Board exam focus — Waves (CBSE & HBSE)

Waves studies the propagation of disturbances that carry energy without transporting matter. The chapter classifies waves as transverse or longitudinal, develops the basic wave terms (wavelength, frequency, amplitude, speed) and the progressive wave equation y = A sin(kx − ωt). It treats the speed of sound in different media, the principle of superposition leading to interference, reflection, standing waves and normal modes in strings and organ pipes, beats, and finally the Doppler effect.

Types of Waves, Wave Terms and the Wave Equation

Transverse and Longitudinal Waves

A wave is a disturbance that travels through a medium transferring energy and momentum without any net transport of the medium itself.

Trap: Sound in air is longitudinal; transverse mechanical waves cannot travel through the bulk of a gas or liquid because these media lack shear rigidity.

Basic Wave Terms

• Wavelength (λ): distance between two consecutive points in the same phase (e.g. two crests).
• Frequency (ν): number of oscillations per second; ν = 1/T.
• Amplitude (A): maximum displacement of a particle from its mean position.
• Wave number: k = 2π/λ (rad m⁻¹); angular frequency ω = 2πν (rad s⁻¹).
• Time period (T): time for one complete oscillation.

Speed of a Wave

The wave speed relates wavelength and frequency:

$$v = \nu\lambda = \frac{\omega}{k}$$

The speed depends only on the properties of the medium, not on frequency or amplitude.

The Progressive Wave Equation

A harmonic wave travelling in the +x direction is:

$$y(x,t) = A\,\sin(kx - \omega t)$$

where k = 2π/λ is the angular wave number and ω = 2π/T is the angular frequency. A wave moving in the −x direction is written y = A sin(kx + ωt).

• The argument (kx − ωt) is the phase of the wave.
• The phase difference between two points separated by Δx is Δφ = kΔx = (2π/λ)Δx.
• The path difference Δx corresponds to phase difference Δφ = (2π/λ)Δx.

Speed of Sound, Superposition, Reflection and Standing Waves

Speed of a Wave in Different Media

On a stretched string (transverse), the speed depends on tension T and linear mass density μ:

$$v = \sqrt{\frac{T}{\mu}}$$

Speed of sound (longitudinal) in a medium is given by Newton–Laplace:

$$v = \sqrt{\frac{E}{\rho}}$$

where E is the relevant modulus of elasticity and ρ the density. For a gas, Laplace's correction (adiabatic process) gives:

$$v = \sqrt{\frac{\gamma P}{\rho}}$$

• Sound travels fastest in solids, slower in liquids, slowest in gases.
• In a gas, v ∝ √T (Kelvin) and is independent of pressure (at constant temperature).

Trap: Newton's formula v = √(P/ρ) gave a value ~15% too low; Laplace corrected it by treating sound propagation as adiabatic, not isothermal, introducing the factor γ.

Principle of Superposition

When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements:

$$y = y_1 + y_2 + \dots$$

This principle leads to interference, standing waves and beats.

Reflection of Waves

• Reflection at a rigid (fixed) boundary causes a phase change of π (crest reflects as trough).
• Reflection at a free (open) boundary causes no phase change.

Standing (Stationary) Waves and Normal Modes

Two identical waves travelling in opposite directions superpose to form a standing wave:

$$y = 2A\sin(kx)\cos(\omega t)$$

• Nodes are points of zero amplitude (fixed); antinodes are points of maximum amplitude. Distance between consecutive nodes = λ/2.

Stretched string fixed at both ends: normal-mode frequencies are

$$\nu_n = \frac{n v}{2L} = \frac{n}{2L}\sqrt{\frac{T}{\mu}}, \quad n = 1, 2, 3, \dots$$

All harmonics (1st, 2nd, 3rd...) are present.

Organ pipes:

Beats and the Doppler Effect

Beats

When two waves of slightly different frequencies (ν₁ and ν₂) superpose, the resultant intensity rises and falls periodically. These periodic variations of loudness are called beats.

$$\text{Beat frequency} = |\nu_1 - \nu_2|$$

• Beats are clearly heard only when |ν₁ − ν₂| ≤ ~10 Hz (limit of human hearing resolution).
• Beats are used to tune musical instruments and to measure an unknown frequency.

Trap: If a known frequency ν is sounded with an unknown frequency producing n beats per second, the unknown frequency is either (ν + n) or (ν − n). Loading one source (e.g. adding wax to a tuning fork) and observing whether beats increase or decrease resolves the ambiguity.

Doppler Effect

The Doppler effect is the apparent change in the observed frequency of a wave due to relative motion between the source and the observer.

The general formula for the apparent frequency of sound is:

$$\nu' = \nu\left(\frac{v \pm v_o}{v \mp v_s}\right)$$

where v is the speed of sound, v_o the observer's speed, and v_s the source's speed.

Sign convention (approach raises pitch, recession lowers it):

• For sound, source motion and observer motion give different results (the medium provides a frame of reference).
• Applications: radar speed guns, astronomical redshift, SONAR, medical ultrasound (echocardiography).

Trap: The Doppler formulas above hold only when speeds are along the line joining source and observer and are much less than the speed of sound. The asymmetry between moving source and moving observer is a common exam point.

Frequently asked questions

Are these Waves notes free?

Yes — the Waves notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Waves notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Waves chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Waves.