Doppler Effect & Beat Frequencies Cheat Sheet

Key Facts

• For sound, the Doppler effect formula is $f' = f\frac{v \pm v_o}{v \mp v_s}$, where $f'$ is observed frequency and $f$ is emitted frequency.
• Use a larger observed frequency when the source and observer move toward each other, so $f' > f$.
• Use a smaller observed frequency when the source and observer move away from each other, so $f' < f$.
• For a moving observer and stationary source, use $f' = f\frac{v \pm v_o}{v}$, with $+$ when the observer moves toward the source.
• For a moving source and stationary observer, use $f' = f\frac{v}{v \mp v_s}$, with $-$ in the denominator when the source moves toward the observer.
• Beat frequency is the absolute difference between two nearby frequencies: $f_{beat} = |f_1 - f_2|$.
• Constructive interference occurs when waves are in phase, and the resulting amplitude is larger than either individual wave.
• Destructive interference occurs when waves are out of phase, and the resulting amplitude is reduced or canceled.

Vocabulary

Doppler effect

The apparent change in observed frequency caused by relative motion between a wave source and an observer.

Observed frequency

The frequency detected by an observer, written as $f'$, which may differ from the emitted frequency.

Source speed

The speed of the object producing the sound wave, written as $v_s$ in Doppler effect formulas.

Observer speed

The speed of the listener or detector, written as $v_o$ in Doppler effect formulas.

Beat frequency

The rate at which loud and soft pulses are heard when two similar frequencies interfere, given by $f_{beat} = |f_1 - f_2|$.

Interference

The combining of two or more waves to produce a larger, smaller, or canceled resultant wave.