CBSE Class 11 Physics Chapter 15 Notes: Waves

Chapter Introduction

Drop a stone in a pond, and ripples spread outward. Pluck a guitar string, and sound fills the room. Speak into a phone, and your voice reaches someone thousands of kilometres away. All of these are waves — disturbances that carry energy from one point to another without transporting matter. Chapter 15 of NCERT Physics for Class 11 covers the mathematics of wave motion, the physics of transverse and longitudinal waves, the principle of superposition, standing waves, beats, and the Doppler effect. These concepts are essential for understanding optics, acoustics, communication systems, and modern physics.

Key Definitions

Detailed Concepts

Transverse and Longitudinal Waves

Displacement Relation in a Progressive Wave

For a wave travelling in the +x direction:

y(x, t) = A sin(kx – ωt + φ)

where A = amplitude, k = 2π/λ = wave number, ω = 2πν = angular frequency, φ = phase constant.

For a wave travelling in the –x direction:

y(x, t) = A sin(kx + ωt + φ)

The argument (kx – ωt + φ) is the phase of the wave. At a fixed point x, the phase changes with time (oscillation). At a fixed time t, the phase changes with x (spatial periodicity).

Wave Speed

v = λν = ω/k

The speed depends on the medium properties, not on frequency (for non-dispersive media):

Speed of sound in air at 0 °C ≈ 331 m/s; increases by ≈ 0.6 m/s per °C rise.

v_T = v₀ + 0.61 T_C (where T_C is temperature in °C)

Principle of Superposition

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

y = y₁ + y₂ + y₃ + ...

This principle is valid for linear media (small amplitudes). It explains:

• Interference: Constructive (in phase) and destructive (out of phase)
• Beats: Two waves with close frequencies
• Standing waves: Two identical waves in opposite directions

Reflection of Waves

When a wave meets a boundary:

Beats

When two sound waves of slightly different frequencies ν₁ and ν₂ are heard simultaneously:

The intensity varies at the beat frequency. The number of beats per second equals the difference of the two frequencies.

Standing Waves

When two identical waves travel in opposite directions:

y₁ = A sin(kx – ωt) and y₂ = A sin(kx + ωt)

By superposition: y = 2A sin(kx) cos(ωt)

This is a standing wave — the spatial pattern (sin kx) and temporal variation (cos ωt) are separated.

• Nodes: sin(kx) = 0 → kx = nπ → x = nλ/2 (spacing between nodes = λ/2)
• Antinodes: sin(kx) = ±1 → kx = (n + ½)π → x = (2n + 1)λ/4

Standing Waves on Strings (Fixed at Both Ends)

Both ends are nodes. Allowed wavelengths: λ_n = 2L/n (n = 1, 2, 3, ...)

Frequencies: ν_n = nv/(2L) = nν₁ (n = 1, 2, 3, ...)

All harmonics are present in a string fixed at both ends.

Standing Waves in Organ Pipes

Open pipe (open at both ends — antinodes at both ends):

Same pattern as string: ν_n = nv/(2L), n = 1, 2, 3, ... → All harmonics present

Closed pipe (closed at one end, open at other — node at closed end, antinode at open end):

Allowed wavelengths: λ_n = 4L/n (n = 1, 3, 5, ... — odd numbers only)

Frequencies: ν_n = nv/(4L), n = 1, 3, 5, ...

Only odd harmonics are present in a closed pipe. This is why a closed pipe has a different timbre than an open pipe.

Doppler Effect

When there is relative motion between source and observer, the observed frequency differs from the actual frequency.

General formula:

ν' = ν × (v + v_o)/(v + v_s)

where:

• v = speed of sound in the medium
• v_o = speed of observer (positive if moving toward the source)
• v_s = speed of source (positive if moving away from the observer)

Special cases:

• Source moving toward stationary observer: ν' = νv/(v – v_s)
• Source moving away from stationary observer: ν' = νv/(v + v_s)
• Observer moving toward stationary source: ν' = ν(v + v_o)/v
• Observer moving away from stationary source: ν' = ν(v – v_o)/v

Limitations: Valid only when v_s and v_o << v (speeds much less than sound). For speeds comparable to v, relativistic corrections are needed.

