Study Notes on the Doppler Effect in Sound

MATRICS PHYSICAL SCIENCE: DOPPLER EFFECT

Introduction

• Developed by the Cape Town Science Centre in collaboration with the Western Cape Education Department (WCED).
• Sponsored by Astron Energy (Pty) Ltd.
• Emphasis on quality education for every child in every classroom and school in the province.

Understanding Sound

Sound Waves

• Physical Interpretation: Sound is understood as a longitudinal wave caused by movement (vibration) and requires a medium (air, water, etc.) to travel through.
• Longitudinal Wave: Characterized by compressions and rarefactions.
• Transverse Wave: Mathematically described where low pressure (rarefaction) is analogous to a trough, and high pressure (compression) is analogous to a crest.

Terminology

• Rarefaction: The region in a wave where the particles are spaced farther apart (low pressure).
• Compression: The region in a wave where the particles are pressed together (high pressure).
• Wavelength ($ ext{λ}$): The distance over which the wave's shape repeats; measured in meters.

Wave Characteristics

Parts of a Wave

1. Frequency (f): Measured in Hertz (Hz), indicating the number of cycles per second.
2. Period (T): Measured in seconds; the time taken for one complete cycle of the wave.
   • Relationship: $f = \frac{1}{T}$
3. Amplitude: Measured in meters; the maximum displacement of the wave from its equilibrium position, perceived as loudness.
   • Small amplitude = softer sound, large amplitude = louder sound.

Relationship Between Frequency, Wavelength, and Speed

• Speed of Sound Equation:
  $v = f imes ext{λ}$
• Wavelength ($ ext{λ}$) is the distance traveling during one period, illustrated graphically as the distance between two consecutive points in phase.

Understanding Doppler Effect

Definition

• The Doppler Effect describes an apparent change in observed frequency (pitch) as a result of relative motion between a sound source and an observer.
• For sound:
  • Moving Away from Observer: Observer hears a lower pitch.
  • Moving Towards Observer: Observer hears a higher pitch.
• For light:
  • Red Shift: Moving away from observer.
  • Blue Shift: Moving towards observer.

Calculating the Change in Frequency

Steps for Solving Doppler Problems

1. Determine what needs to be solved (frequency of sound heard by the listener, $ ext{f}_L$).
2. Identify the sound source and the listener.
3. Define the motion of the source and the listener (stationary or moving).
4. Use the formula and adjust the signs based on the relative motion of the source and listener.

Key Parameters and Equation

• $f_L$: Frequency of the sound heard by the listener (Hz).
• $f_S$: Frequency of the sound emitted by the sound source (Hz).
• $v$: Speed of sound in a specific medium (in air, $v = 340 ext{ m.s}^{-1}$).
• $v_L$: Speed of the listener (m.s-1).
• $v_S$: Speed of the sound source (m.s-1).

Formula

• $f<em>L = \frac{v ext{ ± } v</em>L}{v ext{ ± } v<em>S} f</em>S$
  • The sign depends on whether the source and listener are moving towards or away from each other.

Scenarios in the Doppler Effect

Stationary Source and Listener

• Equal wavefronts, no difference between the frequency emitted and what is heard by the listeners.
  • Parameters:
    • $v<em>L = 0$, $v</em>S = 0$
    • $f<em>L = \frac{v}{v} f</em>S$
    • Conclusion: $f<em>L = f</em>S$.

Stationary Listener and Moving Source

• Listener hears different frequencies based on source movement:
  • Listener A (away) hears a lower frequency; Listener B (toward) hears a higher frequency.
  • Wavefronts further apart for Listener A (less than 1), and closer for Listener B (more than 1).

Moving Listener and Stationary Source

• Listener A moves through more wavefronts per second than Listener B; hence, Listener A hears a higher pitch.
  • Listener A's frequency expression:
    • $f<em>{LA} = \frac{v - v</em>LA}{v} f_S$
  • Listener B's frequency expression:
    • $f<em>{LB} = \frac{v + v</em>LB}{v} f_S$.

Examples and Past Exam Questions

Worked Exam Question (Paper 1 October/November 2019, Q.6)

1. Context: Police car siren emitting sound waves of constant frequency.
2. Detectors P and Q: P located inside the police car, Q next to the road.
3. Observations:
   • Detector P records higher frequency (due to the car approaching).
   • Detector Q records lower frequency (as the car moves away).
4. Analysis and Frequency Calculation:
   • $f = \frac{1}{T}$ calculation for frequencies recorded.
   • $f<em>{L} = \frac{v ± v</em>L}{v ± v_S}$ to find the speed of the police car, where $v = 340 ext{m.s}^{-1}$.
   • Detected periods leading to necessary calculations.

Additional Questions

• Calculate frequency when source is moving.
• Analyze graphs and derive required values.
• Interpret examples involving different scenarios of the Doppler Effect with varying sources and detectors.

Conclusion

• The Doppler Effect plays a significant role in understanding sound and its behavior under various movement conditions. It demonstrates how perception changes based on the relative motion of sources and listeners, aiding in practical applications like radar technology and astronomy.

Key Formulas

1. $f<em>L = \frac{v ± v</em>L}{v ± v<em>S} f</em>S$
2. $f = \frac{1}{T}$

References for Understanding

• Various past exam questions provide scenarios applying the Doppler Effect to enhance learning through practical example Analysis.
• Real-world applications in fields like astrophysics and medical imaging show the Doppler Effect's significance beyond mere physics education.