Lecture Notes: Harmonics in Pipes

Introduction

• Review of open-open tubes/pipes
  • Anti-nodes at both ends
  • Specific wavelengths allowed
  • Formula for wavelengths depending on tube length (L) and harmonic number (N)
  • Fundamental harmonic (N=1), second harmonic (N=2), etc.

Transition to Open-Closed Tubes

• Consideration of a pipe with one open end and one closed end
  • Similar to a soda bottle
  • Open end: allows oscillation (anti-node)
  • Closed end: restricts oscillation (node)

Wavelengths in Open-Closed Tubes

• Fundamental Frequency:
  • Node at closed end, anti-node at open end
  • Simplest wave: from anti-node to node
  • Length (L) is 1/4 of the wavelength: ( \lambda = 4L )
• Second Possible Frequency:
  • One node between anti-node and node
  • Length (L) is 3/4 of the wavelength: ( \lambda = \frac{4L}{3} )
• Third Possible Frequency:
  • Two nodes between anti-node and node
  • Length (L) is 5/4 of the wavelength: ( \lambda = \frac{4L}{5} )

General Formula for Open-Closed Tubes

• Formula: ( \lambda = \frac{4L}{N} ), with N being odd integers (1, 3, 5, ...)
  • Only odd harmonics present
  • Pattern: skips even harmonics

Practical Demonstration

• Testing by blowing over soda bottles
  • Larger tube length (L) results in longer wavelengths and lower frequencies
  • Higher drink level decreases effective tube length, leading to higher pitch
  • Lower drink level increases tube length and results in lower pitch

Conclusion

• Open-closed pipes exhibit unique harmonic patterns with only odd harmonics
• Relationship between tube length and frequency: longer tubes produce lower frequencies