14.4 Sound Interference and Resonance

Learning Objectives

• Describe resonance and beats.
• Define fundamental frequency and harmonic series.
• Contrast open-pipe and closed-pipe resonators.
• Solve problems involving harmonic series and beat frequency.

Key Terms

• Beat
• Beat frequency
• Damping
• Fundamental
• Harmonics
• Natural frequency
• Overtones
• Resonance
• Resonate

Resonance and Beats

• Resonance: Occurs when a system is driven at its natural frequency, causing it to oscillate with larger amplitude.
  • Example: Singing near a piano causes strings with matching frequencies to resonate.
  • Driving Force: Energy added at a certain frequency, potentially different from the system's natural frequency.
• Damping: Energy dissipation over time, leading to amplitude reduction.
• Beat Frequency: Occurs when two waves of similar frequency interfere, causing amplitude to vary over time.
  • Formula: ( f_B = |f_1 - f_2| )

Fundamental Frequency and Harmonics

• Fundamental Frequency: The lowest natural frequency of a system.
• Overtones: Higher natural frequencies; harmonics are integer multiples of the fundamental frequency.
• Standing Waves: Formed due to constructive interference at resonance.

Open-Pipe vs. Closed-Pipe Resonators

• Closed-Pipe Resonators: Closed at one end, resonate at odd multiples of the fundamental frequency (( f_n = \frac{nv}{4L} )).
  • Example: Tuning forks and tubes.
• Open-Pipe Resonators: Open at both ends, resonate at all integer multiples of the fundamental frequency (( f_n = \frac{nv}{2L} )).
  • Examples: Flutes, oboes.
• Difference: Sound from open tubes has more overtones due to even and odd multiples, creating a richer sound.

Solving Problems

• Closed-Pipe Resonator: Calculate tube length for a given fundamental frequency using ( L = \frac{v}{4f_1} ).
• Open-Pipe Resonator: Find overtone frequencies using ( f_n = \frac{nv}{2L} ).
• Beat Frequency: Used by piano tuners to match frequencies by eliminating beats.

Practice Problems

1. Calculating beat frequencies and resonant tube lengths.

Check Your Understanding

• Questions to assess understanding of damping, resonance, and resonators setup.