CBSE Class 11 Physics Revision Notes Chapter 14 Waves

Waves transfer energy and information through a disturbance while the particles of the medium oscillate around their mean positions.

In Class 11 Physics, wave motion connects oscillations with wavelength, frequency, speed, reflection, superposition and beats.

When one particle in an elastic medium is disturbed, its motion affects nearby particles through restoring forces. The disturbance then travels across the medium, although the medium itself does not move from one place to another as a whole.

These CBSE Class 11 Physics Revision Notes Chapter 14 follow the 2026–27 NCERT sequence. They cover wave types, the progressive-wave equation, travelling-wave speed, superposition, reflection, standing waves and beats.

Key Takeaways

• Wave motion: A wave transfers energy without producing permanent transport of matter.
• Travelling-wave speed: v = νλ = ω/k.
• Standing waves: Nodes have zero displacement, while antinodes have maximum displacement.
• Beats: Two nearby frequencies produce periodic changes in loudness with beat frequency |ν1 − ν2|.

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Access Class 11 Physics Chapter 14 Waves Notes in 30 Minutes

These Waves Class 11 Short Notes divide the chapter into connected wave properties and formulas.

The sequence makes these Class 11 Physics Chapter 14 Waves Notes suitable for quick conceptual and numerical revision.

Waves and Transfer of Energy

A wave is a disturbance that travels through space or a material medium.

During wave propagation:

• Energy travels from one region to another.
• Information may also be transmitted.
• Particles of the medium oscillate about their equilibrium positions.
• There is no permanent flow of matter with the wave.

A cork floating on water moves up and down as ripples pass, but it does not travel outward with each ripple. This illustrates the central idea in Wave Motion Class 11 Notes.

Mechanical, Electromagnetic and Matter Waves

Waves may be broadly classified according to whether they require a medium.

The present chapter mainly studies mechanical waves.

Mechanical waves depend on:

• Elastic properties of the medium
• Inertial properties of the medium
• Interaction among neighbouring particles

Pulse, Periodic Wave and Travelling Wave

A pulse is a single short disturbance travelling through a medium.

A periodic wave is produced when the source oscillates repeatedly.

A travelling wave or progressive wave moves continuously from one part of the medium to another.

Each particle oscillates locally while the wave pattern progresses through the medium.

Transverse and Longitudinal Waves

Mechanical waves are classified according to the direction in which the particles oscillate relative to the direction of propagation.

Transverse Waves

In transverse waves, the particles of the medium oscillate perpendicular to the direction of propagation.

Examples include:

• Waves on a stretched string
• A sideways pulse in a spring
• Some seismic waves

Transverse waves have:

• Crests
• Troughs
• Particle motion normal to wave direction

A transverse mechanical wave requires a medium that can sustain shearing stress. Therefore, such waves can propagate through solids and stretched strings but generally not through fluids in bulk.

Longitudinal Waves

In longitudinal waves, particles oscillate parallel to the direction of wave propagation.

Examples include:

• Sound waves in air
• Compression waves in a spring
• Longitudinal waves in a solid rod

Longitudinal waves consist of:

• Compressions: Regions of higher pressure and density
• Rarefactions: Regions of lower pressure and density

Solids, liquids and gases can support longitudinal waves because they can sustain compression.

Transverse and Longitudinal Wave Differences

Water-surface waves involve both vertical and horizontal particle motion. They are therefore more complex than a purely transverse wave.

Displacement Relation in a Progressive Wave

A mathematical description of a progressive wave must depend on both position and time.

The function should describe:

• The shape of the wave at any instant
• The motion of a particle at any fixed position

Equation of a Wave Moving Along Positive x-Direction

A sinusoidal wave moving in the positive x-direction is represented by:

y(x,t) = a sin(kx − ωt + φ)

This wave equation contains:

• y(x,t) = displacement at position x and time t
• a = amplitude
• k = angular wave number
• ω = angular frequency
• φ = initial phase

The combination kx − ωt remains constant for a point of fixed phase moving in the positive x-direction.

Equation of a Wave Moving Along Negative x-Direction

A wave travelling in the negative x-direction is written as:

y(x,t) = a sin(kx + ωt + φ)

The sign before ωt determines the direction:

This sign rule is especially useful when identifying wave direction from a given equation.

