6.2 Resonance in strings, pipes, and cavities

Resonance is a crucial concept in acoustics, affecting everything from musical instruments to building design. It occurs when an object's natural frequency matches an external driving force, resulting in amplified vibrations and enhanced sound production.

Calculating resonant frequencies is essential for understanding acoustic systems. Different formulas apply to strings, pipes, and cavities, with factors like length, tension, and temperature influencing the results. Changing system properties can significantly alter resonance, impacting sound quality and control.

Fundamentals of Resonance

Concept of resonance in acoustics

Top images from around the web for Concept of resonance in acoustics

• 

  Open Source Physics @ Singapore: EJSS SHM model with resonance showing Amplitude vs frequency graphs View original

  Is this image relevant?

• 

  Law of Resonance - Ascension Glossary View original

  Is this image relevant?

• 

  Sound Interference and Resonance: Standing Waves in Air Columns – Fundamentals of Heat, Light ... View original

  Is this image relevant?

• 

  Open Source Physics @ Singapore: EJSS SHM model with resonance showing Amplitude vs frequency graphs View original

  Is this image relevant?

• 

  Law of Resonance - Ascension Glossary View original

  Is this image relevant?

1 of 3

Top images from around the web for Concept of resonance in acoustics

• 

  Open Source Physics @ Singapore: EJSS SHM model with resonance showing Amplitude vs frequency graphs View original

  Is this image relevant?

• 

  Law of Resonance - Ascension Glossary View original

  Is this image relevant?

• 

  Sound Interference and Resonance: Standing Waves in Air Columns – Fundamentals of Heat, Light ... View original

  Is this image relevant?

• 

  Open Source Physics @ Singapore: EJSS SHM model with resonance showing Amplitude vs frequency graphs View original

  Is this image relevant?

• 

  Law of Resonance - Ascension Glossary View original

  Is this image relevant?

1 of 3

• Natural frequency of vibration in an object or system amplifies vibrations when driven at resonant frequency
• Enhances sound production in musical instruments (guitar strings, piano soundboards)
• Crucial for designing acoustic spaces for optimal sound quality (concert halls, recording studios)
• Facilitates noise control and reduction in buildings and vehicles (mufflers, sound barriers)
• Characterized by increased amplitude of vibration, energy transfer between kinetic and potential forms
• Resonant frequency depends on system properties (mass, stiffness, geometry)

Conditions for resonance occurrence

• Driving force at or near the natural frequency with low system damping required
• Strings need fixed ends to create standing waves, tension and linear density affect wave speed
• Open pipes have both ends open to atmosphere, pressure nodes form at open ends
• Closed pipes have one end closed and one open, pressure node at open end, antinode at closed end
• Cavities require enclosed volume of air with rigid walls to reflect sound waves

Resonance Calculations and Analysis

Calculation of resonant frequencies

• Strings: Fundamental frequency $f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$, harmonics $f_n = n f_1$
• Open pipes: Fundamental $f_1 = \frac{v}{2L}$, harmonics $f_n = n f_1$
• Closed pipes: Fundamental $f_1 = \frac{v}{4L}$, harmonics $f_n = (2n-1) f_1$
• Rectangular cavities: $f_{lmn} = \frac{v}{2}\sqrt{(\frac{l}{L_x})^2 + (\frac{m}{L_y})^2 + (\frac{n}{L_z})^2}$
• l, m, n represent mode numbers in cavity equations

Effects of system changes on resonance

• Strings: Longer length lowers frequencies, higher tension raises frequencies, increased linear density lowers frequencies
• Open and closed pipes: Longer length lowers frequencies, temperature changes affect speed of sound altering frequencies
• Rectangular cavities: Increasing any dimension lowers frequencies for that dimension, changing medium inside affects speed of sound
• Adding mass typically lowers resonant frequencies (weighted piano keys)
• Increasing stiffness typically raises resonant frequencies (tighter drum heads)
• Altering boundary conditions can significantly change resonant modes (opening/closing organ pipes)