Doppler Effect

The Doppler effect is the change in the frequency and wavelength that an observer measures when the source of a wave and the observer move relative to one another. Motion that closes the gap squeezes successive wave crests together, so the observed frequency rises; motion that opens the gap stretches them apart, so the observed frequency falls.

You hear this every time an ambulance passes: the siren's pitch is higher while the vehicle approaches and drops noticeably the instant it goes by.

Sound and light behave differently in one important respect. Sound needs a material medium, so it matters physically whether the source or the observer is the one moving through the air. Light needs no medium and always travels at $c$, so only the relative velocity along the line of sight matters.

For electromagnetic waves at IB level you use the approximation, valid whenever $v \ll c$:

$$\frac{\Delta f}{f} = \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}$$

• Be able to state qualitatively how the observed frequency and wavelength change for approach and for recession.
• Be able to apply the fractional-shift approximation to light and other electromagnetic waves.

A wavefront diagram shows circles of constant phase spreading outwards at the wave speed from the point where each crest was emitted. For a stationary source the circles are concentric.

For a moving source each successive circle is centred a little further along the path, so the pattern becomes eccentric: wavefronts bunch together ahead of the source (shorter wavelength, higher observed frequency) and spread out behind it (longer wavelength, lower observed frequency).

Think of the ripples around a swimming duck, crowded in front of it and stretched out behind.

When only the observer moves, the diagram looks completely different: the wavefronts remain concentric because the source and the medium are undisturbed, and the observer simply crosses those unchanging circles at a faster or slower rate.

In an exam you should be able to:

• sketch at least three correctly spaced circular wavefronts and mark the source's velocity with an arrow;
• identify from a given diagram whether the source is moving and in which direction;
• explain the frequency heard at any labelled observer position using the wavefront spacing.

Every element absorbs and emits light at fixed, well-known wavelengths, so the spectrum of a star or galaxy carries a built-in ruler. If a line that sits at $\lambda_0 = 656.3\ \text{nm}$ in the laboratory appears at a longer wavelength, the object is receding (redshift); a shorter observed wavelength means it is approaching (blueshift).

The radial speed follows from:

$$v \approx c\,\frac{\Delta \lambda}{\lambda_0}$$

Astronomers use this to:

• map the rotation of galaxies
• detect the tiny wobble an orbiting planet induces in its star
• measure the orbital speeds of binary star systems

While local movements of stars within our galaxy produce ordinary kinematic Doppler shifts caused by motion through space, the redshift observed in distant galaxies has a different primary origin.

Cosmological redshift is produced by the expansion of space itself: as space expands, the wavelengths of photons travelling through the universe are stretched along with it. This cosmological stretching is physically distinct from a classical Doppler shift, although both are evaluated using the same shift parameter $z = \frac{\Delta \lambda}{\lambda_0}$ for non-relativistic recession speeds.

You must be able to:

• read a rest wavelength and an observed wavelength from data or a spectrum
• decide the direction of motion from the sign of the shift
• calculate the radial speed

At HL you must calculate observed sound frequencies, and the physical difference between a moving source and a moving observer is the key to choosing the right equation.

When a sound source moves through the air, it continues to emit wave crests that expand spherically about the point of emission. Because the source advances during each wave period $T$, every new sphere starts further along the path. This mechanically compresses the physical wavelength in the medium ahead of the source to $\lambda' = \lambda - u_s T$. For approach this gives:

$$f' = f\left(\frac{v}{v - u_s}\right)$$

and $f' = f\left(\frac{v}{v + u_s}\right)$ for recession.

When instead an observer moves through a stationary wave field, the wavelength in the medium is unaffected. The frequency shift arises entirely because the wavefronts sweep past the observer at the relative speed $v_{\text{rel}} = v \pm u_o$, so the observed frequency is:

$$f' = \frac{v_{\text{rel}}}{\lambda} = f\left(\frac{v \pm u_o}{v}\right)$$

The plus sign applies for approach and the minus sign for recession.

A quick sanity check beats memorising signs: approach must always raise the frequency and recession must always lower it.

For example, a $900\ \text{Hz}$ siren approaching at $28\ \text{m s}^{-1}$ is heard above $980\ \text{Hz}$, but only at about $830\ \text{Hz}$ once it has passed.