Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Quick Summary
Waves transfer energy without transferring matter. This chapter covers mechanical waves, sound waves, standing waves, and the Doppler effect. It directly connects to the wave-optics and acoustics topics that appear in NEET.
Wave Classification
| Type | Particle vibration | Examples |
|---|---|---|
| Transverse | Perpendicular to propagation | Light, string waves |
| Longitudinal | Parallel to propagation | Sound, compression waves |
Wave equation: y(x, t) = A sin(kx − ωt + φ)
- A = amplitude, k = 2π/λ (wave number), ω = 2πf, λ = wavelength, f = frequency
Wave speed: v = fλ = ω/k
Speed of Waves
| Medium | Formula |
|---|---|
| String (transverse) | v = √(T/μ) where T = tension, μ = linear mass density |
| Sound in gas | v = √(γP/ρ) |
| Sound at 0°C in air | v ≈ 331 m/s |
| Sound at t°C in air | v = 331 + 0.6t m/s |
Superposition and Interference
Principle of superposition: When two waves overlap, resultant displacement = algebraic sum of individual displacements.
Constructive interference: Phase difference = 0, 2π, 4π... → Maximum amplitude
Destructive interference: Phase difference = π, 3π... → Minimum amplitude
Standing Waves
Formed by superposition of two identical waves travelling in opposite directions.
y = 2A cos(kx) sin(ωt)
- Nodes: Points of zero displacement (2A cos(kx) = 0)
- Antinodes: Points of maximum displacement
Harmonics in a String (both ends fixed)
- Fundamental: f₁ = v/2L
- nth harmonic: fₙ = nv/2L = nf₁
Organ Pipes
| Pipe | Fundamental | Harmonics |
|---|---|---|
| Open (both ends open) | f = v/2L | All harmonics |
| Closed (one end closed) | f = v/4L | Odd harmonics only |
Beats
| Beat frequency = | f₁ − f₂ |
|---|
Maximum beat frequency distinguishable by ear ≈ 10 Hz.
Applications: tuning musical instruments, detecting slight frequency differences.
Doppler Effect
Apparent frequency:
f' = f × (v + v₀) / (v − v_s)
where v = speed of sound, v₀ = speed of observer (+ toward source), v_s = speed of source (+ toward observer)
Sign convention: Take positive direction as source → observer.
Important Questions
One Mark
- What are standing waves?
- Define beat frequency.
- State the principle of superposition.
- What is the speed of sound at 30°C?
- What is a node?
Short Answer (3 marks)
- Distinguish between transverse and longitudinal waves with examples.
- Explain beats. How are they used to tune musical instruments?
- An open organ pipe has length 0.5 m. Find the fundamental frequency. (v = 340 m/s)
Long Answer (5 marks)
- Derive the expression for standing waves formed on a string fixed at both ends. Find the frequencies of harmonics.
- Explain the Doppler effect. Derive the expression for apparent frequency when (a) observer moves toward source, (b) source moves toward observer.
Common Mistakes
- Mixing up closed and open pipe formulas (closed pipe supports only odd harmonics).
- Using incorrect signs in the Doppler formula — define a consistent positive direction first.
- Confusing beats (amplitude variation) with interference (constant pattern).
Board Exam Tips
- Standing waves derivation (string and pipes) is a 5-mark question.
- Doppler effect numerical problems appear in 3-mark and 5-mark sections.
- NEET: Harmonics and organ pipe problems are very common.