Click image to open full size
Standing Waves & Resonance
Nodes, Antinodes, Harmonics, and Resonant Frequencies
Related Tools
Related Labs
Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. Instead of the pattern moving along, fixed points of no motion and maximum motion appear. These patterns matter because they explain how musical instruments produce notes, how bridges and buildings vibrate, and how many physical systems store wave energy. Resonance is closely connected because it causes certain frequencies to produce especially large vibrations.
In a standing wave, nodes are points that remain at zero displacement, while antinodes are points that oscillate with the greatest amplitude. A system resonates when it is driven at one of its natural frequencies, allowing energy to build efficiently over time. On a string fixed at both ends or in an air column, only specific wavelengths fit the boundary conditions, so only certain resonant frequencies are allowed. These allowed patterns are called harmonics or normal modes, and each one has a distinct shape and frequency.
Key Facts
- Standing waves are produced by interference of two identical waves traveling in opposite directions.
- Nodes are points of zero displacement; antinodes are points of maximum displacement.
- For a string fixed at both ends: lambda_n = 2L/n, where n = 1, 2, 3, ...
- Resonant frequencies on a string are f_n = nv/(2L).
- Wave speed is related by v = f lambda.
- For an open-open air column: f_n = nv/(2L), while for an open-closed air column: f_n = nv/(4L) for n = 1, 3, 5, ...
Vocabulary
- Standing wave
- A wave pattern with fixed nodes and antinodes formed by two identical waves moving in opposite directions.
- Resonance
- The large increase in amplitude that occurs when a system is driven at one of its natural frequencies.
- Node
- A point on a standing wave that remains at zero displacement at all times.
- Antinode
- A point on a standing wave where the oscillation amplitude is greatest.
- Harmonic
- One of the allowed standing wave patterns in a system, each with a specific frequency and shape.
Common Mistakes to Avoid
- Thinking standing waves travel down the string, which is wrong because the overall pattern stays fixed while the medium itself oscillates in place.
- Mixing up nodes and antinodes, which is wrong because nodes have zero displacement and antinodes have the maximum displacement.
- Using any wavelength for resonance, which is wrong because only wavelengths that satisfy the boundary conditions can form stable standing waves.
- Assuming all air columns have the same harmonic series, which is wrong because open-open and open-closed tubes allow different wavelength patterns and frequencies.
Practice Questions
- 1 A string of length 1.20 m is fixed at both ends and supports waves with speed 240 m/s. Find the frequencies of the first three harmonics.
- 2 An open-open air column is 0.85 m long. If the speed of sound is 340 m/s, calculate the fundamental frequency and the second harmonic.
- 3 A singer holds a note near a glass and the glass begins vibrating strongly only at one particular pitch. Explain this using natural frequency and resonance.