaching motion always produces a higher observed frequency. For the moving-source equation, approaching means using the minus sign in the denominator (making it smaller, so $f'$ increases). For the moving-observer equation, approaching means using the plus sign in the numerator (making it larger, so $f'$ increases). If your chosen sign makes $f'$ decrease for approaching motion, you have picked the wrong sign.

Technology and Science Applications

Doppler shifts are useful because motion becomes measurable through frequency change, enabling technologies that range from everyday road safety to cutting-edge astronomy.

In medical ultrasound, high-frequency sound waves are reflected off moving blood cells. The frequency shift of the reflected waves reveals blood velocity, which cardiologists use to diagnose heart-valve problems or arterial blockages. In police speed detection, a radar gun emits microwaves and measures the shift of the wave reflected from a moving vehicle. Because the wave reflects off the moving car, the total shift is roughly double the single-shift value for the same speed. In weather radar, meteorologists measure the Doppler shift of raindrops to determine wind speed and direction inside storms.

In astronomy, the light from distant galaxies is red-shifted — its wavelength is stretched toward the red end of the spectrum because the galaxies are receding from us. The greater the red-shift, the faster the recession. This relationship, known as Hubble’s law, provided the first strong evidence that the universe is expanding.

Combined Motion and the General Equation

When both the source and the observer move, the two effects combine into a single equation that captures both shifts at once.

The general Doppler equation for sound in a medium is:

$f' = f \times \dfrac{v \pm v_o}{v \mp v_s}$

In the numerator, use plus when the observer moves towards the source and minus when moving away. In the denominator, use minus when the source moves towards the observer and plus when moving away. Notice that the signs in the numerator and denominator are independent — they do not have to be the same. The NSW HSC typically tests only one moving object at a time, but understanding the combined case deepens your intuition for the sign conventions.

When the Doppler Equation Breaks Down

The standard Doppler equations assume the source speed is less than the wave speed in the medium.

If a source moves faster than the wave speed — for example, a supersonic aircraft moving faster than the speed of sound — the wavefronts pile up on top of each other and form a shock wave, known as a sonic boom. In this regime, the simple Doppler formula no longer applies because the wavefronts cannot outrun the source. Similarly, if the observer moves faster than the wave speed, the observer outruns the wavefronts and the observed frequency becomes undefined in the simple model.

For all HSC problems, you can safely assume that both source and observer speeds are much smaller than the wave speed, so the standard equations are valid. This assumption is built into the derivation of the equations and is worth stating when explaining your reasoning.