Formula
Progressive wave equation
y(x,t) = A sin(kx - ωt + φ), where k = 2π/λ (wave number) and ω = 2π/T = 2π ν. v = ω/k = ν λ.
-- NCERT Class 11 Physics, Ch. 14, p. 281Progressive Wave Displacement
8 MCQs6 revision cards9-step worked example
Source: NCERT Oscillations and WavesOfficial key: NTA-verifiedLast updated: 27 Sep 2026
Progressive Wave Displacement, explained for NEET
A progressive (travelling) wave carries energy from one point to another without net displacement of the medium. The displacement relation that describes this wave is the single equation NEET expects you to read, manipulate, and extract parameters from.
The displacement relation. For a sinusoidal wave travelling along the +x direction, the displacement of a medium particle at position x and time t is:
y(x, t) = A sin(ωt − kx + φ₀)
where A is amplitude, ω = 2πf is angular frequency, k = 2π/λ is the angular wave number, and φ₀ is the initial phase constant (NCERT Class 11 Physics Chapter 14, page 283).
Reading the equation — what NEET asks. Given this relation, you should be able to extract: amplitude (coefficient of sine), frequency (f = ω/2π), wavelength (λ = 2π/k), wave speed (v = ω/k = fλ), time period (T = 2π/ω), and direction of propagation (−kx → +x direction; +kx → −x direction).
The sign convention trap. The sign in front of kx decides the wave's travel direction. A term (ωt − kx) means the wave moves in the +x direction. A term (ωt + kx) means −x direction. Aspirants frequently reverse this. The mnemonic: the signs of ωt and kx are opposite for +x travel.
Phase and particle velocity. The argument (ωt − kx + φ₀) is the phase of the wave at position x and time t. Particle velocity is ∂y/∂t, not the wave speed v = ω/k. These are different quantities — particle velocity depends on position and time; wave speed is constant for a given medium.
Watch-out: When the equation is given in cosine form, y = A cos(ωt − kx), the same parameter-extraction rules apply — amplitude, ω, k, v are read identically. Do not assume cosine form changes the wave speed or direction.
Can you answer these Progressive Wave Displacement MCQs?
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
MCQ 1CalculationPractice
A progressive wave has a time period of 0.02 s and travels with speed 300 m/s. What is the angular wave number k (the coefficient of x) in its displacement equation y = A sin(ωt − kx)?
Show answer and why every option is right or wrong
Answer: A. A is correct. Step 1: frequency f = 1/T = 1/0.02 = 50 Hz. Step 2: using v = fλ, wavelength λ = v/f = 300/50 = 6 m. Step 3: k = 2π/λ = 2π/6 = π/3 rad/m ≈ 1.05 rad/m (NCERT Class 11 Physics Chapter 14, page 283, defines k = 2π/λ and v = fλ).
Why B is wrong: B is wrong — 1/6 rad/m omits the 2π factor in k = 2π/λ; k measures radians per metre, not cycles per metre.
Why C is wrong: C is wrong — 6/(2π) rad/m inverts the relation, using k = λ/(2π) instead of k = 2π/λ.
Why D is wrong: D is wrong — 100π rad/m is the angular frequency ω = 2πf (units rad/s), mistakenly reported as k; ω comes from the t-coefficient, k from the x-coefficient, and they are not interchangeable.
MCQ 2Easy RecallPractice
A wave is described by y = 5 sin(4πt + 2πx) cm. In which direction does this wave travel?
Show answer and why every option is right or wrong
Answer: C. C is correct. The displacement relation has the form y = A sin(ωt + kx). When the signs of the ωt and kx terms are the same (both positive here), the wave travels in the −x direction (NCERT Class 11 Physics Chapter 14, page 283).
Why A is wrong: A is wrong because +x propagation requires opposite signs for ωt and kx, i.e. y = A sin(ωt − kx). Here both signs are positive, indicating −x travel.
Why B is wrong: B is wrong because y is the transverse displacement of particles, not the direction of wave propagation. The wave travels along the x-axis.
Why D is wrong: D is wrong because the direction of propagation is fully determined by the sign convention in the wave equation — no additional medium information is needed.