Amplitude and Phase

The amplitude a is the maximum displacement of a particle from its equilibrium position.

The quantity:

kx − ωt + φ

is the phase of the wave.

The amplitude and phase serve different purposes:

• Amplitude determines maximum displacement.
• Phase determines the state of oscillation at a given position and time.
• Initial phase φ gives the phase at x = 0 and t = 0.

Two particles are in the same phase when their displacements and directions of motion are identical.

Wavelength and Angular Wave Number

The wavelength λ is the minimum distance between two points in the same phase.

It may be measured between:

• Two consecutive crests
• Two consecutive troughs
• Two consecutive compressions
• Two consecutive rarefactions

The angular wave number is:

k = 2π/λ

Therefore:

λ = 2π/k

The SI unit of k is rad m⁻¹.

The phase difference between two points separated by distance Δx is:

Δφ = kΔx

Therefore:

Δφ = 2πΔx/λ

Time Period, Frequency and Angular Frequency

The time period T is the time taken by a particle to complete one oscillation.

Frequency ν is the number of oscillations per second:

ν = 1/T

Angular frequency is:

ω = 2πν

Therefore:

ω = 2π/T

The SI unit of frequency is hertz, while angular frequency is measured in rad s⁻¹.

Speed of a Travelling Wave

Wave speed describes how fast a point of fixed phase, such as a crest, travels through the medium.

General Wave-Speed Relation

For:

kx − ωt = constant

Differentiating with respect to time:

k(dx/dt) − ω = 0

Therefore:

v = ω/k

Using:

ω = 2πν

and:

k = 2π/λ

We get:

v = νλ

Thus:

Wave speed = frequency × wavelength

The source fixes the frequency, while the medium determines the wave speed. Once v and ν are known, wavelength is fixed.

Wave Speed and Particle Speed

Wave speed and particle speed are different quantities.

A particle can momentarily have zero velocity while the wave continues moving through it.

Speed of Waves in Different Media

The speed of a mechanical wave depends on the restoring property and inertia of the medium.

In general:

Wave speed ∝ √(elastic property/inertial property)

Speed of Transverse Waves on a Stretched String

For a stretched string:

v = √(F/μ)

Here:

• F = tension in the string
• μ = linear mass density

The linear mass density is:

μ = m/L

Therefore:

v = √(FL/m)

The speed:

• Increases when tension increases
• Decreases when linear mass density increases
• Does not directly depend on the source frequency

Once the speed is fixed, a change in frequency produces a corresponding change in wavelength.

Speed of Longitudinal Waves

For a longitudinal wave in a fluid:

v = √(B/ρ)

Here:

• B = bulk modulus
• ρ = mass density

For a longitudinal wave in a solid rod:

v = √(Y/ρ)

Here, Y is Young’s modulus.

Solids generally transmit sound faster than gases because their elastic moduli are much larger.

Speed of Sound in a Gas

Sound is a longitudinal wave that travels through compressions and rarefactions.

For an ideal gas, the speed of sound is:

v = √(γP/ρ)

Here:

• γ = Cp/Cv
• P = gas pressure
• ρ = gas density

Using the ideal-gas relation:

P = ρRT/M

We get:

v = √(γRT/M)

Therefore, sound speed in an ideal gas:

• Increases with the square root of absolute temperature
• Decreases with the square root of molar mass

Newton’s Formula and Laplace Correction

Newton initially assumed that compressions and rarefactions during sound propagation were isothermal.

He obtained:

v = √(P/ρ)

This result was lower than the measured speed.

Laplace explained that pressure changes occur too rapidly for heat exchange. Therefore, the changes are adiabatic.

For an adiabatic process:

Bulk modulus = γP

Hence:

v = √(γP/ρ)

The inclusion of γ is known as the Laplace correction.

Principle of Superposition of Waves

When two or more waves overlap, each wave continues to travel independently.

Resultant Displacement

The principle of superposition states that the resultant displacement equals the algebraic sum of individual displacements.

For two waves:

y(x,t) = y1(x,t) + y2(x,t)

For several waves:

y = y1 + y2 + y3 + ...

After overlapping, the waves continue with their original forms if the medium behaves linearly.

