Doppler — Both Moving

When BOTH source and observer move, use the COMBINED Doppler formula.

f' = f·(v ± v_o)/(v ∓ v_s). Top sign: observer approaches source. Bottom sign: source approaches observer.

Approach geometry: + in top, − in bottom ⇒ shift UP.

Receding geometry: − in top, + in bottom ⇒ shift DOWN.

Sign convention is the trickiest part — drawing a diagram helps.

Special case: if source and observer move at same velocity, f' = f (no relative motion).

Applications: aircraft tracking radar (both moving), astronomy (both rotating), military sonar (both ships moving).

Relativistic effects matter when speeds approach c — modified formulas for light.