Waves is Chapter 14 of NCERT Class 11 Physics (Part II) and the second half of Unit 10, Oscillations and Waves, in both the NEET UG and JEE Main syllabus. These notes cover transverse and longitudinal waves, the progressive wave equation, wave speed, Laplace’s correction, superposition, reflection, standing waves in strings and pipes, and beats, with a free PDF.
Waves Notes for NEET and JEE Main
Section-wise key points from NCERT Chapter 14, formula tables, high-yield facts and common mistakes for 2027 revision.
Waves at a Glance
| Item | Details |
|---|---|
| Syllabus unit | Unit 10: Oscillations and Waves (same unit number in NEET UG and JEE Main) |
| Exams that cover it | NEET UG and JEE Main Paper 1 (B.E./B.Tech.) |
| NCERT textbook | Physics Part II, Class 11, Chapter 14 (rationalised book) |
| Other chapter in the unit | Oscillations (Chapter 13) |
| Syllabus topics | Wave motion; longitudinal and transverse waves; speed of the travelling wave; displacement relation for a progressive wave; principle of superposition; reflection of waves; standing waves in strings and organ pipes; fundamental mode and harmonics; beats |
| Experimental skills (Unit 20) | Speed of sound in air at room temperature using a resonance tube |
| Official chapter | NCERT Chapter 14 |
| Notes PDF | View PDF |
The topic list follows the NEET UG 2026 syllabus released by the National Medical Commission (NMC) on 22 December 2025 and the JEE Main 2026 syllabus published by the National Testing Agency (NTA). Both stay the reference until the 2027 syllabi are out. The full lists are on the NEET syllabus and JEE Main syllabus pages. The Doppler effect is not in the rationalised NCERT chapter or in either syllabus, so it is not covered here.
Transverse and Longitudinal Waves
A mechanical wave carries energy, not matter, through a material medium; elastic forces couple neighbouring oscillating parts of the medium.
| Feature | Transverse wave | Longitudinal wave |
|---|---|---|
| Particle motion | Perpendicular to the direction of propagation | Along the direction of propagation |
| Elastic property needed | Shear modulus | Bulk modulus |
| Media | Solids and strings (not in fluids) | Solids, liquids and gases |
| Example | Wave on a stretched string | Sound in air (compressions and rarefactions) |
- Mechanical waves need a medium and are governed by Newton’s laws; electromagnetic waves need no medium.
- A wave is not motion of matter as a whole: wind moves air from place to place, while a sound wave only compresses and rarefies layers of air.
- A progressive (travelling) wave moves from one point of the medium to another.
Displacement Relation for a Progressive Wave
A sinusoidal wave travelling in the positive x direction is y(x, t) = a sin(kx − ωt + φ); for the negative x direction the phase is kx + ωt + φ.
| Term | Symbol and relation | Meaning |
|---|---|---|
| Amplitude | a | Maximum displacement of an element of the medium |
| Phase | kx − ωt + φ | Describes the state of motion at x and t |
| Wavelength | λ | Distance between two consecutive points in the same phase |
| Angular wave number | k = 2π/λ | SI unit rad m⁻¹ |
| Period | T = 2π/ω | Time for one complete oscillation of an element |
| Frequency | ν = 1/T = ω/2π | Oscillations per second (Hz) |
- In a harmonic progressive wave all particles have the same amplitude but different phases at a given instant.
Speed of a Travelling Wave
Wave speed is v = λ/T = λν = ω/k, and it is set by the elastic and inertial properties of the medium, not by the source’s velocity.
| Wave | Speed formula | Notes |
|---|---|---|
| Transverse wave on a string | v = √(T/µ) | T = tension, µ = linear mass density; independent of frequency |
| Longitudinal wave in a fluid | v = √(B/ρ) | B = bulk modulus, ρ = density |
| Longitudinal wave in a solid bar | v = √(Y/ρ) | Y = Young’s modulus |
| Sound in a gas: Newton’s formula | v = √(P/ρ) | Assumes isothermal changes; gives about 280 m s⁻¹ in air at STP |
| Sound in a gas: Laplace correction | v = √(γP/ρ) | Changes are adiabatic; with γ = 7/5 gives 331.3 m s⁻¹ in air at STP |
- Newton’s value is about 15% lower than the measured 331 m s⁻¹, because sound compressions are too fast for heat to flow; the process is adiabatic, not isothermal.
- Sound travels faster in solids and liquids because their elastic moduli are much higher.
- At a fixed temperature P/ρ is constant, so the speed of sound in a gas does not depend on pressure; it rises with temperature and with humidity.
Principle of Superposition and Interference
When waves overlap, the resultant displacement of each element is the algebraic sum of the displacements due to each wave: y = Σ f<sub>i</sub>(x − vt).
- Wave pulses keep their identities after crossing each other.
- Two waves of equal amplitude a and frequency travelling in the same direction with phase difference φ combine into y = 2a cos(φ/2) sin(kx − ωt + φ/2).