MCQ 3Direct ApplicationPractice
For the progressive wave y = 0.1 sin(200πt − 2πx/0.3) m, the wave speed is:
Show answer and why every option is right or wrong
Answer: A. A is correct. Here ω = 200π rad/s and k = 2π/0.3 rad/m. Wave speed v = ω/k = (200π)/(2π/0.3) = (200π × 0.3)/(2π) = 30 m/s (NCERT Class 11 Physics Chapter 14, page 283).
Why B is wrong: B is wrong because it doubles the correct answer — a common arithmetic slip from misreading k as π/0.3 instead of 2π/0.3.
Why C is wrong: C is wrong because this uses ω directly as the speed. Wave speed is v = ω/k, not ω alone. ω has units rad/s, not m/s.
Why D is wrong: D is wrong because 0.3 m is the wavelength (the denominator in the kx term), not the wave speed.
MCQ 4Direct ApplicationPractice
A progressive wave is represented by y = A sin(kx − ωt). The particle velocity at any point is given by:
Show answer and why every option is right or wrong
Answer: A. A is correct. Particle velocity = ∂y/∂t = ∂/∂t [A sin(kx − ωt)] = −Aω cos(kx − ωt). The chain rule gives a factor of −ω from differentiating the argument with respect to t (NCERT Class 11 Physics Chapter 14, page 283).
Why B is wrong: B is wrong because ω/k is the wave speed (phase velocity), not the particle velocity. Particle velocity varies with position and time; wave speed is constant.
Why C is wrong: C is wrong because differentiating with respect to t brings out −ω, not +k. The factor k appears when differentiating with respect to x (slope of the wave), not with respect to t.
Why D is wrong: D is wrong because this is the displacement y itself, not its time derivative. Particle velocity requires differentiation with respect to time.
MCQ 5Direct ApplicationPractice
The equation of a wave is y = 10 sin(πt/2 − πx/4) mm. The phase difference between two points separated by 2 m at the same instant is:
Show answer and why every option is right or wrong
Answer: B. B is correct. At a fixed time, the phase at position x is (constant − πx/4). The phase difference between two points separated by Δx = 2 m is Δφ = k × Δx = (π/4) × 2 = π/2 rad (NCERT Class 11 Physics Chapter 14, page 283).
Why A is wrong: A is wrong because π/4 is the value of k itself (in rad/m), not the phase difference. Phase difference = k × Δx, so you must multiply k by the separation distance.
Why C is wrong: C is wrong because π rad would require a separation of 4 m (since Δφ = (π/4) × 4 = π), not 2 m.
Why D is wrong: D is wrong because 2π rad corresponds to a full wavelength separation (λ = 2π/k = 8 m), not 2 m.
MCQ 6Easy RecallPractice
Which of the following represents a wave travelling in the +x direction?
Show answer and why every option is right or wrong
Answer: B. B is correct. y = A sin(kx − ωt) = −A sin(ωt − kx). This is the standard form for +x propagation since the phase (kx − ωt) has opposite signs for x and t terms — equivalent to (ωt − kx) with a phase shift of π (NCERT Class 11 Physics Chapter 14, page 283).
Why A is wrong: A is wrong because y = A sin(ωt + kx) has the same sign for both ω and k terms, which represents a wave travelling in the −x direction.
Why C is wrong: C is wrong because this is identical in form to option A (kx + ωt = ωt + kx), which represents −x propagation.
Why D is wrong: D is wrong because a logarithmic function does not represent a physically valid wave — it diverges and does not satisfy the wave equation. Progressive waves require bounded, periodic functions.
MCQ 7CalculationPractice
A transverse wave on a string is described by y(x, t) = 0.05 sin(40πt − 2πx) m. If the linear mass density of the string is 0.01 kg/m, the tension in the string is:
Show answer and why every option is right or wrong
Answer: D. D is correct. Step 1: wave speed v = ω/k = 40π/2π = 20 m/s. Step 2: from v = √(T/μ), tension T = μv² = 0.01 × 20² = 4 N (NCERT Class 11 Physics Chapter 14, pages 281 and 6).
