NCERT Solutions for Class 11 Physics Chapter 15 – Waves
Chapter 15 of Class 11 Physics, Waves, is a comprehensive chapter that explores how energy and information travel through matter and space without bulk transport of material. Building on the oscillations studied in Chapter 14, this chapter extends periodic motion to propagation through a medium. Students learn to classify waves as transverse (vibration perpendicular to propagation — light, strings) or longitudinal (vibration parallel — sound waves). The mathematical description of a progressive wave — y(x,t) = A sin(kx − ωt + φ) — introduces wave number, angular frequency, and phase, enabling calculation of velocity, wavelength, and frequency. A central topic is the speed of sound in a medium — derived from elasticity and density — along with Newton's and Laplace's corrections. The Principle of Superposition explains interference, beats, and standing waves. Standing waves formed in strings and air columns (open and closed pipes) produce harmonics and overtones, explaining the physics of musical instruments. Beats — the periodic variation in loudness when two nearly equal frequencies are combined — have applications in musical tuning and radar technology. Finally, the Doppler Effect describes the apparent change in frequency when source or observer moves relative to the medium, with applications in ambulance sirens, sonar, and astrophysics. NCERT solutions for Chapter 15 provide detailed answers to all exercises in this rich chapter.
NCERT Solutions PDF – Class 11 Physics Chapter 15 (All Exercises)
Important Formulas – Chapter 15: Waves
| Formula | Expression | Description |
|---|---|---|
| Progressive Wave Equation | y(x,t) = A sin(kx − ωt + φ) | A=amplitude; k=wave number; ω=angular frequency |
| Wave Speed | v = λf = ω/k | λ = wavelength; f = frequency |
| Wave Number | k = 2π/λ | Spatial frequency; unit: rad/m |
| Angular Frequency | ω = 2πf = 2π/T | Temporal frequency; unit: rad/s |
| Speed in stretched string | v = √(T/μ) | T = tension (N); μ = linear mass density (kg/m) |
| Speed of sound (Newton) | v = √(B/ρ) | B = bulk modulus; ρ = density |
| Speed of sound (Laplace) | v = √(γP/ρ) = √(γRT/M) | γ correction for adiabatic process; ≈343 m/s in air at 20°C |
| Standing Wave (string) | y = 2A sin(kx) cos(ωt) | Superposition of two waves in opposite directions |
| Harmonics in string (both ends fixed) | f_n = n·v/(2L); n = 1,2,3... | Fundamental (n=1): f₁ = v/2L |
| Open Pipe harmonics | f_n = n·v/(2L); n = 1,2,3... | All harmonics present |
| Closed Pipe harmonics | f_n = n·v/(4L); n = 1,3,5... | Only odd harmonics; fundamental f₁ = v/4L |
| Beats Frequency | f_beat = |f₁ − f₂| | Amplitude varies at beat frequency |
| Doppler Effect | f' = f·(v + v_o)/(v − v_s) | v_o = observer speed; v_s = source speed; signs change with direction |
Subtopics Explained – Chapter 15: Waves
Types of Waves
Mechanical waves need a medium (sound, water waves, seismic waves). Electromagnetic waves do not (light, radio waves). Transverse waves have oscillation perpendicular to propagation — they can be polarised. Longitudinal waves (like sound) have oscillation parallel to propagation — they consist of compressions and rarefactions and cannot be polarised.
Mathematical Representation of a Progressive Wave
The equation y(x,t) = A sin(kx − ωt + φ) fully describes a sinusoidal wave. The phase difference between two points separated by Δx is kΔx. The wave moves in the +x direction for (kx − ωt) and in −x for (kx + ωt). Particle velocity (dy/dt) and wave speed (dx/dt) are very different quantities — a common source of confusion in exams.
Speed of Sound: Newton and Laplace
Newton assumed isothermal compression, giving v = √(P/ρ) ≈ 280 m/s — too low. Laplace corrected it by noting that rapid compressions in sound are adiabatic, giving v = √(γP/ρ) ≈ 332 m/s (matches experiment). Speed of sound increases with temperature as v ∝ √T and is independent of pressure at constant temperature.
Superposition and Interference
When two waves overlap, the resultant displacement is the algebraic sum (superposition principle). Constructive interference (in phase) gives maximum amplitude; destructive interference (out of phase by π) gives zero amplitude. This explains the formation of beats and standing waves.
