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.
The 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.
Deriving the Formula
- Relative Speed: If the observer moves towards the source with speed $u_o$, the wave speed relative to the observer becomes $v + u_o$.
- Observed Frequency: The frequency measured by the observer is $f' = f \frac{v + u_o}{v}$.
An observer runs towards a stationary siren emitting sound at 500 Hz with a speed of $10 \text{ m s}^{-1}$. The speed of sound is $340 \text{ m s}^{-1}$.
What frequency does the observer hear?
Solution
- Using the formula:
$$f' = f \frac{v + u_o}{v}$$
$$ = 500 \frac{340 + 10}{340}$$
$$ = 500 \frac{350}{340} \approx 514.7 \text{ Hz}$$
- The observer hears a frequency of approximately 515 Hz.
Why Does the Frequency Change?
The key to understanding the Doppler effect lies in the relative motion between the source and the observer.
Wavefront Diagrams
- Stationary Source and Observer: Wavefronts are evenly spaced, and the observer measures the emitted frequency.
- Moving Source: Wavefronts are compressed in front of the source and spread out behind it.
- Observer in front measures a higher frequency (shorter wavelength).
- Observer behind measures a lower frequency (longer wavelength).
- Moving Observer: The observer encounters wavefronts more frequently when moving towards the source and less frequently when moving away.
- How does the observed frequency change when a sound source moves towards a stationary observer?
- What is the difference between the Doppler effect for sound and light waves?
- How can the Doppler effect be used to determine the speed of a galaxy moving away from Earth?
