Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Waves — Key Facts

A wave is a disturbance that transfers energy from one point to another without the transfer of matter. Waves are central to both NEET Physics (optics, sound, modern physics) and everyday phenomena.

Wave Equation: $$v = f\lambda$$ where v = wave speed, f = frequency, λ = wavelength

Types of Waves:

Type	Definition	Examples
Mechanical waves	Require a medium to travel	Sound waves, water waves, waves on strings
Electromagnetic waves	Do not require a medium; travel at c = 3 × 10⁸ m/s in vacuum	Light, radio waves, X-rays
Transverse waves	Particle displacement perpendicular to wave propagation direction	Light, waves on string, ripples
Longitudinal waves	Particle displacement parallel to wave propagation direction	Sound waves, spring waves

Key Formulas:

• Wave on string: v = √(T/μ) where T = tension, μ = linear mass density = m/L
• Sound in gas: v = √(γP/ρ) = √(γRT/M)
• Period: T = 1/f; Angular frequency: ω = 2πf
• Equation of travelling wave: y = A sin(ωt − kx) where k = 2π/λ

Doppler Effect: When source and observer move relative to each other, the observed frequency changes: $$f’ = f \cdot \frac{(v \pm v_o)}{(v \mp v_s)}$$

• Upper signs: motion toward each other → frequency increases
• Lower signs: motion away → frequency decreases
• If source moves toward stationary observer: denominator has (v − vₛ)
• If observer moves toward stationary source: numerator has (v + vₒ)

⚡ Exam tip: For light waves (Doppler effect in light), the formula differs: f’ = f√((1+β)/(1−β)) where β = v_sound/v_light. Redshift (stars moving away) shows redshift — light shifted toward longer (redder) wavelengths.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Waves — NEET/JEE Study Guide

Superposition and Interference: When two or more waves pass through the same point simultaneously, the resultant displacement is the algebraic sum of individual displacements.

• Constructive interference: Crests meet crests → amplitude increases → path difference = nλ
• Destructive interference: Crests meet troughs → amplitude decreases → path difference = (2n+1)λ/2

Beats: When two waves of slightly different frequencies superpose, the resultant amplitude waxes and wanes periodically. $$f_{beat} = |f_1 - f_2|$$ Beats are heard as a pulsing effect — used in tuning musical instruments.

Stationary Waves (Standing Waves): Formed by superposition of two identical waves travelling in opposite directions.

String Fixed at Both Ends: $$L = \frac{n\lambda}{2} \implies f_n = \frac{nv}{2L}$$

• Fundamental (n=1): f₁ = v/(2L)
• First overtone (n=2): f₂ = v/L
• Second overtone (n=3): f₃ = 3v/(2L)

Open Pipe (Both ends open): L = n(λ/2) — same as string fixed at both ends (both ends are antinodes) $$f_n = \frac{nv}{2L}$$

Closed Pipe (One end closed): L = n(λ/4) for odd n only (n = 1, 3, 5, …) $$f_n = \frac{nv}{4L}$$ Only odd harmonics are present (fundamental, third harmonic, fifth harmonic…).

Doppler Effect — Extended Formula: For source and observer both moving along the line joining them: $$f’ = f \times \frac{(v \pm v_o)}{(v \mp v_s)}$$

⚡ NEET 2020 Qn: A source of sound of frequency 512 Hz moves toward a stationary observer at 20 m/s. Speed of sound = 340 m/s. Find apparent frequency. f’ = 512 × (340+0)/(340−20) = 512 × 340/320 = 544 Hz. Since source moves toward observer, frequency increases.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Waves — Comprehensive Notes

Derivation of Stationary Wave Frequencies:

For a string fixed at both ends, the boundary conditions require nodes at both ends. A stationary wave on a string of length L requires: $$\sin(kL) = 0 \implies kL = n\pi \implies k = \frac{n\pi}{L}$$

Since k = 2π/λ: $$\frac{2\pi}{\lambda} \cdot L = n\pi \implies \lambda = \frac{2L}{n}$$

Since v = fλ: $$f_n = \frac{v}{\lambda} = \frac{nv}{2L}$$

Energy in Wave Motion:

For a progressive wave carrying energy:

• Kinetic energy: KE = ½ mω²A² (maximum when displacement is zero)
• Potential energy: PE = ½ mω²A² (maximum when displacement is maximum)
• At any point, KE and PE are in phase for transverse waves
• Energy propagates with the wave; at a fixed point, energy oscillates

Wave Intensity: $$I = \frac{P}{4\pi r^2}$$ For sound waves: I ∝ A² and I ∝ f² Doubling the frequency quadruples the intensity if amplitude is constant.

Wave Propagation in Different Media:

Medium	Wave Type	Speed Formula
String (transverse)	Mechanical	v = √(T/μ)
Sound in gas	Longitudinal	v = √(γP/ρ) = √(γRT/M)
Sound in liquid	Longitudinal	v = √(B/ρ) where B = bulk modulus
Sound in solid (rod)	Longitudinal	v = √(Y/ρ) where Y = Young’s modulus

The speed of sound in air at 25°C (298 K) ≈ 346 m/s (approximately 330 m/s at 0°C). γ (adiabatic index) for air = 1.4.

Shock Waves: When a source moves faster than the wave speed in the medium (v_s > v):

• Mach number M = v_s/v_wave
• M > 1: Supersonic speed
• Shock wave front forms a cone (Mach cone)
• Sonic boom: Sharp explosive sound heard when cone sweeps past observer
• Example: Jet plane breaking the sound barrier

Doppler Effect Derivation (Sound): When source moves toward stationary observer:

• Source approaches at speed vₛ
• Wavelength decreases: λ’ = λ − vₛT = λ − vₛ/f
• New frequency: f’ = v/λ’ = v/(λ − vₛ/f) = vf/(v − vₛ) = f × v/(v − vₛ)

When observer moves toward stationary source:

• Observer approaches at speed vₒ
• Relative speed increases: v’ = v + vₒ
• f’ = (v + vₒ)/λ = f(v + vₒ)/v = f × (v + vₒ)/v

Light as an Electromagnetic Wave: Light waves are transverse EM waves requiring no medium. Speed of light in vacuum: c = 3 × 10⁸ m/s (exactly 299,792,458 m/s by definition) Relationship: c = fλ Visible light: λ = 400–700 nm (violet ~400 nm, red ~700 nm)

NEET Pattern Analysis: Waves contributes 2 questions per year in NEET Physics. Key areas: Doppler effect (especially moving source vs moving observer), stationary waves on strings and pipes, wave equation calculations, and wave interference (beats). The distinction between open and closed pipe harmonics is frequently tested.

⚡ NEET 2023 Qn: An open organ pipe and a closed organ pipe of the same length produce 4 beats per second when sounding their fundamental notes together. The speed of sound is 340 m/s. Length of pipe = 0.5 m. The closed pipe’s fundamental frequency = v/(4L) = 340/(4×0.5) = 170 Hz. Open pipe’s fundamental frequency = v/(2L) = 340/(2×0.5) = 340 Hz. Difference = 170 Hz, not 4 Hz… The question likely involves higher harmonics where beats = 4 Hz.