Waves in Pipes and Strings
Class 12 Physics Notes • NEB Syllabus 2083
Syllabus, 14 solved multiple choice questions, step-by-step numericals, handwritten and typed PDF notes and frequently asked questions on stationary waves, organ pipes, end correction and vibrating strings.
1. Syllabus
Unit: Waves and Optics
Chapter 8: Waves in Pipes and Strings (6 hours)
These notes follow the NEB Class 12 Physics syllabus (+2 Physics, Nepal). Practise the multiple choice questions first, then work through the numericals, and finish with the PDF notes for quick revision.
2. Multiple Choice Questions
Question 1. What is the fundamental frequency of a closed organ pipe of length \(l\) if the speed of sound in air is \(v\)?
- a) \(\dfrac{v}{2l}\)
- b) \(\dfrac{v}{l}\)
- c) \(\dfrac{v}{4l}\)
- d) \(\dfrac{2v}{l}\)
Correct Answer: c) \(\dfrac{v}{4l}\)
Reason: A closed pipe has a node at the closed end and an antinode at the open end. In the fundamental mode the pipe length is one quarter of a wavelength:
Question 2. Which of the following statements is true for an open organ pipe?
- a) It produces only even harmonics.
- b) It produces only odd harmonics.
- c) It produces both even and odd harmonics.
- d) It does not produce harmonics.
Correct Answer: c) It produces both even and odd harmonics.
Reason: An open pipe has antinodes at both ends. Its fundamental frequency is \(f_1=\dfrac{v}{2l}\) and the other modes are
Since \(n\) can be any integer, both even and odd harmonics are present.
Question 3. The end correction in an organ pipe is due to:
- a) Friction of air molecules
- b) Effect of temperature
- c) Effect of pressure difference
- d) Air bulging beyond the pipe’s end
Correct Answer: d) Air bulging beyond the pipe’s end
Reason: The antinode at the open end does not form exactly at the physical end of the tube. Because of the inertia of the air, the vibrating air column bulges slightly outward, so the acoustic length of the pipe is a little longer than its geometric length. This difference is the end correction.
Question 4. The third harmonic of a closed pipe corresponds to which overtone?
- a) First overtone
- b) Second overtone
- c) Third overtone
- d) Fourth overtone
Correct Answer: a) First overtone
Reason: A closed pipe produces only odd harmonics: \(f_1,\ 3f_1,\ 5f_1,\dots\) Overtones are the frequencies higher than the fundamental, so the first frequency above the fundamental, \(3f_1\), is the first overtone.
Question 5. If the length of a vibrating string is doubled while keeping tension and linear density constant, the frequency:
- a) Doubles
- b) Remains the same
- c) Becomes half
- d) Becomes four times
Correct Answer: c) Becomes half
Reason: The fundamental frequency of a stretched string (Mersenne’s law) is
If \(l\) becomes \(2l\), the frequency becomes \(\dfrac f2\).
Question 6. A string fixed at both ends vibrates in its fundamental mode. What is the nature of its displacement at the centre?
- a) Zero displacement
- b) Maximum displacement
- c) Half of maximum displacement
- d) Twice the maximum displacement
Correct Answer: b) Maximum displacement
Reason: In the fundamental mode a single loop forms. The fixed ends are nodes (zero displacement) and there is one antinode, with maximum displacement, exactly at the centre of the string.
Question 7. Two strings of the same material but different lengths and diameters are stretched under the same tension. Which factor most affects their vibrational frequency?
- a) Tension
- b) Density
- c) Length and diameter
- d) Shape of the string
Correct Answer: c) Length and diameter
Reason: The linear density is \(\mu=\pi r^2\rho=\dfrac{\pi d^2}{4}\rho\). Substituting in \(f=\dfrac1{2l}\sqrt{T/\mu}\):
Tension and material (density) are the same, so the frequency depends inversely on both the length \(l\) and the diameter \(d\).
Question 8. A flute and a guitar string produce the same frequency of sound. Which statement must be true?
- a) They have the same mode of vibration.
- b) They have the same amplitude.
- c) They produce the same fundamental frequency.
- d) They have the same length.
Correct Answer: c) They produce the same fundamental frequency.
Reason: Two instruments playing the same note have the same pitch, that is, the same fundamental frequency. They need not have the same length, mode of vibration or amplitude; they only have to be tuned to vibrate at the same base rate.