- Resultant amplitude is 2a cos(φ/2).
- Constructive interference: φ = 0 or an integral multiple of 2π (amplitude 2a). Destructive interference: φ = π (amplitude zero).
Reflection of Waves
A wave reflected at a rigid boundary returns with a phase reversal; at an open boundary it reflects without a phase change.
| Boundary | Reflected wave for y<sub>i</sub> = a sin(kx − ωt) | Phase change |
|---|---|---|
| Rigid boundary or closed end | y<sub>r</sub> = −a sin(kx + ωt) | π (phase reversal) |
| Open boundary | y<sub>r</sub> = a sin(kx + ωt) | None |
Standing Waves and Normal Modes
Two identical waves travelling in opposite directions produce a standing wave, y(x, t) = [2a sin kx] cos ωt, with fixed nodes and antinodes.
- Nodes: points of zero displacement; antinodes: points of maximum displacement.
- Distance between consecutive nodes or consecutive antinodes = λ/2.
- In a stationary wave, all particles between two nodes have the same phase but different amplitudes.
- Each allowed frequency of a string or air column is a normal mode and a resonant frequency of the system.
| System | Allowed frequencies | Fundamental | Harmonics present |
|---|---|---|---|
| String fixed at both ends (length L) | ν = nv/2L, n = 1, 2, 3, … | v/2L | All harmonics |
| Pipe closed at one end | ν = (n + ½)v/2L, n = 0, 1, 2, … | v/4L | Odd harmonics only |
| Pipe open at both ends | ν = nv/2L, n = 1, 2, 3, … | v/2L | All harmonics (each end is an antinode) |
- The lowest frequency is the fundamental mode or first harmonic; the mode with n = 2 on a string is the second harmonic.
- NCERT example: a 30.0 cm pipe open at both ends (v = 330 m s⁻¹) has a fundamental of 550 Hz, so a 1.1 kHz source excites its second harmonic; with one end closed it would not resonate, since only odd harmonics occur.
Beats
Beats are the periodic waxing and waning of loudness heard when two waves of slightly different frequencies and comparable amplitudes are superposed.
- Beat frequency: ν<sub>beat</sub> = ν₁ − ν₂ (the difference of the two frequencies).
- Musicians tune instruments against each other until no beats are heard.
Most-asked Facts for NEET and JEE Main: Waves
Revise these one-line facts from the NCERT chapter just before the exam.
- v = λν = ω/k; k = 2π/λ; ω = 2πν.
- Speed on a stretched string: v = √(T/µ).
- Speed of sound in a gas: v = √(γP/ρ) (Laplace correction).
- Newton’s isothermal formula gives about 280 m s⁻¹ for air at STP; Laplace’s gives 331.3 m s⁻¹.
- Transverse waves need shear modulus; longitudinal waves need bulk modulus.
- Reflection at a rigid end: phase change π; at an open end: no phase change.
- Node to node distance = λ/2.
- String fixed at both ends and pipe open at both ends: all harmonics, fundamental v/2L.
- Pipe closed at one end: odd harmonics only, fundamental v/4L.
- Beat frequency = ν₁ − ν₂.
- A wave transfers energy, not matter.
Common Mistakes to Avoid
Watch for these mix-ups when you answer questions on this chapter.
- Saying the speed of sound in air changes with pressure at constant temperature. P/ρ stays constant, so it does not.
- Using Newton’s formula v = √(P/ρ) for sound in air. The correct one includes γ (Laplace correction).
- Giving a closed pipe even harmonics. A pipe closed at one end has only odd harmonics.
- Taking node-to-node spacing as λ. It is λ/2; node to adjacent antinode is λ/4.
- Thinking wave speed depends on the source’s speed or on frequency. It depends on the medium.
- Confusing particle velocity (∂y/∂t) with wave speed v.
- Expecting transverse waves in the bulk of a gas or liquid. Fluids have no shear modulus, so they carry only longitudinal waves.
Waves Notes PDF and Related Pages
Download the AglaSem Class 11 Physics notes for Oscillations and Waves (key notes and important questions with answers) free: View PDF. The same PDF covers Oscillations. It also has older topics, such as the Doppler effect, that are not in the rationalised NCERT chapter.
- All chapters: Physics notes for NEET and JEE Main
- Full topic list: NEET syllabus and JEE Main syllabus
- Practice: NEET question papers and JEE Main question papers
- Source text: NCERT Chapter 14 (NCERT, Class 11 Physics Part II)
Physics Notes for NEET and JEE Main: All Chapters
- Units and Measurement
- Motion in a Straight Line
- Motion in a Plane
- Laws of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermodynamics
- Kinetic Theory of Gases
- Oscillations
- Waves
- Thermal Properties of Matter
- Electric Charges and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Electromagnetic Waves
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics
All chapters with key points and notes PDFs are on the Physics notes for NEET and JEE Main page.