Standing Waves in Strings and Pipes
Standing waves are the most consequential outcome of the superposition principle: when a wave reflects at a boundary and interferes with its own incident counterpart, energy stops travelling and settles into fixed nodes and antinodes.
This quiz tests the structural logic that IIT JEE demands:
- Boundary conditions determine which wavelengths survive.
- Strings fixed at both ends, pipes open at both ends, and pipes closed at one end each admit a distinct harmonic series.
- Wave speed on a stretched string obeys ##v=\sqrt{T/\mu}##, and in a gas column ##v=\sqrt{\gamma RT/M}##.
- Every frequency follows from ##f=v/\lambda##.
Identify the boundary conditions first, derive the allowed wavelengths second, and only then substitute numbers. Good luck!
Q1. A standing wave is formed in a medium when:
A standing wave requires two identical waves of equal amplitude travelling in opposite directions through the same medium, typically produced by reflection at a boundary.
Q2. At a rigid boundary such as a string clamped to a wall, the reflected wave undergoes a phase change of:
Reflection at a rigid boundary involves a phase inversion of ##\pi## radians, so incident and reflected displacements cancel and a displacement node is forced at the boundary.
Q3. At the open end of an air column, which of the following is true?
At a free boundary the reflected wave returns without phase inversion, so displacement is maximum (antinode) while pressure is a node because the open end communicates with constant atmospheric pressure.
Q4. The distance between two adjacent nodes in a standing wave is:
Adjacent nodes are always separated by ##\lambda/2##, and the distance from a node to the nearest antinode is ##\lambda/4##, independent of medium, frequency or boundary type.
Q5. A string of length ##L## fixed at both ends vibrates in its fundamental mode. The wavelength is:
For a string fixed at both ends, the fundamental has nodes at both ends with one antinode in between, so ##L=\lambda/2##, giving ##\lambda=2L##.
Q6. The wave speed on a stretched string is given by:
The speed on a stretched string obeys ##v=\sqrt{T/\mu}##, where ##T## is tension and ##\mu## the linear mass density.
Q7. If the tension in a vibrating string is quadrupled while the linear density is unchanged, the fundamental frequency becomes:
Since ##f\propto\sqrt{T}##, quadrupling ##T## multiplies the frequency by ##\sqrt{4}=2##.
Q8. A pipe open at both ends has a fundamental frequency ##f##. Its first overtone (second harmonic) frequency is:
An open pipe admits all harmonics, so the second harmonic is ##2f##. The first overtone of an open pipe is its second harmonic.
Q9. A pipe closed at one end has a fundamental frequency ##f##. Its first overtone frequency is:
A closed pipe admits only odd harmonics, so the first overtone is the third harmonic, ##3f##.
Q10. Which harmonic series is supported by a pipe closed at one end?
The closed end forces a displacement node and the open end an antinode, permitting only odd harmonics ##f, 3f, 5f, \ldots##
Q11. End correction in an organ pipe accounts for:
The displacement antinode actually forms slightly beyond the open end, so an end correction is added to the effective length of the pipe.
Q12. The speed of sound in a gas column is given by:
The speed of sound in a gas column follows ##v=\sqrt{\gamma RT/M}##, where ##\gamma## is the ratio of specific heats, ##T## the temperature and ##M## the molar mass.
Q13. When the temperature of a gas in an organ pipe is raised, the resonance frequency of a given mode:
Since ##v=\sqrt{\gamma RT/M}##, raising ##T## increases the wave speed and hence the resonance frequency of each mode.
Q14. In a sonometer experiment, the fundamental frequency of a vibrating wire is doubled by:
With ##f=\dfrac{1}{2L}\sqrt{T/\mu}##, quadrupling ##T## at constant ##L## and ##\mu## doubles the frequency.
Q15. In a standing wave, energy:
In a standing wave the spatial envelope is fixed, so energy does not travel along the medium but is trapped between nodes and antinodes.