Waves
'The wave is the fundamental entity, the primary mode of communication of energy through the universe.' — Wave Mechanics
1. Chapter Overview
WAVES are disturbances that TRANSPORT ENERGY without transporting matter. This chapter focuses on MECHANICAL WAVES (waves that require a medium — sound, springs, water waves). You will learn about TRANSVERSE and LONGITUDINAL waves, the WAVE EQUATION, SUPERPOSITION, STANDING WAVES in strings and pipes, BEATS, and the DOPPLER EFFECT.
2. Types of Waves
| Type | Medium Required? | Examples |
|---|---|---|
| Mechanical | YES | Sound, water waves, seismic waves |
| Electromagnetic | NO | Light, radio, X-rays |
| Matter waves | NO | Electron waves (quantum) |
Mechanical Wave Classification
- Transverse Waves: Particles vibrate PERPENDICULAR to wave direction (string waves, EM waves)
- Longitudinal Waves: Particles vibrate PARALLEL to wave direction (sound waves, spring compression waves)
Key Terms
- Crest: Maximum positive displacement (transverse)
- Trough: Maximum negative displacement (transverse)
- Compression: Region of high pressure/density (longitudinal)
- Rarefaction: Region of low pressure/density (longitudinal)
- Wavelength (λ): Distance between TWO successive crests/compressions
- Amplitude (A): Maximum displacement from equilibrium
3. Wave Speed
Speed on a Stretched String
- v = √(T/μ) where T = tension, μ = linear mass density (mass/length)
- Independent of frequency!
Speed of Sound in a Gas
- v = √(γP/ρ) = √(γRT/M)
- γ = adiabatic index (1.4 for air)
- At STP: v_air ≈ 332 m/s at 0°C, ≈ 340 m/s at 20°C
Factors Affecting Sound Speed
- Temperature: v ∝ √T (higher T = faster sound)
- Density: v ∝ 1/√ρ (lighter medium = faster sound)
- Humidity: v increases with humidity (water vapour is less dense)
4. Wave Equation and Progressive Waves
Equation of a Progressive Wave
- y(x, t) = A sin(ωt — kx + φ₀)
- ω = 2πf (angular frequency)
- k = 2π/λ (wave number)
- v = ω/k = fλ (wave speed)
Particle Velocity vs Wave Velocity
- Particle velocity: v_p = dy/dt = Aω cos(ωt — kx)
- Wave velocity: v_w = ω/k (constant for a given medium)
- These are DIFFERENT concepts!
Worked Problem
Q: A wave is described by y = 0.02 sin(100t — 2x) m. Find amplitude, frequency, wavelength, and wave speed. A: Comparing with y = A sin(ωt — kx): A = 0.02 m, ω = 100 rad/s, k = 2 rad/m. f = ω/2π = 100/2π = 15.92 Hz. λ = 2π/k = 2π/2 = π = 3.14 m. v = ω/k = 100/2 = 50 m/s.
5. Superposition Principle
- Principle: When TWO or more waves overlap, the NET displacement is the SUM of individual displacements
- y_net = y₁ + y₂ + y₃ + ...
Interference
- Constructive: Waves in PHASE (path difference = nλ) → amplitude DOUBLES
- Destructive: Waves OPPOSITE phase (path difference = (2n+1)λ/2) → amplitude ZERO
- Resultant amplitude: R = √(A₁² + A₂² + 2A₁A₂cosφ)
6. Reflection of Waves
Reflection at a Boundary
| Boundary | String Pulse | Phase Change |
|---|---|---|
| Rigid (fixed end) | Inverted | π radians (180°) |
| Free end | Not inverted | No phase change |
| Sound from rigid wall | Compression as compression? | Depends on density |
Standing (Stationary) Waves
- Formed when two IDENTICAL waves travel in OPPOSITE directions
- y(x,t) = 2A sin(kx) cos(ωt)
- Nodes: Points of ZERO displacement (2A sin(kx) = 0)
- Antinodes: Points of MAXIMUM displacement (2A sin(kx) = ±2A)
Standing Waves in a String (Fixed at Both Ends)
- Allowed wavelengths: λ_n = 2L/n (n = 1, 2, 3...)
