🧠 Core Concepts

CBSENEETJEE

Build your mental model from scratch. Wave nature → Superposition → Standing Waves → Sound → Doppler. Every concept explained with the reasoning a JEE examiner expects.

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Exam Insight

JEE loves asking "what type of wave" questions disguised as numerical problems. Always identify: Is it mechanical or EM? Transverse or longitudinal? That determines which formula applies.

What is a Wave?

A wave is a disturbance that carries energy without transporting matter. The medium particles oscillate about their mean positions — the energy travels, the particles don't.

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Thinking Step

If a particle travels with the wave, it's NOT a wave — it's particle motion. In a wave, the pattern travels. This distinction is tested every year in NEET assertion-reason questions.

Amplitude (A)

metres (m)

Max displacement from mean. Related to energy: E ∝ A²

Wavelength (λ)

metres (m)

Distance between two consecutive in-phase points

Time Period (T)

seconds (s)

Time for one complete oscillation: T = 1/f

Frequency (f)

hertz (Hz)

Oscillations per second. f = ω/2π

Wave Speed (v)

m/s

v = fλ = λ/T. Medium property, not amplitude.

Wave Number (k)

rad/m

k = 2π/λ. Spatial frequency analogue of ω

Transverse vs Longitudinal Waves

Transverse Waves

Particle oscillation perpendicular to wave propagation direction.

Examples: Light waves, waves on strings, seismic S-waves

y(x,t) = A sin(kx - ωt)

Longitudinal Waves

Particle oscillation parallel to wave propagation direction. Creates compressions & rarefactions.

Examples: Sound waves, seismic P-waves, waves in springs

s(x,t) = s₀ sin(kx - ωt)

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Common Mistake Alert

Sound waves are LONGITUDINAL, not transverse. In a pressure wave, compressions correspond to pressure maxima but displacement MINIMA. This inverse relationship trips up 60% of students in NEET.

The Wave Equation Decoded

Progressive Wave

y(x, t) = A sin(kx − ωt + φ)

k = 2π/λ | ω = 2πf | v = ω/k = fλ

• kx − ωt = phase at position x, time t

• +kx − ωt → wave travels in +x direction

• −kx − ωt → wave travels in −x direction

• Particle velocity: ∂y/∂t = −Aω cos(kx − ωt)

• Particle acceleration: ∂²y/∂t² = −Aω² sin(kx − ωt) = −ω²y → SHM!

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Strategy Tip

To find wave speed from equation: v = coefficient of t / coefficient of x. For y = A sin(4x − 8t), v = 8/4 = 2 m/s. This shortcut works every time — saves 30 seconds.

Speed of Waves in Different Media

String

v = \sqrt(T/μ)

T = tension (N), μ = linear mass density (kg/m). v independent of frequency!

Solid Rod

v = \sqrt(Y/ρ)

Y = Young's modulus, ρ = density. Fastest in solids.

Liquid/Gas

v = \sqrt(B/ρ)

B = Bulk modulus. For gas: B = γP (Laplace). Newton used B = P (wrong!).

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Exam Insight — Newton vs Laplace

Newton assumed isothermal (B = P → v = 280 m/s at STP). Wrong! Laplace corrected it to adiabatic (B = γP → v = 332 m/s). γ for air = 1.4. JEE always asks: why was Newton wrong? Answer: Sound is adiabatic, not isothermal — the compressions happen too fast for heat exchange.

The Superposition Principle

When two or more waves meet at a point, the resultant displacement is the algebraic sum of individual displacements. Waves pass through each other without affecting each other.

Superposition

y = y₁ + y₂ = A₁sin(kx−ωt) + A₂sin(kx−ωt+φ)

Resultant Amplitude: A = √(A₁² + A₂² + 2A₁A₂cosφ)

Maximum (Constructive): A_max = A₁ + A₂ when φ = 2nπ

Minimum (Destructive): A_min = |A₁ − A₂| when φ = (2n+1)π

Phase difference: φ = (2π/λ) × Δx (path diff → phase diff)

Constructive vs Destructive Interference

✅ Constructive Interference

Crests meet crests. Energy concentrates.

Path diff: Δx = nλ (n = 0,1,2...) Phase diff: φ = 2nπ

Intensity: I_max = (√I₁ + √I₂)²

❌ Destructive Interference

Crests meet troughs. Energy redistributes.

Path diff: Δx = (2n-1)λ/2 Phase diff: φ = (2n-1)π

Intensity: I_min = (√I₁ − √I₂)²

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Common Mistake Alert

Destructive interference does NOT violate energy conservation. Energy is NOT destroyed — it is redistributed. Where intensity is minimum, it appears at maximum elsewhere. Total energy is conserved. This is a favourite assertion-reason in JEE.

Coherence — Why It Matters

For stable interference, sources must be coherent: same frequency and constant phase difference. Non-coherent sources (like two bulbs) produce no stable pattern — the phase differences fluctuate rapidly.

