Standing Waves Strings Pipes

8 MCQs4 revision cards9-step worked example

Source: NCERT Oscillations and WavesPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

The high-frequency trap in standing waves on strings and pipes is confusing which systems allow all harmonics and which allow only odd ones. A closed-end pipe supports only odd harmonics — yet students routinely plug in even harmonic numbers and lose marks.

Standing waves on a string fixed at both ends. When a transverse wave reflects at both fixed ends, superposition produces standing waves. Both ends are displacement nodes. The allowed frequencies are f_n = nv/(2L), where n = 1, 2, 3, … and v is the wave speed on the string (NCERT Class 11 Physics Chapter 14, page 12). All harmonics — fundamental, second, third, and so on — are present.

Standing waves in an open-open pipe. Both ends are displacement antinodes (pressure nodes). The formula is identical to the string: f_n = nv/(2L), with v now being the speed of sound. All harmonics are present (NCERT Class 11 Physics Chapter 14, page 13).

Standing waves in a closed-end pipe. One end is a displacement node (closed) and the other is a displacement antinode (open). The boundary asymmetry forces only odd harmonics: f_n = (2n−1)v/(4L), giving frequencies f, 3f, 5f, … (NCERT Class 11 Physics Chapter 14, page 13). Even harmonics (2f, 4f, …) are physically impossible in this geometry.

Why the closed pipe is different. The closed end demands a node and the open end demands an antinode. Fitting a standing wave between a node and the nearest antinode requires an odd number of quarter-wavelengths. That constraint eliminates every even harmonic.

The trap on paper. When a question says "pipe closed at one end," students who default to f_n = nv/(2L) — the string/open-pipe formula — will include forbidden even harmonics and pick a wrong option. Read the boundary condition first, then choose the formula.

Practice MCQs

Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.

MCQ 1Easy RecallPractice

A string fixed at both ends has a fundamental frequency of 200 Hz. What is the frequency of its third harmonic?

MCQ 2Easy RecallPractice

Which harmonics are present in a pipe closed at one end?

MCQ 3Direct ApplicationPractice

A pipe open at both ends has a fundamental frequency f. If one end is now closed (pipe length unchanged), the new fundamental frequency is:

MCQ 4Direct ApplicationPractice

The second overtone of an open-open pipe has a frequency of 900 Hz. What is the fundamental frequency of this pipe?

MCQ 5Direct ApplicationPractice

The second overtone of a pipe closed at one end has a frequency of 1250 Hz. What is the fundamental frequency?

MCQ 6Easy RecallPractice

A string of length 1.0 m is fixed at both ends and vibrates in its fundamental mode. The positions of nodes are at:

MCQ 7CalculationPractice

An open-open pipe of length L and a closed-end pipe of length L have the same speed of sound. The ratio of the fundamental frequency of the open pipe to that of the closed pipe is:

MCQ 8CalculationPractice

A closed-end pipe resonates at frequencies 850 Hz and 1190 Hz with no resonant frequency between them. What is the fundamental frequency of the pipe?

Quick recall before you leave

Worked Example

Pattern: Open-open vs closed-end pipe harmonic comparison (based on NEET pattern: compare frequency ratios across pipe types).