Standing Waves and Resonance
Standing waves form when two identical waves travel in opposite directions. Nodes (zero displacement) and antinodes (maximum displacement) are formed at fixed positions. In a string fixed at both ends: λ = 2L/n. In pipes, boundary conditions (open/closed ends) determine which harmonics are allowed. The lowest resonant frequency is the fundamental frequency or first harmonic.
Beats
When two sound waves of slightly different frequencies f₁ and f₂ are superposed, the resultant has a frequency (f₁+f₂)/2 but an amplitude that oscillates at |f₁−f₂| — the beat frequency. Beats are heard as periodic loud-soft variations in sound. Musicians use beats to tune instruments: when beats disappear, the frequencies match.
Doppler Effect
The Doppler Effect is the apparent change in frequency due to relative motion between source and observer. If they approach each other, apparent frequency is higher (blue shift in light); if receding, lower (red shift). The general formula accounts for both source and observer moving in the medium. Applications include police speed guns, ultrasound diagnostics, and astronomical redshift measurements.
| Resource Name | Description | Best For |
|---|---|---|
| NCERT Solutions | Detailed answers and explanations for NCERT textbook questions across all classes and subjects. | Homework, assignments, and exam preparation |
| NCERT Solutions for Class 11 | Chapter-wise solutions for all Class 11 subjects including Physics, Chemistry, Mathematics, Biology, and English. | Class 11 board exam preparation |
| NCERT Solutions for Class 11 Physics | Step-by-step solutions covering all chapters such as Motion, Laws of Motion, Work Energy and Power, Thermodynamics, and Waves. | Concept building and numerical problem-solving |
| NCERT Exemplar Class 11 Physics | Advanced and application-based questions designed to strengthen conceptual understanding and analytical skills. | JEE, NEET, Olympiads, and higher-order practice |
| Physics Formula | Chapter-wise collection of important formulas, equations, and derivations for quick revision. | Last-minute revision and numerical practice |
Quick Reference Table – Standing Waves in Different Systems
| System | Boundary Condition | Harmonics Present | Fundamental Frequency | nth Harmonic |
|---|---|---|---|---|
| String (both ends fixed) | Nodes at both ends | All (1,2,3...) | v/2L | nv/2L |
| Open Pipe | Antinodes at both ends | All (1,2,3...) | v/2L | nv/2L |
| Closed Pipe (one end closed) | Node at closed, antinode at open | Odd only (1,3,5...) | v/4L | (2n−1)v/4L |
| Open pipe vs Closed pipe | — | Open has all; closed has odd | f_open = 2×f_closed (same L) | — |
NCERT Solutions for Class 11 Physics Chapter 15: Waves – FAQs
A wave is a disturbance that travels through a medium or space and transfers energy from one location to another without transporting matter permanently. In this chapter, students learn how waves carry energy through vibrations of particles. The concept is fundamental because it explains sound, light, and many other natural phenomena. NCERT questions often focus on wave properties and energy transfer mechanisms. Understanding this topic helps students appreciate how communication systems, musical instruments, and various physical processes depend on wave motion for energy transmission.
Transverse waves involve particle vibrations perpendicular to the direction of wave propagation, while longitudinal waves involve vibrations parallel to the direction of propagation. Examples of transverse waves include waves on a stretched string, whereas sound waves are longitudinal. This distinction is one of the most important concepts in the chapter and is frequently tested in examinations. Students should learn the characteristics, examples, and particle motion associated with each type. A clear understanding of these wave forms helps in studying sound, optics, and advanced wave phenomena.
Wavelength is the distance between two consecutive points in the same phase of a wave. Frequency is the number of complete vibrations occurring per second, while wave speed refers to the rate at which the wave travels through a medium. These three quantities are closely related and form the basis of many numerical questions in NCERT exercises. Students must understand their physical significance and relationships to solve wave problems effectively. Mastering these concepts also helps in understanding sound waves and electromagnetic waves in higher classes.
The principle of superposition states that when two or more waves travel through the same medium simultaneously, the resultant displacement at any point equals the algebraic sum of individual displacements. This principle explains interference and many other wave phenomena. NCERT introduces this concept to help students understand how waves interact with one another. Questions based on constructive and destructive effects often appear in examinations. Understanding superposition provides a strong foundation for advanced topics related to sound and light waves.
Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. These waves appear stationary because certain points remain fixed while others oscillate with maximum amplitude. NCERT explains standing waves using examples such as vibrating strings and air columns. Questions based on nodes and antinodes are commonly asked in examinations. Understanding standing waves helps students analyze musical instruments and resonance phenomena. This topic also serves as an important link between oscillations and wave motion.