The Doppler effect

Frequency changes from motion

The Doppler effect is the change in observed frequency when a source and observer move relative to each other. A familiar example is the changing pitch of a siren as an ambulance passes. The pitch is higher as it approaches and lower as it moves away.

The sound source emits waves at frequency $f_s$. The observer detects frequency $f_o$. Motion changes the spacing or encounter rate of wavefronts.

Stationary source and moving observer

If the source is stationary and the observer moves toward it, the observer meets wavefronts more frequently. The observed frequency increases.

For sound speed $v$, observer speed $v_o$, and source frequency $f_s$,

ight)$$ when the observer moves toward the source. If the observer moves away, $$f_o=f_sleft(rac{v-v_o}{v} ight)$$ A compact sign convention is often written $$f_o=f_sleft(rac{vpm v_o}{v} ight)$$ where the plus sign is for moving toward the source. ## Moving source and stationary observer If the source moves toward a stationary observer, the wavefronts are emitted closer together in front of the source, reducing wavelength. Since sound speed in the medium remains $v$, the observed frequency increases. For a source moving toward the observer, $$f_o=f_sleft(rac{v}{v-v_s} ight)$$ For a source moving away, $$f_o=f_sleft(rac{v}{v+v_s} ight)$$ The moving source changes wavelength; the moving observer changes wavefront encounter rate. ## General Doppler formula for sound A common one-dimensional formula is $$f_o=f_sleft(rac{vpm v_o}{vmp v_s} ight)$$ The signs are chosen so that motion toward each other increases frequency and motion away decreases frequency. Careful problem solving is more reliable than memorizing signs. Ask whether the observed frequency should increase or decrease, then choose signs accordingly. ## Source speed and shock waves If a source moves at or above the speed of sound, wavefronts pile up. At the speed of sound, this produces a strong pressure buildup. Above the speed of sound, a shock wave forms, creating a sonic boom. The Mach number is $$M=rac{v_s}{v}$$ where $v_s$ is source speed and $v$ is sound speed. Supersonic motion has $M>1$. ## Doppler effect for light Light also exhibits a Doppler effect, but because light does not require a medium and because relativity matters, the formula differs. For low speeds, the fractional shift is approximately $$rac{Delta f}{f}approx rac{v_{rel}}{c}$$ with sign depending on approach or recession. Astronomers use redshift and blueshift to measure motion of stars, galaxies, and gas. ## Applications Doppler radar measures vehicle speed and storm motion. Medical Doppler ultrasound measures blood flow. Astronomers detect exoplanets by observing periodic Doppler shifts in starlight. Bats use frequency shifts in echoes during navigation and hunting. The Doppler effect turns wave frequency into a motion measurement tool. ## The big idea The Doppler effect occurs because relative motion changes the rate at which wavefronts are received. For sound, moving observers and moving sources affect the observed frequency in different ways because sound travels through a medium. The effect explains changing siren pitch, sonic booms, radar, ultrasound, and astronomical velocity measurements.