1
Given
An organ pipe open at both ends has a fundamental frequency of 480 Hz. The speed of sound is 340 m/s.
2
Required
(a) Find the length of the pipe. (b) If one end is now closed, find the fundamental frequency and the frequency of the 2nd overtone.
3
Concept
Open-open pipe: all harmonics, f_n = nv/(2L). Closed-end pipe: only odd harmonics, f_n = (2n−1)v/(4L). Closing one end changes the boundary condition from antinode-antinode to node-antinode.
4
Formula
Open pipe fundamental: f₁ = v/(2L) → L = v/(2f₁) Closed pipe fundamental: f₁' = v/(4L) Closed pipe 2nd overtone = 5th harmonic: f = 5v/(4L)
5
Substitution
L = 340/(2 × 480) = 340/960 f₁' = 340/(4 × 0.354) 2nd overtone = 5 × 340/(4 × 0.354)
6
Calculation
L = 340/960 ≈ 0.354 m f₁' = 340/(4 × 0.354) = 340/1.417 ≈ 240 Hz 2nd overtone of closed pipe = 5 × 240 = 1200 Hz **Note on exact values:** 480, 340, and the integer multipliers (2, 4, 5) are given or exact values — they do not limit significant figures. The division 340/960 is carried to three significant figures consistent with the input data.
7
Final answer
(a) L ≈ 0.354 m (about 35.4 cm) (b) Closed-pipe fundamental = 240 Hz; 2nd overtone = 1200 Hz Cross-check: 240 Hz is exactly half of 480 Hz, consistent with closing one end halving the fundamental. The 2nd overtone (5th harmonic) of the closed pipe at 1200 Hz is an odd multiple of 240 Hz. Both checks confirm the result.
8
Common trap
Using f_n = nv/(2L) for the closed pipe gives the 2nd overtone as 3 × 480 = 1440 Hz (the open pipe's 3rd harmonic), which is wrong. The closed pipe's 2nd overtone is the 5th harmonic at (2×3−1)v/(4L) = 5f₁', not 3f₁. Always check whether the pipe is open or closed before selecting the formula.
9
Similar NEET-style question
A pipe closed at one end has a fundamental frequency of 150 Hz. An open pipe of the same length is placed nearby. What is the fundamental frequency of the open pipe, and which is the lowest harmonic of the open pipe that matches a resonant frequency of the closed pipe? *Hint:* The open pipe's fundamental is 2 × 150 = 300 Hz. Its harmonics are 300, 600, 900, 1200, … Hz. The closed pipe's harmonics are 150, 450, 750, 1050, … Hz. Find the first common frequency. ---

Before solving, remember these

Formula

Standing waves on string (fixed-fixed)

Frequencies of standing waves: ν_n = n v / (2L) for n = 1, 2, 3, ... ν_1 is the fundamental (first harmonic); ν_2 = 2ν_1 is the second harmonic, etc.

-- NCERT, p. 12

Formula

Standing waves in pipes

Open at both ends: ν_n = n v/(2L), all harmonics present. Closed at one end: ν_n = (2n-1) v/(4L), only odd harmonics.

-- NCERT, p. 13

Formulas

10 formulas — click to collapse

Beat frequency

f_{beat} = ∣ f_{1} - f_{2} ∣

When two waves of nearly equal frequencies superpose, amplitude oscillates at the difference frequency.

Symbol	Quantity	SI Unit
f_beat	beat frequency	Hz
f1, f2	superposed frequencies	Hz

Valid when

Linear superposition
f1, f2 close in value

Period of simple pendulum (small angle)

T = 2 π \frac{L}{g}

Period of simple pendulum of length L. Holds for small amplitudes (sin theta ~ theta).

Symbol	Quantity	SI Unit
T	period	s
L	pendulum length	m
g	gravity	m/s^2

Valid when

Small angular amplitude (typically <15°)
Massless string
Point bob

SHM displacement

x (t) = A cos (ω t + ϕ); ω = 2 π f

Displacement in simple harmonic motion. Velocity = -A*omega*sin(omega*t+phi); a = -omega^2 * x.

Symbol	Quantity	SI Unit
A	amplitude	m
omega	angular frequency	rad/s
phi	phase	rad
T	period	s
f	frequency	Hz

Valid when

Restoring force linear (F = -kx)
No damping

Total energy in SHM

E = \frac{1}{2} k A^{2} = \frac{1}{2} m ω^{2} A^{2}

Total mechanical energy is constant. Oscillates between KE (max at x=0) and PE (max at x=±A).

Symbol	Quantity	SI Unit
E	total energy	J
k	spring constant	N/m
A	amplitude	m
m	mass	kg
omega	angular frequency	rad/s

Valid when

Conservative SHM (no damping)
Elastic regime

Period of mass-spring oscillator

T = 2 π \frac{m}{k}

Period of horizontal spring with mass m, spring constant k. Independent of amplitude.

