Doppler effect

The change in the **frequency (pitch) a listener hears** when the source (or listener) is **moving** — higher on approach, lower on recession.

**Higher** pitch — the wavefronts bunch up, shortening the wavelength and raising the frequency you hear.

**Lower** pitch — the wavefronts stretch out, lengthening the wavelength and lowering the frequency you hear.

**No** — the source always emits the same f. Only the **observed** frequency f' changes.

$f' = f\left(\dfrac{v}{v \pm v_{s}}\right)$ — **minus** approaching, **plus** receding. **Given** in the data booklet.

The **minus** sign — it makes the denominator smaller, so f' is **larger** (higher pitch).

The source moves forward between emitting each crest, so the **wavefronts ahead bunch together** → shorter wavelength → higher frequency.

**High and flat** (approaching) → a **sharp step down** (passing) → **low and flat** (receding). Not a smooth slope.

You hear it **above** its true pitch while it approaches, then a **sudden drop** to **below** its true pitch as it passes and recedes.

Drawing the heard pitch **sliding down smoothly**. It actually steps **down sharply** at the instant the source passes.

The change in the **wavelength (and frequency)** of light you receive when its **source moves toward or away** from you.

The observed wavelength is **stretched longer** (shifted toward red) because the source is **receding** (moving away).

The observed wavelength is **squashed shorter** (shifted toward blue) because the source is **approaching** (moving toward you).

$\dfrac{\Delta f}{f} = \dfrac{\Delta\lambda}{\lambda} \approx \dfrac{v}{c}$ — **given** in the data booklet (valid for v ≪ c).

Rearrange to **v = (Δλ ÷ λ) × c**, where Δλ = observed − laboratory wavelength and c = 3.0 × 10⁸ m s⁻¹.

The **change** in wavelength: observed wavelength − laboratory (true) wavelength — NOT the whole wavelength.

**Red = Receding** (away, longer λ); **Blue = approaching** (toward, shorter λ).

The **Universe is expanding**, so distant galaxies are **receding** from us — their light is shifted to longer (redder) wavelengths.

The **approaching edge is blueshifted** (shorter λ) and the **receding edge is redshifted** (longer λ) at the same time.

A **bigger** shift means a **faster** source — Δλ is proportional to v (Δλ/λ = v/c).

Only when the source speed **v is much smaller than c** (the speed of light).

Using the **whole observed wavelength** instead of the **change Δλ** (observed − lab) on the top of the fraction.

The **calculation** of the shift: the observed frequency f' (sound) or the fractional shift Δf/f (light), not just the qualitative 'higher/lower'.

$f' = f\,\dfrac{v}{v \mp u_s}$ — use **−** approaching, **+** receding.

The **minus** sign (smaller denominator) ⇒ bigger fraction ⇒ **higher** f'.

$\dfrac{\Delta f}{f} = \dfrac{\Delta\lambda}{\lambda} \approx \dfrac{v}{c}$.

The wavelength is **longer**, so the source is **receding** (moving away).

The wavelength is **shorter**, so the source is **approaching**.

Light needs **no medium**, so there is no medium speed v of sound; use Δλ/λ ≈ v/c instead.

Only when the source speed is **much less than c** ($v \ll c$); it is an approximation.

$f' = 512 \times \dfrac{340}{340-30} = 561$ Hz (higher).

$v = c \times 0.02 = 6 \times 10^{6}$ m s⁻¹ (about 0.02c).

Decide **physically**: approaching ⇒ higher f' (red/blue: blueshift = approaching). Then match the algebra to it.

Sound gives the **actual** f'; light gives the **fractional** shift Δλ/λ (multiply by λ or f for the change).