CBSE Class 11 · Physics · NCERT Physics Part-II
Waves: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A mechanical wave requires a…
- In a transverse wave the particle…
- The wave equation y(x, t) = A sin(kx…
- Wave speed v = fλ = ω/k; for a…
- When two or more waves meet, the…
- Standing waves form when two…
- Beats occur when two waves of…
- Formulas and facts
- Travelling wave equation
- k = 2π/λ
- ω = 2π/T = 2πf
- v = fλ = ω/k
- v = √(T/μ)
- v = √(γP/ρ)
- Standing wave on a string fixed at…
- Worked examples
- Speed of a wave on a wire
- Fundamental frequency of a closed pipe
- Beat frequency
- Watch out for
- Confusing wavelength with amplitude
- Using f = n v/(2L) for a closed pipe
- Forgetting that wave speed depends…
- Sign errors in the Doppler formula
- Treating nodes and antinodes in…
What this chapter is about
This chapter introduces you to wave motion, one of the most important ideas in physics. A wave carries energy and information from one place to another without carrying matter along with it. When you throw a stone into a still pond, ripples spread outward, yet the water itself does not travel across the pond; each bit of water simply moves up and down. This distinction between matter transport and energy transport is central to understanding waves.
Key ideas
- A mechanical wave requires a material medium and arises because of the medium's elasticity and inertia; electromagnetic waves need no medium.
- In a transverse wave the particle displacement is perpendicular to the wave velocity; in a longitudinal wave it is parallel.
- The wave equation y(x, t) = A sin(kx − ωt + φ) describes a sinusoidal wave travelling in the +x direction with amplitude A, angular wave number k = 2π/λ, angular frequency ω = 2πf, and initial phase φ.
- Wave speed v = fλ = ω/k; for a string under tension T with linear mass density μ, v = √(T/μ).
- When two or more waves meet, the resultant displacement is the algebraic sum of individual displacements (principle of superposition).
- Standing waves form when two identical waves travelling in opposite directions superpose, producing nodes (zero displacement) and antinodes (maximum displacement) at fixed positions.
- Beats occur when two waves of slightly different frequencies superpose; the beat frequency equals the magnitude of the difference of the two frequencies.
- The Doppler effect is the apparent change in frequency when source or observer or both move relative to the medium.
Formulas and facts to remember
- Travelling wave equation: y = A sin(kx − ωt) for motion in +x; y = A sin(kx + ωt) for motion in −x.
- Wave number and wavelength: k = 2π/λ.
- Angular frequency and period: ω = 2π/T = 2πf.
- Wave speed: v = fλ = ω/k.
- Speed on a stretched string: v = √(T/μ), where T is tension (N) and μ is mass per unit length (kg/m).
- Speed of sound in a gas (Newton-Laplace): v = √(γP/ρ), where γ = Cp/Cv, P is pressure, ρ is density.
- Standing wave on a string fixed at both ends: frequencies fₙ = n v/(2L), n = 1, 2, 3, …; first harmonic (fundamental) n = 1.
- Open pipe (both ends open): fₙ = n v/(2L), n = 1, 2, 3, …; closed pipe (one end closed): fₙ = n v/(4L), n = 1, 3, 5, … (odd harmonics only).
- Beat frequency: f_beat = |f₁ − f₂|.
- Doppler effect for sound (observer and source along the line of propagation): f′ = f (v + vₒ)/(v − vₛ), taking signs so that motion towards each other increases frequency.
Common mistakes
- Confusing wavelength with amplitude → wavelength is the spatial period (distance for one complete cycle), amplitude is the maximum displacement from equilibrium.
- Using f = n v/(2L) for a closed pipe → closed pipes support only odd harmonics; the correct formula is f = n v/(4L) with n = 1, 3, 5, …
- Forgetting that wave speed depends on the medium, not on frequency or amplitude → changing frequency changes wavelength, not speed in a given medium.
- Sign errors in the Doppler formula → decide a positive direction first and apply signs consistently; motion towards increases frequency, motion away decreases it.
- Treating nodes and antinodes in standing waves as travelling points → they are fixed in space.
Quick revision
- A wave transfers energy, not matter, through a medium.
- v = fλ; for a string v = √(T/μ).
- Standing waves have fixed nodes and antinodes; frequencies depend on boundary conditions.
- Beat frequency equals the absolute difference of the two interfering frequencies.
- Doppler effect: approach raises apparent frequency; recession lowers it.
- Open pipe has all harmonics; closed pipe has only odd harmonics.