Symbol	Quantity	SI Unit
T	period	s
m	mass	kg
k	spring constant	N/m

Valid when

Hooke's law spring
No damping
Small enough amplitude to stay in elastic regime

Standing wave in closed-end pipe

f_{n} = \frac{( 2 n - 1 ) v}{4 L}

Pipe closed at one end has only odd harmonics: f, 3f, 5f, ...

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	sound speed	m/s
L	pipe length	m

Valid when

Closed at one end (open at other)
End correction neglected

Standing wave in open-open pipe

f_{n} = \frac{n v}{2 L}

Pipe open at both ends has all harmonics. Same formula as string.

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	sound speed	m/s
L	pipe length	m

Valid when

Open at both ends
End correction neglected

Standing wave frequencies on fixed-fixed string

f_{n} = \frac{n v}{2 L}

Allowed frequencies on string fixed at both ends. n=1 fundamental; harmonics 2f, 3f, ...

Symbol	Quantity	SI Unit
f_n	n-th harmonic	Hz
v	wave speed on string	m/s
L	string length	m
n	harmonic number	-

Valid when

String fixed at both ends
Wave speed v as defined above

Speed of sound in gas (Newton-Laplace)

v = \frac{γ P}{ρ}

Speed of sound in gas. Adiabatic index gamma, pressure P, density rho. Increases with sqrt(T).

Symbol	Quantity	SI Unit
v	speed of sound	m/s
gamma	adiabatic index	-
P	pressure	Pa
rho	density	kg/m^3

Valid when

Ideal gas
Adiabatic compression/expansion of sound waves

Wave speed on string

v = \frac{T}{μ}

Speed of transverse wave on string under tension T, linear mass density mu.

Symbol	Quantity	SI Unit
v	wave speed	m/s
T	tension	N
mu	linear mass density	kg/m

Valid when

Stretched uniform string
Small amplitude

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

6 items — click to collapse

Trap

Pendulum: mass-dependence claim

Category: Overthinking

Student writes T as depending on bob mass. Simple pendulum T = 2π√(L/g); independent of m.

When it triggers

Question changes pendulum bob mass and asks for new period.

How to avoid

Mass cancels in derivation (gravitational mass = inertial mass). Mass changes the bob's KE and PE proportionally; period unaffected.

Trap

SHM period: amplitude dependence claim

Category: Overthinking

Student claims SHM period depends on amplitude. For ideal SHM (Hooke's law spring or simple pendulum at small angle), period is INDEPENDENT of amplitude.

When it triggers

Question gives changes in amplitude and asks for new period.

How to avoid

T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — neither depends on A. Only at large pendulum angles does T pick up a small amplitude correction.

Trap

Pipe harmonics: open vs closed end

Category: Similar Terms

Student includes even harmonics in a closed-end pipe. Closed pipe has only ODD harmonics (f, 3f, 5f, ...).

When it triggers

Question describes pipe closed at one end (e.g. resonance tube).

How to avoid

Open both ends: all harmonics, f_n = nv/(2L). Closed one end: odd only, f_n = (2n-1)v/(4L). Fundamental of closed pipe is HALF that of open pipe of same L.

Mistake

Includes bob mass in simple pendulum period formula.

Root cause: concept gap

Correction

Simple pendulum T = 2π√(L/g) — independent of mass. Equivalence of inertial and gravitational mass cancels m.

Mistake

Claims SHM period depends on amplitude.

Root cause: concept gap

Correction

Ideal SHM: T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — no amplitude dependence. Doubling amplitude does not change period.

Mistake

Includes even harmonics in a closed-end pipe.

Root cause: concept gap

Correction

Closed-end pipe has only ODD harmonics (f, 3f, 5f, ...). Open-both-ends pipe has all (f, 2f, 3f, ...). Reason: closed end has displacement node and pressure antinode.