Question 9. A closed pipe has a fundamental frequency of \(220\ \mathrm{Hz}\). What will be the frequency of its first overtone?
- a) \(220\ \mathrm{Hz}\)
- b) \(440\ \mathrm{Hz}\)
- c) \(660\ \mathrm{Hz}\)
- d) \(880\ \mathrm{Hz}\)
Correct Answer: c) \(660\ \mathrm{Hz}\)
Reason: A closed pipe has only odd harmonics, so the first overtone is the third harmonic:
Question 10. A string of length \(1.5\ \mathrm{m}\) is stretched under tension and vibrates with a fundamental frequency of \(150\ \mathrm{Hz}\). What is the speed of the wave along the string?
- a) \(150\ \mathrm{m\,s^{-1}}\)
- b) \(225\ \mathrm{m\,s^{-1}}\)
- c) \(300\ \mathrm{m\,s^{-1}}\)
- d) \(450\ \mathrm{m\,s^{-1}}\)
Correct Answer: d) \(450\ \mathrm{m\,s^{-1}}\)
Reason: In the fundamental mode \(l=\dfrac\lambda2\), so \(\lambda=2l=3.0\ \mathrm{m}\).
Question 11. The fundamental frequency of a stretched string is \(100\ \mathrm{Hz}\). If the tension is increased by a factor of 4, what will be the new frequency?
- a) \(100\ \mathrm{Hz}\)
- b) \(200\ \mathrm{Hz}\)
- c) \(400\ \mathrm{Hz}\)
- d) \(50\ \mathrm{Hz}\)
Correct Answer: b) \(200\ \mathrm{Hz}\)
Reason: By the law of tension, \(f\propto\sqrt T\):
Question 12. A closed pipe resonates at \(300\ \mathrm{Hz}\) and \(500\ \mathrm{Hz}\) but not at \(400\ \mathrm{Hz}\). What is the fundamental frequency of the pipe?
- a) \(100\ \mathrm{Hz}\)
- b) \(200\ \mathrm{Hz}\)
- c) \(300\ \mathrm{Hz}\)
- d) \(500\ \mathrm{Hz}\)
Correct Answer: a) \(100\ \mathrm{Hz}\)
Reason: A closed pipe has only odd harmonics, so the two resonances are consecutive odd harmonics, \(nf_1\) and \((n+2)f_1\). Their difference is \(2f_1\):
Check: the harmonics are 100, 300, 500, \(\dots\) Hz.
Question 13. Two strings of the same length but different tensions are vibrating. The first string has a tension of 4 times that of the second string. If the fundamental frequency of the second string is \(f\), what is the frequency of the first string?
- a) \(f\)
- b) \(2f\)
- c) \(4f\)
- d) \(8f\)
Correct Answer: b) \(2f\)
Reason: For the same length and material, \(f\propto\sqrt T\):
Question 14. A pipe closed at one end produces a sound of wavelength \(1.2\ \mathrm{m}\) in its fundamental mode. What is the length of the pipe if the velocity of sound is \(340\ \mathrm{m\,s^{-1}}\)?
- a) \(0.15\ \mathrm{m}\)
- b) \(0.30\ \mathrm{m}\)
- c) \(0.60\ \mathrm{m}\)
- d) \(1.20\ \mathrm{m}\)
Correct Answer: b) \(0.30\ \mathrm{m}\)
Reason: In the fundamental mode of a closed pipe, the length is one quarter of the wavelength:
The speed of sound is not needed because the wavelength is already given.
3. Numerical and Subjective Questions
a) Define harmonics and overtones.
Harmonics: integral multiples of the lowest (fundamental) frequency of an oscillating system. If the fundamental is \(f_1\), the harmonics are \(f_1,2f_1,3f_1,\dots,nf_1\). The fundamental is the first harmonic.
Overtones: all the frequencies present in the vibration that are higher than the fundamental. The first frequency above the fundamental is the first overtone, the next is the second overtone, and so on.
b) Differentiate between harmonics and overtones with respect to closed and open organ pipes.
