Class 11 · Chapter 15

Waves

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Waves Formula Sheet

10 formulas across 3 topics in Waves.

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Updated 2026-07-04 · v1.0.1

Wave Equation & Wave Speed

A travelling wave carries phase at speed v = fλ. On a string the speed is set by tension and mass density; in a gas, by pressure and γ.

y = A sin(ωt − kx); v = fλ = ω/k

Travelling wave equation

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
k	angular wave number (2π/λ)	rad/m	[L⁻¹]
ω	angular frequency (2πf)	rad/s	[T⁻¹]
λ	wavelength	m	[L]

Valid when

(ωt − kx) travels toward +x; (ωt + kx) toward −x
Particle velocity = −v × (slope of the wave curve) — particle and wave velocities are different things

Common mistakes

Maximum PARTICLE speed is ωA, unrelated to the WAVE speed v — a favourite trap

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v = √(T/μ)

Speed of a wave on a string

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	tension in the string	N	[MLT⁻²]
μ	mass per unit length	kg/m	[ML⁻¹]

Valid when

Speed depends on the MEDIUM only — changing frequency changes λ, not v

Common mistakes

A wave passing from a thin to a thick string keeps its FREQUENCY; speed and wavelength both drop

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v = √(γP/ρ) = √(γRT/M) (Laplace)

Speed of sound in a gas

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Newton's isothermal formula √(P/ρ) corrected by Laplace: sound compressions are adiabatic

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
γ	adiabatic exponent of the gas	dimensionless	[M⁰L⁰T⁰]
M	molar mass	kg/mol	[MN⁻¹]

Valid when

v ∝ √T — sound is faster on hot days; INDEPENDENT of pressure at fixed temperature (P/ρ fixed)
≈ 332 m/s in air at 0 °C, rising ≈ 0.61 m/s per °C

Common mistakes

Humidity RAISES the speed of sound — moist air is lighter than dry air

Read the concept →

Worth remembering

Sound in solids > liquids > gases (steel ≈ 15× air) — stiffness wins over density

Superposition, Standing Waves & Beats

Waves add. In phase they reinforce, out of phase they cancel, and two counter-propagating waves lock into a standing pattern of nodes and antinodes.

I = I₁ + I₂ + 2√(I₁I₂) cosφ; I(max)/I(min) = (√I₁ + √I₂)²/(√I₁ − √I₂)²

Interference of two waves

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
φ	phase difference between the waves	rad	[M⁰L⁰T⁰]
I₁, I₂	individual intensities	W/m²	[MT⁻³]

Valid when

Constructive: φ = 2nπ (path difference nλ); destructive: φ = (2n+1)π (path difference (2n+1)λ/2)
Intensity ∝ (amplitude)²

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f(n) = n v/(2L) = (n/2L)√(T/μ), n = 1, 2, 3...

Harmonics of a stretched string (fixed both ends)

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Standing wave condition: L = nλ/2

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
L	length of the vibrating string	m	[L]
n	harmonic number	dimensionless	[M⁰L⁰T⁰]

Valid when

ALL harmonics present; nth harmonic has n loops, (n+1) nodes counting both ends

Common mistakes

Sonometer/tuning: f ∝ √T — to raise a string's pitch by 2× the tension must go up 4×

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Open pipe: f(n) = nv/2L (all n); Closed pipe: f(n) = nv/4L (odd n only)

Organ pipe harmonics

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Open end = antinode, closed end = node

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
L	length of the air column	m	[L]

Valid when

Closed pipe supports ONLY odd harmonics — its fundamental is half the open pipe's
End correction: effective length = L + 0.6r (closed), L + 1.2r (open, both ends)

Common mistakes

Resonance tube (two-position method): λ = 2(L₂ − L₁) — end correction cancels automatically

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p₀ = B k s₀; pressure wave leads displacement by π/2

Pressure wave vs displacement wave

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p = −B (∂s/∂x): differentiating the displacement wave shifts its phase by 90°

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
p₀	pressure amplitude	N/m²	[ML⁻¹T⁻²]
s₀	displacement amplitude	m	[L]
B	bulk modulus of the medium	N/m²	[ML⁻¹T⁻²]
k	angular wave number	rad/m	[L⁻¹]

Valid when

A displacement NODE is a pressure ANTINODE and vice versa — the closed end of a pipe is where pressure swings hardest

Common mistakes

The ear detects PRESSURE variation — that is why a closed end (displacement node) is where sound feels loudest inside a pipe

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f(beat) = |f₁ − f₂|

Beat frequency

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
f₁, f₂	the two close frequencies	Hz	[T⁻¹]

Valid when

Audible only when the difference is small (≲ 10 Hz)

Common mistakes

Tuning-fork logic: loading with wax LOWERS its frequency; filing RAISES it — use the beat change to decide which fork is higher

Read the concept →

Worth remembering

Node spacing in any standing wave is λ/2; node-to-adjacent-antinode is λ/4
Between two adjacent nodes all particles move in phase; across a node the phase flips by π

NEET rotates pipe-harmonic and beats questions annually; JEE Advanced merges beats with Doppler or with slowly changing tension.

Doppler Effect

Relative motion between source and observer reshapes the received frequency. One master formula with a strict sign convention covers every case.

f' = f (v ± v(o)) / (v ∓ v(s))

Doppler effect (sound)

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Moving source compresses wavefronts ahead of it; a moving observer sweeps up wavefronts faster

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
f'	observed frequency	Hz	[T⁻¹]
v	speed of sound in the medium	m/s	[LT⁻¹]
v(o)	observer's speed	m/s	[LT⁻¹]
v(s)	source's speed	m/s	[LT⁻¹]

Valid when

Signs: pick the upper signs when the motion is TOWARD the other party (approach raises f')
Speeds along the line joining source and observer only

Common mistakes

Source moving vs observer moving are NOT symmetric — a source at speed v/2 toward you gives f' = 2f, but an observer at v/2 toward a still source gives only 1.5f
No frequency change when motion is perpendicular to the line of sight (at the instant of crossing)

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Beats heard by the source from its wall echo: Δf = f' − f, with f' via a double Doppler shift

Doppler with reflection (echo problems)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
Δf	beat frequency between direct and reflected sound	Hz	[T⁻¹]

Valid when

Treat the wall as an observer first (receives f₁), then as a stationary source re-emitting f₁

Common mistakes

Forgetting the second shift — reflection problems ALWAYS involve two Doppler steps

Read the concept →

Worth remembering

Doppler changes FREQUENCY, never the wave speed — v belongs to the medium

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