Past Year Questions

11 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse

NEET 2025

A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to: f

12f

22 3f

NTA Answer: Option 3(final)

NEET 2025

Two identical point masses P and Q, suspended from two separate massless springs of spring constants k 1 and k 2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is k k

22 k k 2 1

3k 1

4k 2 k k 2 1

NTA Answer: Option 1(final)

NEET 2024Revised key

 π If x =5sinπt + m represents the motion of a particle executing simple harmonic motion, the amplitude  3 and time period of motion, respectively, are

15 cm, 2 s

25 m, 2 s

35 cm, 1 s

45 m, 1 s

NTA Answer: Option 2(revised_final)

NEET 2024Revised key

If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half x its original length, then the new time period of oscillation is times its original time period. Then the value of 2 x is:

32 3

NTA Answer: Option 2(revised_final)

NEET 2023

The ratio of frequencies of fundamental harmonic produced by an open pipe to that of closed pipe having the same length is

12 : 1

21 : 3

33 : 1

41 : 2

NTA Answer: Option 1(final)

NEET 2023

The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is π2 π2

1– ms–2

2ms–2 8 16 π2 π2

3– ms–2

4ms–2 16 8

NTA Answer: Option 3(final)

NEET 2022

If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is

11 : 2

21 : 1

32 : 1

41 : 2

NTA Answer: Option 4(final)

NEET 2022

Two pendulums of length 121 cm and 100 cm start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is:

211

410

NTA Answer: Option 2(final)

NEET 2021

A spring is stretched by 5 cm by a force 10 N. The time period of the oscillations when a mass of 2 kg is suspended by it is

10.628 s

20.0628 s

36.28 s

43.14 s

NTA Answer: Option 1(final)

NEET 2020

U U U U A U B Sfl U U U 6 Hz fl 25. In a guitar, two strings A and B made of same flS U U – B fl U U material are slightly out of tune and produce beats , flS fl U of frequency 6 Hz. When tension in B is slightly decreased, the beat frequency increases to 7 Hz. 7 Hz – A fl 530 Hz , B If the frequency of A is 530 Hz, the original fl — frequency of B will be :

1537 Hz

2523 Hz

3524 Hz

4536 Hz

NTA Answer: Option 3(final)

NEET 2020

U fl U flS U fl U 29. The phase difference between displacement and U — acceleration of a particle in a simple harmonic motion is :

1zero

2rad 3

3rad 2

4rad 2

NTA Answer: Option 2(final)

How NEET usually asks this

6 recurring patterns from past papers — click to collapse

Two strings/forks slightly out of tune produce beats; find one frequency given the other and beat frequency.

Direct ApplicationEasy

Common distractors

treats beat as sum

Adds frequencies instead of subtracting

Given pendulum length and g, find T. Or given mass change, observe T unchanged.

Direct ApplicationEasy

Common distractors

expects mass dependence

Assumes T depends on bob mass

Open-open pipe (all harmonics) vs closed-end pipe (odd only). Compare frequency ratios.

InterpretationMedium

Common distractors

treats closed pipe like open

Includes even harmonics in closed pipe

Phase between displacement, velocity, acceleration in SHM. v leads x by π/2; a leads x by π.

InterpretationEasy

Common distractors

uses pi 2 instead of pi

Confuses v-x phase with a-x phase

Given spring extension/compression with given force, find k, then find period when given mass m. T = 2pisqrt(m/k).

Multi StepEasy

Common distractors

forgets 2pi factor

Drops 2*pi

uses amplitude in period

Believes T depends on amplitude

Given tension change, find new wave speed on string. v ∝ sqrt(T).

Direct ApplicationEasy

Common distractors

uses linear tension scaling

Treats v ∝ T not sqrt(T)

Sources

NCERT refs: Class 11 Physics Chapter 14, p.12 | Class 11 Physics Chapter 14, p.13

Test yourself on this topic with real past-paper questions:

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Standing Waves Strings Pipes

Lesson

Practice MCQs

Quick recall before you leave

f_n = nv/(2L), n = 1, 2, 3, …

f_n = (2n−1)v/(4L), n = 1, 2, 3, …

v = √(T/μ)

Consecutive closed-pipe resonances differ by 2f₁

Worked Example

Before solving, remember these

Formulas

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Valid when

Exam Traps & Common Mistakes

When it triggers

How to avoid

When it triggers

How to avoid

When it triggers

How to avoid

Correction

Correction

Correction

Past Year Questions

How NEET usually asks this

Common distractors

Common distractors

Common distractors

Common distractors

Common distractors

Common distractors

Sources

Free NEET study resources