Doppler Effect for Light & Sound Reference Cheat Sheet

Key Facts

• For any wave, speed, frequency, and wavelength are related by $v = f\lambda$.
• For sound in still air, the Doppler formula is $f' = f\frac{v + v_o}{v - v_s}$ when the observer moves toward the source and the source moves toward the observer.
• If a sound source and observer move closer together, the observed frequency increases, so $f' > f$.
• If a sound source and observer move farther apart, the observed frequency decreases, so $f' < f$.
• For light at low relative speeds, the approximate Doppler shift is $\frac{\Delta \lambda}{\lambda} \approx \frac{v_r}{c}$, where $v_r$ is positive for recession.
• Redshift is defined by $z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$.
• A positive redshift $z > 0$ means the source is moving away, while a negative redshift $z < 0$ means the source is moving toward the observer.
• For relativistic light Doppler shift along the line of sight, $\frac{f_{\text{obs}}}{f_{\text{emit}}} = \sqrt{\frac{1 - \beta}{1 + \beta}}$ for a receding source, where $\beta = \frac{v}{c}$.

Vocabulary

Doppler effect

The Doppler effect is the change in observed frequency or wavelength caused by relative motion between a wave source and an observer.

Observed frequency

Observed frequency is the frequency measured by an observer, often written as $f'$ or $f_{\text{obs}}$.

Redshift

Redshift is an increase in observed wavelength, described by $z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}$.

Blueshift

Blueshift is a decrease in observed wavelength that occurs when a light source moves toward an observer.

Wave speed

Wave speed is the speed at which a wave travels through a medium or through space, related by $v = f\lambda$.

Relative velocity

Relative velocity is the velocity of one object as measured from another object's frame of reference.