Doppler effect — all cases - source moving, observer moving, both: Visual walkthrough

We only ever need two ideas, and we will earn both:

Wavelength — the distance between two neighbouring crests.
Crest-arrival speed — how fast those crests sweep into your ear.

Frequency heard $=$ (arrival speed) $\div$ (wavelength you meet). Everything else is bookkeeping.

Step 1 — What a "wavefront" even is

WHAT. A source of sound (a horn, a bird, a struck bell) makes the air pressure rise and fall over and over. Each time the pressure peaks, that peak spreads outward as an expanding sphere — a crest. Draw it as a circle. One new circle is born every period $T$ .

WHY start here. Before any symbol appears we must picture the thing being counted. The pitch you hear is literally how many circles hit your ear each second — that is the true frequency $f$ , and $T = 1/ f$ is the wait between births.

PICTURE. Below, a still source (magenta dot) has released circles at times $0, T, 2 T, 3 T$ . Because the source hasn't moved, every circle shares the same centre — they are evenly spaced like ripples in a still pond.

Figure — Doppler effect — all cases - source moving, observer moving, both

The spacing between neighbouring circles is the wavelength $λ$ . In one period $T$ a crest flies a distance $v T$ , so:

$gap between crests λ = crest speed v \times time between births T = \frac{v}{f}$

This is exactly Wavelength and frequency relation. Nobody is moving yet, so this $λ$ is our baseline.

Step 2 — The counting rule (the heart of everything)

WHAT. Stand your ear at a fixed point. Count how many crests pass per second.

WHY. That count is the frequency you perceive. It equals how fast crests come at you, divided by how far apart they are:

$f^{'} = \frac{speed crests sweep past your ear}{distance between crests you meet}$

PICTURE. A ruler laid along the line from source to ear. If crests are $λ$ apart and rush past at speed $u$ , then in one second a length $u$ of "crest-train" goes by, containing $u / λ$ crests.

Every case below just plugs the right $u$ and the right $λ_{met}$ into this one rule.

Step 3 — Observer moves, source still

WHAT. The source sits still, so the circles stay evenly spaced: $λ_{met} = λ = v / f$ (unchanged). Now the observer walks toward the source at speed $v_{o}$ .

WHY the wavelength does NOT change. The source isn't chasing anything — the picture from Step 1 is frozen in the air. Only the observer's motion matters, and it changes how fast crests reach the ear, not their spacing.

PICTURE. Crests still travel outward at $v$ . But the ear runs into them at its own speed $v_{o}$ . So the closing speed — see Relative velocity — is $v + v_{o}$ .

$f^{'} = \frac{u}{λ _{met}} = \frac{v + v _{o} crests come at you faster}{same old spacing v / f} = f \frac{v + v _{o}}{v}$

Term by term: $v$ is the crest's own speed; $+ v_{o}$ is your extra speed into them; the bottom $v / f$ is the untouched wavelength. Bigger top $\Rightarrow$ higher pitch.

Away case. If you walk away, you flee the crests, closing speed is $v - v_{o}$ : $f^{'} = f \frac{v - v _{o}}{v} (lower pitch) .$

Degenerate check. If $v_{o} = 0$ you get $f^{'} = f$ — no motion, no shift. Good. If you run away at $v_{o} = v$ (impossibly fast for sound, but algebraically), $f^{'} = 0$ : you keep pace with a crest and never meet the next one.

Step 4 — Source moves, observer still

WHAT. Now YOU stand still and the source drives toward you at $v_{s}$ . Here the crest speed into your ear is still just $v$ (the medium sets that, not the source) — but the spacing shrinks.

WHY the spacing shrinks. Between making one crest and the next, the source itself moves forward by $v_{s} T$ . So the next crest is born closer to the previous one. Each circle's centre is nudged forward. The circles bunch up in front, spread out behind.

PICTURE. Watch the offset centres. In front of the moving source the crest gap is squeezed; behind it, stretched. This is the classic "ambulance" picture.