Important Derivations

1. Wave Equation from Newton's Law (String)

Consider a small element dx of a string under tension T. The net transverse force on the element:

By Newton's second law: F_y = μ dx (∂²y/∂t²)

This is the wave equation with v = √(T/μ)

2. Standing Wave from Superposition

y₁ = A sin(kx – ωt), y₂ = A sin(kx + ωt)

y = y₁ + y₂ = 2A sin(kx) cos(ωt)

Using sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

Nodes: sin(kx) = 0 → kx = nπ → x = nλ/2

3. Doppler Effect (Source Moving)

When source moves toward observer at speed v_s:

In time T_s (one period), source moves v_s T_s toward observer.

Apparent wavelength: λ' = (v – v_s)T_s = (v – v_s)/ν

Apparent frequency: ν' = v/λ' = vν/(v – v_s)

Common Mistakes

1. Confusing transverse and longitudinal waves in different media: Transverse mechanical waves cannot propagate in gases (gases cannot sustain shear). Sound in air is always longitudinal. 2. Wrong sign in Doppler formula: Memorise the sign convention — velocities toward each other increase the observed frequency. When in doubt, use the general formula with correct signs. 3. Mixing up nodes and antinodes: Nodes have zero displacement; antinodes have maximum displacement. In a string, fixed ends are nodes; open ends of pipes are antinodes. 4. Forgetting odd-only harmonics in closed pipe: A closed pipe supports only n = 1, 3, 5, 7, ... — NOT n = 2, 4, 6. This is one of the most commonly tested distinctions. 5. Using wave speed formula for the wrong medium: v = √(T/μ) for strings, v = √(B/ρ) for fluids, v = √(γRT/M) for ideal gas. Do not mix them.

Numerical Practice

Example 1: Wave Speed on a String

A string of mass 0.02 kg and length 2 m is under tension 800 N. Find the speed of transverse waves.

μ = m/L = 0.02/2 = 0.01 kg/m

v = √(T/μ) = √(800/0.01) = √80000 ≈ 283 m/s

Example 2: Beats

Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. How many beats are heard in 5 seconds?

Beats in 5 seconds = 4 × 5 = 20 beats

Example 3: Closed Organ Pipe

A closed organ pipe of length 0.5 m resonates at its fundamental frequency. If speed of sound is 340 m/s, find the fundamental frequency and the frequency of the first overtone.

Fundamental: ν₁ = v/(4L) = 340/(4 × 0.5) = 340/2 = 170 Hz

First overtone (3rd harmonic): ν₃ = 3ν₁ = 3 × 170 = 510 Hz

Example 4: Doppler Effect

A car sounding a horn of frequency 400 Hz approaches a stationary observer at 20 m/s. Speed of sound is 340 m/s. Find the apparent frequency.

ν' = νv/(v – v_s) = 400 × 340/(340 – 20) = 400 × 340/320 = 400 × 1.0625 = 425 Hz

Exam Tips

1. Wave equation: The derivation of the wave equation on a string (v = √(T/μ)) from Newton's law is a 5-mark question. Practise with a clear diagram. 2. Standing waves: Know the conditions for nodes and antinodes on strings and in pipes. Drawing the standing wave pattern with nodes (N) and antinodes (A) labelled carries marks. 3. Open vs closed pipe: Be ready to compare the two — all harmonics vs odd-only harmonics. This is a 2–3 mark comparison question.

5. Doppler effect: Practise the four special cases. State the sign convention you are using before solving. 6. Wave speed formulas: Memorise v = √(T/μ) for strings, v = √(B/ρ) for fluids, v = √(γRT/M) for gases. Know which applies where. 7. Always draw diagrams: Standing wave patterns, reflection at fixed/free ends, Doppler wavelength compression — diagrams are essential.

Quick Revision Notes

Standing wave key: Nodes at λ/2 spacing; antinodes at λ/4 from nodes

String: All harmonics (1, 2, 3, ...)

Closed pipe: Odd harmonics only (1, 3, 5, ...)

Doppler: Approaching → higher freq; Receding → lower freq