The relation depends on whether the pipe produces all harmonics or only the odd ones.
| Pipe | Fundamental | First overtone | Second overtone | \(n\)th overtone |
|---|---|---|---|---|
| Open (all harmonics) | \(f_1\) (first harmonic) | \(2f_1\) (second harmonic) | \(3f_1\) (third harmonic) | \((n+1)\)th harmonic |
| Closed (odd harmonics only) | \(f_1\) (first harmonic) | \(3f_1\) (third harmonic) | \(5f_1\) (fifth harmonic) | \((2n+1)\)th harmonic |
c) A closed organ pipe has a fundamental frequency of \(200\ \mathrm{Hz}\). What is the length of the pipe? (Speed of sound \(=340\ \mathrm{m\,s^{-1}}\))
For a closed pipe, \(f_1=\dfrac{v}{4l}\), so
d) A string has linear density \(0.01\ \mathrm{kg\,m^{-1}}\) and is under a tension of \(144\ \mathrm{N}\). What will be its fundamental frequency if its length is \(0.6\ \mathrm{m}\)?
a) Explain the formation of stationary waves in closed and open organ pipes.
Stationary (standing) waves form when two progressive waves of the same frequency and amplitude, travelling in opposite directions, superpose. In a pipe, a sound wave travelling along the pipe is reflected at the end and the incident and reflected waves interfere.
Closed pipe: the wave is reflected at the rigid closed end. Air particles cannot move there, so a node forms at the closed end, while an antinode forms at the open end where the particles vibrate freely.
Open pipe: at the open end a compression is reflected as a rarefaction. Particles at both open ends have maximum displacement amplitude, so antinodes form at both ends.
b) How do boundary conditions affect the formation of nodes and antinodes?
Rigid or closed boundary: air molecules cannot move past a solid barrier, so the displacement must be zero and a node is formed.
Free or open boundary: air molecules are free to oscillate against the open atmosphere, so the displacement amplitude is maximum and an antinode is formed.
c) How does changing the length of an organ pipe affect the pitch of the sound produced? Explain using wave theory.
The pipe length restricts the allowed wavelengths of the stationary waves. For an open pipe \(f=\dfrac{v}{2l}\) and for a closed pipe \(f=\dfrac{v}{4l}\), so in both cases \(f\propto\dfrac1l\).
Pitch is the perception of frequency. When the length increases, the allowed wavelength increases; since \(v=f\lambda\) and \(v\) is constant in air, the frequency decreases. A longer pipe therefore gives a lower pitch and a shorter pipe gives a higher pitch.
d) An open organ pipe is \(85\ \mathrm{cm}\) long. Calculate its fundamental frequency. (Speed of sound \(=340\ \mathrm{m\,s^{-1}}\))
a) Derive the expressions for the fundamental frequency and overtones in (i) a closed organ pipe and (ii) an open organ pipe.
Let \(l\) be the length of the pipe and \(v\) the speed of sound.
(i) Closed organ pipe (node at the closed end, antinode at the open end)
Fundamental (first harmonic): a quarter of a wave fits in the pipe.
First overtone (third harmonic): three quarters of a wave fit.
Second overtone (fifth harmonic): five quarters of a wave fit.
Only odd harmonics are present: \(f_1:f_3:f_5=1:3:5\).
(ii) Open organ pipe (antinodes at both ends)
Fundamental (first harmonic): half a wave fits (node in the middle).
First overtone (second harmonic): one full wave fits.
Second overtone (third harmonic): one and a half waves fit.
All harmonics are present: \(f_1:f_2:f_3=1:2:3\).
b) A closed pipe resonates with a tuning fork of frequency \(256\ \mathrm{Hz}\) in its first harmonic. What is the length of the pipe? (\(v=343\ \mathrm{m\,s^{-1}}\))
a) Why are only odd harmonics present in closed pipes, while all harmonics are present in open pipes?
A closed pipe needs a node at one end and an antinode at the other. This is possible only when the pipe length contains an odd number of quarter wavelengths (\(\lambda/4,\ 3\lambda/4,\ 5\lambda/4,\dots\)), so only the odd harmonics \(1,3,5,\dots\) times the fundamental occur.
An open pipe needs antinodes at both ends. This is possible whenever the length contains a whole number of half wavelengths (\(\lambda/2,\ 2\lambda/2,\ 3\lambda/2,\dots\)), so all harmonics \(1,2,3,\dots\) occur.
b) Compare the harmonic content in a pipe open at both ends and in a pipe closed at one end, and explain how this affects the quality of sound.
Harmonic content: an open pipe produces \(f_1,2f_1,3f_1,4f_1,\dots\) (all harmonics); a closed pipe produces \(f_1,3f_1,5f_1,7f_1,\dots\) (odd harmonics only).
