Fundamental Mode Harmonics

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 trap that costs marks on harmonics questions: confusing which pipe supports which harmonics.

A string fixed at both ends or a pipe open at both ends supports all harmonics. The allowed frequencies are fₙ = nv/(2L), where n = 1, 2, 3, … The n = 1 mode is the fundamental (first harmonic). The n = 2 mode is the second harmonic (first overtone), and so on. Every integer multiple of the fundamental is present.

A pipe closed at one end behaves differently. The closed end forces a displacement node; the open end has an antinode. This boundary asymmetry permits only odd harmonics: fₙ = (2n−1)v/(4L), giving frequencies f, 3f, 5f, … No even harmonics exist. The fundamental frequency of a closed pipe is exactly half that of an open pipe of the same length.

For a string, the wave speed is v = √(T/μ), where T is tension and μ is linear mass density. So the fundamental of a string fixed at both ends is f₁ = (1/2L)√(T/μ). Changing tension or mass density shifts all harmonics proportionally.

The high-frequency trap (NCERT Class 11 Physics Chapter 14, pages 12–13): When a question describes a "resonance tube closed at one end" and asks for the next resonance after the fundamental, a common wrong answer picks 2f (the second harmonic). The correct answer is 3f — the next allowed mode in a closed pipe is the third harmonic. NEET has tested this distinction repeatedly (2023, 2025 papers).

Watch for the phrasing: "overtone" versus "harmonic." The first overtone of a closed pipe is the third harmonic (3f), not the second.

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 1Direct ApplicationPractice

A string of length 1.0 m is fixed at both ends. If the wave speed on the string is 200 m/s, the frequency of the third harmonic is:

MCQ 2Direct ApplicationPractice

A pipe open at both ends has a fundamental frequency of 300 Hz. If one end is now closed, the new fundamental frequency is:

MCQ 3Easy RecallPractice

Which of the following harmonics is NOT possible in a pipe closed at one end?

MCQ 4CalculationPractice

The first overtone of a closed pipe has the same frequency as the third harmonic of an open pipe. If the length of the open pipe is 30 cm, the length of the closed pipe is:

MCQ 5Easy RecallPractice

A string fixed at both ends vibrates in its second harmonic. The number of nodes (including the endpoints) is:

MCQ 6Easy RecallPractice

The ratio of the fundamental frequency of a pipe open at both ends to that of a pipe closed at one end, both of the same length, is:

MCQ 7Direct ApplicationPractice

A wire of length 1.0 m is stretched between two fixed supports with tension 100 N. If the mass per unit length of the wire is 0.01 kg/m, the fundamental frequency of vibration is:

MCQ 8Direct ApplicationPractice

An organ pipe closed at one end resonates at its fundamental frequency of 250 Hz. Its next higher resonant frequency is:

Quick recall before you leave

Worked Example

Pattern: Open-open vs closed-end pipe harmonic comparison (based on NEET pattern: pipe harmonics).

1
Given
- L = 0.50 m - v = 340 m/s
2
Required
(a) First three harmonics of open-open pipe. (b) First three allowed modes of closed-end pipe.
3
Concept
Open pipe: all harmonics, fₙ = nv/(2L). Closed pipe: odd harmonics only, fₙ = (2n−1)v/(4L) (NCERT Class 11 Physics Chapter 14, pages 12–13).
4
Formula
- Open: fₙ = nv/(2L) - Closed: fₙ = (2n−1)v/(4L)
5
Substitution
(a) Open pipe: - f₁ = 1 × 340 / (2 × 0.50) = 340/1.0 - f₂ = 2 × 340 / (2 × 0.50) = 680/1.0 - f₃ = 3 × 340 / (2 × 0.50) = 1020/1.0 (b) Closed pipe: - n = 1: f = (2×1 − 1) × 340 / (4 × 0.50) = 1 × 340/2.0 - n = 2: f = (2×2 − 1) × 340 / (4 × 0.50) = 3 × 340/2.0 - n = 3: f = (2×3 − 1) × 340 / (4 × 0.50) = 5 × 340/2.0
6
Calculation
(a) Open pipe: f₁ = 340 Hz, f₂ = 680 Hz, f₃ = 1020 Hz (ratio 1 : 2 : 3). (b) Closed pipe: f₁ = 170 Hz, f₃ = 510 Hz, f₅ = 850 Hz (ratio 1 : 3 : 5). Note: The integers 1, 2, 3, 5 are exact counting numbers and do not limit significant figures. The given values L = 0.50 m (2 sig figs) and v = 340 m/s (3 sig figs as conventionally treated) govern precision. Results are reported to 2–3 significant figures accordingly.
7
Final answer
| Mode | Open pipe | Closed pipe | |------|-----------|-------------| | Fundamental | 340 Hz | 170 Hz | | Next allowed | 680 Hz (2nd harmonic) | 510 Hz (3rd harmonic) | | Third allowed | 1020 Hz (3rd harmonic) | 850 Hz (5th harmonic) | Key observation: The closed pipe's fundamental is exactly half the open pipe's. The closed pipe skips all even harmonics.
8
Common trap
For part (b), a common error is writing the second mode of the closed pipe as 340 Hz (2 × 170), treating it like an open pipe. The second harmonic does not exist in a closed pipe — the next mode jumps to the third harmonic (3 × 170 = 510 Hz).
9
Similar NEET-style question
A resonance tube closed at one end has a fundamental frequency of 200 Hz. What are the frequencies of the next two resonances? *(Answer: 600 Hz and 1000 Hz — odd multiples 3f and 5f.)* ---

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)

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

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Fundamental Mode Harmonics

Lesson

Practice MCQs

Quick recall before you leave

Standing wave on fixed-fixed string

Standing wave in open-open pipe

Standing wave in closed-end pipe

Wave speed on string

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

Free NEET study resources