Lecture Notes: Sound Waves - Beats

Introduction

• Topic: Beats (Lecture 4)
• Lecturer: Alak Pandey
• Objective: Understand the concept of beats and beat frequency.

Key Concepts

• Definition of Beats:
  • Beats are formed by the superposition of two sound waves that have a small difference in frequency.
  • Notation: Let F1 and F2 be the frequencies of the two sound waves.

Frequency Difference Conditions

• For beats to be heard, the frequency difference should be small:
  • Limit values:
    • According to Sr. D.C. Pandey: |F2 - F1| < 10 Hz
    • According to Sr. H.C. Verma: |F2 - F1| < 16 Hz
    • Other sources may suggest 8 Hz as a limit.

Superposition of Waves

• Conditions for Beats:
  • Both sound waves travel in the same direction.
  • Both waves should have the same amplitude but different frequencies.

Difference from Standing Waves

• Comparison with Standing Waves:
  • Standing waves involve waves traveling in opposite directions with the same frequency.
  • In beats, waves travel in the same direction and have small differences in frequency.

Visualization of Beats

• Visualization of beats can be done with wave equations:
  • Example equations:
    • Wave 1: y1 = A sin(ω1t - k1x)
    • Wave 2: y2 = A sin(ω2t - k2x)
    • The net wave function: y_net = y1 + y2
• Shows how waves interfere to create beats.

Beat Frequency

• Beat Frequency (f_beats):
  • Formula: f_beats = |F2 - F1|
  • Represents the number of beats heard per second.
• Derivation:
  • Beats occur when frequencies differ, leading to periodic increases and decreases in sound intensity.
  • Example: If f1 = 440 Hz and f2 = 440 +/- n Hz (where n is small), we categorize, it results in a beat frequency of n per second.

Examples and Applications

1. Example Problem 1:

   • A tuning fork at 512 Hz produces 4 beats per second with another fork.
   • Adjusting tension decreases beat frequency indicating potential changes in initial frequency calculations.

2. Example Problem 2:

   • Two tuning forks, where A has a frequency of 440 Hz. Loading B changes beat frequency from 4 to 6 beats/sec, resulting in B's frequency variation.

3. Scrapping Concept of Frequency:

   • Two tuning forks have identical frequencies; scrapping one results in increased beats, illustrating frequency dependence.

Conclusion

• Beats arise from the interference of sound waves with slightly different frequencies, central to understanding wave behavior in acoustics.
• The formula for beat frequency relates directly to the difference in frequencies of the interfering waves.
• Understanding beats enhances the grasp of sound phenomena and applications in acoustics studies.