Section 8.6: Standing Waves
A standing wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. Unlike traveling waves, the energy does not propagate; nodes remain stationary while antinodes oscillate maximally.
Where:
\( A \) = amplitude of individual waves
\( k \) = wave number \( (2\pi/\lambda) \)
\( \omega \) = angular frequency \( (2\pi f) \)
Nodes: points of zero displacement \( (\sin(kx)=0) \)
Antinodes: points of maximum displacement \( (\sin(kx)=\pm 1) \)
Example: String Fixed at Both Ends
A string of length 1 m is fixed at both ends. If the string vibrates in its 2nd harmonic (n=2) with speed of wave 200 m/s, find its frequency.
Wavelength of 2nd harmonic: \( \lambda = \frac{2L}{n} = \frac{2 \cdot 1}{2} = 1 \text{ m} \)
Frequency: \( f = \frac{v}{\lambda} = \frac{200}{1} = 200 \text{ Hz} \)
Practice Problems
- A string 0.8 m long fixed at both ends vibrates in the 3rd harmonic. Wave speed is 160 m/s. Find the frequency.
- A pipe closed at one end has a length of 0.85 m. Find the wavelength of the fundamental mode.
- Explain why nodes in standing waves do not move while antinodes oscillate.
- Two waves traveling in opposite directions produce a standing wave with amplitude 5 cm. What is the maximum displacement at an antinode?
- Derive the condition for standing waves in a string fixed at both ends.