Why A is wrong: A is wrong because 400 N = v² = 20² leaves out the linear mass density; T = μv² = 0.01 × 400 = 4 N.
Why B is wrong: B is wrong because this results from using v = 40 m/s (taking ω/k as 40π/π = 40, misreading k as π instead of 2π). Always identify k as the full coefficient of x.
Why C is wrong: C is wrong because it uses v = ω = 40π directly without dividing by k: μ(40π)² = 16π² N. Wave speed is v = ω/k, not ω.
MCQ 8Easy RecallPractice
Two progressive waves are given by y₁ = A sin(ωt − kx) and y₂ = A cos(ωt − kx). The phase difference between them is:
Show answer and why every option is right or wrong
Answer: C. C is correct. cos(θ) = sin(θ + π/2). Therefore y₂ = A sin(ωt − kx + π/2), which leads by π/2 relative to y₁ (NCERT Class 11 Physics Chapter 14, page 283).
Why A is wrong: A is wrong because sin and cos are not in phase — they are shifted by π/2. They would be in phase only if both were sine (or both cosine) with the same argument.
Why B is wrong: B is wrong because the phase shift between sine and cosine is exactly π/2, not π/4. There is no factor of 1/2 involved in the sin-to-cos conversion.
Why D is wrong: D is wrong because a phase difference of π would make the two waves anti-phase (y₂ = −y₁), which is not the case for sine and cosine.
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Progressive Wave Displacement: quick recall before you leave
How do you solve a Progressive Wave Displacement question? A worked example
- 1
Given
A transverse wave travelling along a string is described by:
y(x, t) = 3.0 × 10⁻² sin(36t − 1.80x) m
where t is in seconds and x in metres. - 2
Required
Find: (a) amplitude, (b) wavelength, (c) frequency, (d) wave speed, (e) direction of propagation.
- 3
Concept
The standard progressive wave equation y = A sin(ωt − kx) allows direct extraction of all wave parameters by comparing coefficients (NCERT Class 11 Physics Chapter 14, page 283).
- 4
Formula
• A = coefficient of sine• ω = coefficient of t → f = ω/(2π), T = 2π/ω• k = coefficient of x → λ = 2π/k• v = ω/k• Sign of kx term determines direction
- 5
Substitution
Comparing y = 3.0 × 10⁻² sin(36t − 1.80x) with y = A sin(ωt − kx):
• A = 3.0 × 10⁻² m• ω = 36 rad/s• k = 1.80 rad/m - 6
Calculation
(a) Amplitude A = 3.0 × 10⁻² m = 3.0 cm
(b) Wavelength λ = 2π/k = 2π/1.80 = 3.49 m
(c) Frequency f = ω/(2π) = 36/(2π) = 5.73 Hz
(d) Wave speed v = ω/k = 36/1.80 = 20 m/s
(e) The term (36t − 1.80x) has opposite signs for t and x → wave travels in +x direction.
Note: 2π is a mathematical constant (exact) and does not limit significant figures. The given values (36, 1.80) each have 3 significant figures, so answers are reported to 3 significant figures. - 7
Final answer
| Parameter | Value |
|-----------|-------|
| Amplitude | 3.00 × 10⁻² m |
| Wavelength | 3.49 m |
| Frequency | 5.73 Hz |
| Wave speed | 20.0 m/s |
| Direction | +x | - 8
Common trap
Confusing ω with k: using the coefficient of t (36) as the wave number and the coefficient of x (1.80) as angular frequency. This gives an incorrect wave speed of 1.80/36 = 0.05 m/s. Always identify the coefficient by which variable (t or x) it multiplies.
- 9
Similar NEET-style question
A progressive wave is given by y = 0.04 sin(600t − 6x) m. Find the wavelength and wave speed. [Answer: λ = 2π/6 ≈ 1.05 m; v = 600/6 = 100 m/s]
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What to remember before solving Progressive Wave Displacement questions
More in Oscillations and Waves: 6 exam traps and mistakes · 10 formulas · 6 question patterns from its other lessons.
Progressive Wave Displacement questions from past NEET papers
No question in our NEET 2020–2025 set targets this topic directly.
Sources
NCERT refs: Class 11 Physics Chapter 14, p.283
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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