Interference of Two Waves

Consider two waves of equal amplitude and frequency:

y1 = a sin(kx − ωt)

y2 = a sin(kx − ωt + φ)

Their resultant is:

y = 2a cos(φ/2) sin(kx − ωt + φ/2)

The resultant amplitude is:

A = 2a cos(φ/2)

For constructive interference:

φ = 0, 2π, 4π, ...

A = 2a

For destructive interference:

φ = π, 3π, 5π, ...

A = 0

Interference results from the superposition of coherent waves.

Reflection of Waves

The reflection of waves occurs when a travelling wave reaches a boundary and returns through the original medium.

The reflected wave has the same frequency as the incident wave.

Reflection from a Fixed End

At a fixed end, the displacement must remain zero.

The reflected pulse is inverted.

Therefore:

• A crest returns as a trough.
• A trough returns as a crest.
• The reflected wave undergoes a phase change of π.

If the incident wave is:

yi = a sin(kx − ωt)

The reflected wave may be written as:

yr = −a sin(kx + ωt)

Reflection from a Free End

At a free end, the end is allowed to move.

The reflected pulse is not inverted.

Therefore:

• A crest returns as a crest.
• A trough returns as a trough.
• There is no phase reversal.

The reflected wave may be written as:

yr = a sin(kx + ωt)

Standing Waves and Normal Modes

Standing waves form when two waves of the same amplitude, frequency and speed travel in opposite directions and superpose.

The wave pattern does not travel through the medium.

Formation of Standing Waves

Consider:

y1 = a sin(kx − ωt)

y2 = a sin(kx + ωt)

Their resultant is:

y = 2a sin(kx) cos(ωt)

The position-dependent amplitude is:

A(x) = 2a sin(kx)

Different particles oscillate with different amplitudes.

A standing wave has no net transfer of energy from one end to the other.

Nodes and Antinodes

Nodes and antinodes are fixed positions in a standing-wave pattern.

At a node:

sin(kx) = 0

Therefore:

kx = nπ

x = nλ/2

The displacement is always zero.

At an antinode:

|sin(kx)| = 1

Therefore:

kx = (2n + 1)π/2

x = (2n + 1)λ/4

The amplitude is maximum and equal to 2a.

Important separations are:

All particles between two successive nodes oscillate in the same phase. Particles in neighbouring segments oscillate in opposite phases.

Standing Waves on a Stretched String

For a string fixed at both ends, both ends must be nodes.

If string length is L:

L = nλn/2

Therefore:

λn = 2L/n

The allowed frequencies are:

νn = nv/2L

Here:

n = 1, 2, 3, ...

The fundamental frequency is:

ν1 = v/2L

The higher frequencies are:

ν2 = 2ν1

ν3 = 3ν1

These are called harmonics.

Using v = √(F/μ):

νn = n/(2L) √(F/μ)

Standing Waves in Air Columns

In an open organ pipe, both ends behave approximately as displacement antinodes.

For an open pipe:

νn = nv/2L

All harmonics are present.

In a pipe closed at one end:

• The closed end is a displacement node.
• The open end is a displacement antinode.

The allowed frequencies are:

νn = (2n − 1)v/4L

Here:

n = 1, 2, 3, ...

Only odd harmonics are present.

The fundamental frequency is:

ν1 = v/4L

Beats

Beats are periodic variations in loudness produced when two sound waves of slightly different frequencies superpose.

Formation of Beats

Consider:

y1 = a sin(2πν1t)

y2 = a sin(2πν2t)

Their resultant is:

y = 2a cos[π(ν1 − ν2)t] sin[π(ν1 + ν2)t]

The sound has an approximate frequency:

νaverage = (ν1 + ν2)/2

Its amplitude varies slowly with time:

A = 2a cos[π(ν1 − ν2)t]

The sound becomes alternately loud and faint.

Beat Frequency and Beat Period

The beat frequency is:

νbeat = |ν1 − ν2|

It gives the number of beats heard per second.

The beat period is:

Tbeat = 1/νbeat

Therefore:

Tbeat = 1/|ν1 − ν2|

Beats are useful for comparing the frequencies of two sound sources. An instrument is tuned by adjusting its frequency until the beats disappear.

Waves Formula Sheet

These Waves Class 11 Formulas bring together the main progressive-wave, reflection, standing-wave and beat relations.

These Physics Class 11 Chapter 14 Notes connect each equation with the corresponding wave behaviour.

Important Differences for Quick Revision

Useful Links for Class 11 Physics