- Frequencies: f_n = nv/2L = n√(T/μ)/2L
- n = 1: Fundamental (first harmonic)
- n = 2: First overtone (second harmonic)
- n = 3: Second overtone (third harmonic)
Standing Waves in a Pipe
| Pipe Type | Open Ends | Closed End |
|---|---|---|
| Open at both ends | f_n = nv/2L | ALL harmonics present |
| Closed at one end | f_n = nv/4L | ONLY ODD harmonics |
7. Beats
- Formation: When TWO waves of SLIGHTLY DIFFERENT frequencies travel together
- Beat frequency: f_beat = |f₁ — f₂|
- The amplitude (loudness) varies periodically
- Applications: Tuning musical instruments, detecting frequency differences
Worked Problem
Q: Two tuning forks of frequencies 256 Hz and 260 Hz sound together. Find beat frequency. A: f_beat = |260 — 256| = 4 Hz. You hear 4 loudness maxima per second.
8. Doppler Effect
- Definition: APPARENT change in frequency due to RELATIVE MOTION between source and observer
- Formula: f' = f(v + v₀)/(v — v_s)
- v = speed of sound in the medium
- v₀ = velocity of observer (positive TOWARD source)
- v_s = velocity of source (positive AWAY from observer)
Special Cases
| Situation | Apparent Frequency |
|---|---|
| Source moving toward stationary observer | f' = fv/(v — v_s) — INCREASED |
| Source moving away from stationary observer | f' = fv/(v + v_s) — DECREASED |
| Observer moving toward stationary source | f' = f(v + v₀)/v — INCREASED |
| Observer moving away from stationary source | f' = f(v — v₀)/v — DECREASED |
Applications
- Radar speed guns (police tracking)
- Medical ultrasound (blood flow measurement)
- Astronomy: REDSHIFT (stars moving away from Earth)
- Weather radar (Doppler radar)
9. Common Mistakes
- Particle velocity ≠ wave velocity: Particles oscillate; the WAVE propagates. A cork on water bobs up and down while the wave moves horizontally
- Nodes are NOT points of minimum amplitude: They are points of PERMANENTLY ZERO amplitude
- Beat frequency is |f₁ — f₂|, NOT (f₁ + f₂)/2: The average frequency is what you hear; the beat is the VARIATION
- Doppler effect depends on RELATIVE velocity: Moving source vs. moving observer give DIFFERENT formulas
- Closed pipe has ONLY odd harmonics: This gives the characteristic 'mellow' sound of a clarinet
10. CBSE Exam Focus
- Progressive wave equation — derivation and numericals (3/5-mark)
- Standing waves in strings — harmonics (5-mark)
- Standing waves in open and closed organ pipes (5-mark)
- Beats — formation, beat frequency (3-mark)
- Doppler effect — derivation and numericals (5-mark)
- Speed of sound — factors affecting it
11. Key Formulas
- v = fλ = ω/k
- v_string = √(T/μ)
- v_sound = √(γP/ρ) = √(γRT/M)
- y(x,t) = A sin(ωt — kx + φ₀)
- Standing wave: y = 2A sin(kx) cos(ωt)
- f_n (string) = nv/2L
- f_n (open pipe) = nv/2L
- f_n (closed pipe) = nv/4L (n odd)
- f_beat = |f₁ — f₂|
- f' = f(v ± v₀)/(v ∓ v_s) (Doppler)
12. Self-Test (5+ Q&A)
Q1: A string of length 1 m has linear density 0.01 kg/m and tension 100 N. Find fundamental frequency. A: v = √(T/μ) = √(100/0.01) = 100 m/s. f₁ = v/2L = 100/2 = 50 Hz.
Q2: Two waves y₁ = 0.05 sin(200t — 5x) and y₂ = 0.05 sin(200t + 5x) interfere. Find the standing wave equation and node positions. A: y = 2×0.05 sin(5x) cos(200t) = 0.1 sin(5x) cos(200t). Nodes where sin(5x) = 0 → x = nπ/5.
Q3: A source moving at 30 m/s emits 500 Hz sound. Find apparent frequency for a stationary observer when source approaches. (v = 340 m/s) A: f' = fv/(v — v_s) = 500×340/(340 — 30) = 170000/310 = 548.4 Hz.
Q4: Find the third harmonic of a 50 cm long open organ pipe (v = 340 m/s). A: f₃ = 3v/2L = 3×340/(2×0.5) = 1020 Hz.
Q5: What are beats and how are they formed? A: Beats are periodic variations in loudness produced by superposition of two waves of slightly different frequencies. Beat frequency = |f₁ — f₂|.
13. Conclusion
Waves are FUNDAMENTAL to physics — they transport energy and information across distances. Mechanical waves in strings and pipes lead to musical instruments. Standing waves explain why a guitar string produces specific notes. Beats help in tuning. The Doppler effect has APPLICATIONS ranging from radar to astronomy. While this chapter focuses on mechanical waves, the concepts of wavelength, frequency, superposition, and interference EXTEND to light (wave optics) and quantum mechanics.
