The Doppler Effect for Sound and Frequency Shifts
Doppler effect
The Doppler effect is the change in the observed frequency of a wave when there is relative motion between the source and the observer
This phenomenon is crucial for understanding sound and light waves, with applications ranging from radar technology to astronomy.
The Doppler Effect for Sound Waves
Moving Source
When a sound source moves, the wavefronts it emits become compressed in the direction of motion and spread out behind it.
Common MistakeThe speed of the wave is determined by the medium (e.g., air) and not by the motion of the source or observer.
Deriving the Formula
- Wavefront Compression: If the source moves towards a stationary observer with speed $u_s$, the wavefronts are compressed.
- New Wavelength: The distance between wavefronts becomes $\lambda' = \frac{v - u_s}{f}$, where $v$ is the speed of sound.
- Observed Frequency: The observer measures a frequency $f' = \frac{v}{\lambda'} = f \frac{v}{v - u_s}$.
A car with a siren emitting sound at 500 Hz moves towards a stationary observer at $20 \text{ m s}^{-1}$. The speed of sound is $340 \text{ m s}^{-1}$.
What frequency does the observer hear?
Solution
- Using the formula:
$$f' = f \frac{v}{v - u_s} $$
$$= 500 \frac{340}{340 - 20}$$
$$ = 500 \frac{340}{320} \approx 531.25 \text{ Hz}$$
- The observer hears a frequency of approximately 531 Hz.
Moving Observer
When the observer moves, the speed of the wave relative to the observer changes.