Quality (timbre): the quality of a sound depends on the number and relative intensities of the overtones present. An open pipe has a richer mixture of overtones, so its sound is fuller and richer. A closed pipe lacks the even harmonics, so its tone is hollower and plainer than that of an open pipe of the same fundamental frequency.
c) An open pipe resonates in its second harmonic at \(680\ \mathrm{Hz}\). What is its length? (\(v=340\ \mathrm{m\,s^{-1}}\))
For an open pipe \(f_n=n\dfrac{v}{2l}\), so \(f_2=\dfrac vl\):
d) Calculate the frequency of the third harmonic of a closed pipe of length \(0.4\ \mathrm{m}\). (\(v=340\ \mathrm{m\,s^{-1}}\))
a) Explain the concept of end correction in resonance tubes.
The antinode at the open end of a pipe does not form exactly at the geometric opening. The air just outside the pipe is partly restrained by the surrounding air, so the point of maximum displacement (the acoustic antinode) lies slightly beyond the end of the tube. This small extra distance is the end correction \(e\). It depends on the radius \(r\) of the pipe and is approximately
b) Derive the expression for the effective length of a pipe including end correction.
Let \(l\) be the geometric length of the pipe and \(e\) the end correction.
Closed pipe: there is one open end, so one correction applies and the air column behaves as if it were longer by \(e\):
Open pipe: there are two open ends, and an antinode forms slightly outside each, so the correction is applied twice:
c) A resonance tube has an actual length of \(0.75\ \mathrm{m}\). If the end correction is \(0.03\ \mathrm{m}\), what is the effective length of the pipe?
A resonance tube is closed at one end, so
(If the tube were open at both ends, \(L=l+2e=0.81\ \mathrm{m}\).)
a) Mention the factors affecting the velocity of transverse waves along a stretched string.
The velocity depends on two factors:
b) Explain the laws of vibration of a stretched string.
The laws of transverse vibration (Mersenne’s laws) follow from the fundamental frequency \(f=\dfrac1{2l}\sqrt{\dfrac T\mu}\):
c) How do length, tension and mass per unit length affect the fundamental frequency of transverse vibration of a string?
From \(f=\dfrac1{2l}\sqrt{\dfrac T\mu}\):
d) A string of length \(1.2\ \mathrm{m}\) has a mass of \(0.01\ \mathrm{kg}\) and a tension of \(100\ \mathrm{N}\). Find the velocity of the transverse wave.
Figure: Third harmonic of a string fixed at both ends (N = node, A = antinode)
a) A string is vibrating in its third harmonic. Sketch the wave pattern and explain the number of nodes and antinodes formed.
The sketch is shown in the figure above. In the third harmonic the string vibrates in 3 segments (loops).
For a string fixed at both ends and vibrating in its \(n\)th harmonic, there are \(n\) loops. The fixed ends are always nodes.
Nodes: the two fixed ends plus two points inside the string, so there are 4 nodes.
Antinodes: one at the centre of each of the 3 loops, so there are 3 antinodes.
In general, the \(n\)th harmonic has \(n\) antinodes and \((n+1)\) nodes.
b) Find the frequency of the fundamental mode of a stretched string \(1\ \mathrm{m}\) long, under a tension of \(150\ \mathrm{N}\) and mass per unit length \(0.005\ \mathrm{kg\,m^{-1}}\).
c) A stretched string vibrates in its second overtone at \(300\ \mathrm{Hz}\). What is the fundamental frequency?
A string fixed at both ends has all harmonics, so: fundamental \(=f_1\), first overtone \(=2f_1\), second overtone \(=3f_1\).
4. View PDF Notes
i. Handwritten Notes
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5. Frequently Asked Questions
What are waves in pipes and strings in Class 12 Physics?
What topics are included in Chapter 8 Waves in Pipes and Strings of NEB Class 12 Physics?
What are mechanical waves and what are the types of mechanical waves?
What are some examples of mechanical waves?
What is the difference between mechanical waves and electromagnetic (non-mechanical) waves?
What are stationary waves, nodes and antinodes?
What are the frequencies of a closed organ pipe?
What are the frequencies of an open organ pipe?
What is the difference between harmonics and overtones?
What is end correction in an organ pipe?
What is the velocity of transverse waves along a stretched string?
What are the laws of vibration of a stretched string?
Why does a closed pipe have only odd harmonics while an open pipe has all harmonics?
How does the length of an organ pipe change the pitch?
Where can I get Class 12 Physics Chapter 8 Waves in Pipes and Strings notes in PDF?
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