$λ^{'} = v / f λ - source’s forward step v_{s} T = \frac{v}{f} - \frac{v _{s}}{f} = \frac{v - v _{s}}{f}$

Term by term: $λ = v / f$ is what the spacing would be; $v_{s} T = v_{s} / f$ is how far the source crept between births; subtracting gives the squeezed front gap. Now feed it to the counting rule with crest speed $u = v$ :

$f^{'} = \frac{u}{λ ^{'}} = \frac{v}{( v - v _{s} ) / f} = f \frac{v}{v - v _{s}}$

Term by term: top $v$ = crests still fly at $v$ ; bottom $v - v_{s}$ = the compressed spacing. Smaller bottom $\Rightarrow$ higher pitch.

Receding case. Source flees $\Rightarrow$ crests stretch, $λ^{'} = (v + v_{s}) / f$ : $f^{'} = f \frac{v}{v + v _{s}} (lower pitch — the tone drops as it passes) .$

Step 5 — Why the two effects live on different lines

WHAT. Compare Steps 3 and 4. Observer motion changed the top (arrival speed). Source motion changed the bottom (wavelength). They are not the same lever.

WHY it matters. People expect "toward at speed $x$ " to give one answer no matter who moves. It doesn't, because the two are physically different mechanisms — one is additive on speed, one is a squeeze on distance.

PICTURE. A split panel: left, the ear speeding into fixed circles (top-line effect); right, the source dragging circles closer together (bottom-line effect).

Step 6 — Both move: stacking the two levers

WHAT. Let the observer move (top) and the source move (bottom) at once. Just combine the two independent changes.

WHY you can just multiply/stack. The observer changed only $u$ ; the source changed only $λ_{met}$ . They touch different parts of the counting rule $f^{'} = u / λ_{met}$ , so they don't interfere — plug both in.

PICTURE. Source and ear on one line, both moving toward each other: front-crests squeezed to $(v - v_{s}) / f$ , and the ear rushing into them at $v + v_{o}$ .

$f^{'} = \frac{u}{λ _{met}} = \frac{v + v _{o}}{( v - v _{s} ) / f} = f \frac{v \pm v _{o}}{v \mp v _{s}}$

Every combination, no gaps:

Observer	Source	Top sign	Bottom sign	Pitch
toward	toward	$+ v_{o}$	$- v_{s}$	highest ↑↑
toward	away	$+ v_{o}$	$+ v_{s}$	mixed
away	toward	$- v_{o}$	$- v_{s}$	mixed
away	away	$- v_{o}$	$+ v_{s}$	lowest ↓↓

Step 7 — The wall the formula hits: $v_{s} \to v$

WHAT. Let the source speed up until $v_{s} = v$ . The denominator $v - v_{s} \to 0$ , so $f^{'} \to \infty$ .

WHY that's not a real infinity. All the front crests are born at the same forward-racing point — they pile onto a single surface. The formula assumed a tidy line of separate crests; that assumption dies here.

PICTURE. The offset circles from Step 4, pushed until they all touch along a cone. That stacked wall is the shock front.

Worked examples (each traced to a step)

The one-picture summary

One line, two knobs: the observer knob tweaks the top (arrival speed $v \pm v_{o}$ ); the source knob tweaks the bottom (met wavelength $\to v \mp v_{s}$ ). Turn both toward-each-other and both push pitch up.

Recall Feynman retelling — explain it to a 12-year-old

A drummer bangs a drum every second; each bang is a ring spreading through the air. If you run at the drummer, you smash into rings faster — you hear bangs more often (top of the fraction goes up). If instead the drummer walks at you while banging, each new ring starts closer than the last, so the rings bunch up and you catch them more often (bottom of the fraction shrinks). Running away, or the drummer walking away, spreads the rings out and slows the bangs. That's it: closer-in-any-way = faster bangs = higher pitch. If the drummer ever runs as fast as the rings themselves, all the rings stack into one wall — that's the sonic boom, and the neat little formula gives up.

For the medium-less, fully symmetric relativistic version, see Doppler effect of light.