C.5.2 Quantitative description of frequency shifts (HL only)

The Doppler Effect for Sound and Frequency Shifts

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.

Deriving the Formula

1. Wavefront Compression: If the source moves towards a stationary observer with speed $u_s$, the wavefronts are compressed.
2. New Wavelength: The distance between wavefronts becomes ${\lambda }^{\prime }=\frac{v-u_s}{f}$, where $v$ is the speed of sound.
3. Observed Frequency: The observer measures a frequency $f^{\prime }=\frac{v}{{\lambda }^{\prime }}=f\frac{v}{v-u_s}$.

Moving Observer

When the observer moves, the speed of the wave relative to the observer changes.

Deriving the Formula

1. Relative Speed: If the observer moves towards the source with speed $u_o$, the wave speed relative to the observer becomes $v+u_o$.
2. Observed Frequency: The frequency measured by the observer is $f^{\prime }=f\frac{v+u_o}{v}$.

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.

Self-Assessment