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Thinking Step

JEE question: "Two sources have intensities I and 4I. Find I_max and I_min."
A₁ = √I, A₂ = 2√I.
A_max = 3√I → I_max = 9I
A_min = √I → I_min = I
Ratio I_max:I_min = 9:1. Always remember: work with amplitudes, square for intensity.

How Standing Waves Form

Two identical waves travelling in opposite directions superpose to form a standing (stationary) wave. No energy is transported. Pattern of nodes and antinodes is fixed in space.

Standing Wave Equation

y = 2A sin(kx) cos(ωt)

Amplitude = 2A sin(kx) — varies with position, oscillates with time

Node

y = 0 always
sin(kx)=0

Antinode

max amplitude
sin(kx)=±1

Node

λ/2 separation
between nodes

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Strategy Tip — Node/Antinode Memory

Node: particle NEVER moves (displacement = 0). Antinode: particle has MAXIMUM displacement. Between consecutive node & antinode: λ/4. Between consecutive nodes: λ/2. Write this once before exam — saves confusion.

Vibrating Strings — Harmonics

Both ends fixed → displacement nodes at both ends.

Frequency of nth Harmonic

fₙ = n/(2L) · √(T/μ) n = 1, 2, 3...

All harmonics present. f₁ = fundamental, f₂ = 1st overtone = 2nd harmonic

n=1: Fundamental (1st Harmonic)

L = λ/2 f₁ = v/2L

1 loop, 0 intermediate nodes

n=2: 2nd Harmonic (1st Overtone)

L = λ f₂ = v/L = 2f₁

2 loops, 1 intermediate node

n=3: 3rd Harmonic (2nd Overtone)

L = 3λ/2 f₃ = 3v/2L = 3f₁

3 loops, 2 intermediate nodes

Organ Pipes — Open & Closed

Open Pipe (both ends open)

Antinodes at both ends. Supports all harmonics.

fₙ = nv/(2L) n = 1,2,3...

Frequency ratio: 1 : 2 : 3 : 4 ...

Closed Pipe (one end closed)

Node at closed end, antinode at open end. Only ODD harmonics.

fₙ = (2n−1)v/(4L) n = 1,2,3...

Frequency ratio: 1 : 3 : 5 : 7 ...

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Common Mistake Alert — Overtone vs Harmonic

In a closed pipe: 1st harmonic = fundamental (n=1). 2nd harmonic = 1st overtone (n=2 for open, but doesn't exist for closed). The 1st overtone of a CLOSED pipe is the 3rd harmonic, NOT the 2nd. This destroys marks in NEET every year.

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Exam Insight — End Correction

Real organ pipes: open end is antinode but slightly beyond the pipe. End correction e ≈ 0.6r (r = radius). Effective length L_eff = L + e (open end) or L + 2e (both ends open). End correction is tested in CBSE board numericals.

Sound as a Pressure Wave

Sound is a longitudinal mechanical wave in a medium. It propagates as alternating compressions (high pressure) and rarefactions (low pressure).

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Thinking Step — The Anti-Intuitive Fact

Where displacement is maximum → pressure change is MINIMUM (at antinodes of displacement = nodes of pressure).
Where displacement is zero → pressure change is MAXIMUM (at nodes of displacement = antinodes of pressure).
Displacement and pressure waves are 90° (π/2) out of phase. This is a guaranteed JEE question.

Speed of Sound

v = √(γP/ρ) = √(γRT/M)

γ = adiabatic index, R = 8.314 J/mol·K, M = molar mass, T in Kelvin

• v ∝ √T (temperature) — at higher T, particles move faster, sound propagates faster

• v independent of pressure (at const T) — ρ ∝ P, so P/ρ = const

• v ∝ 1/√M — lighter molecules → faster sound (H₂ has highest v)

• Effect of humidity: water vapour (M=18) replaces N₂(28)/O₂(32) → lower M → higher v. Humid air has HIGHER speed of sound.

Intensity & Decibel Scale

Intensity

I = P/A = 2π²f²A²ρv

I ∝ A² ∝ f² | Reference: I₀ = 10⁻¹² W/m²

Decibel Level

β = 10 log₁₀(I/I₀) dB

Every 10 dB increase → 10× increase in intensity

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Strategy Tip — dB Shortcuts

• +10 dB → I × 10
• +20 dB → I × 100
• +3 dB → I × 2 (approx)
• 2 sources of equal intensity → level increases by 3 dB (not doubles). This is a standard JEE trap.

Beat Phenomenon

When two sound waves of slightly different frequencies (f₁ and f₂) superpose, the resultant amplitude varies periodically. This periodic variation in intensity is called beats.

Beat Frequency

f_beat = |f₁ − f₂|

Number of beats per second = |f₁ − f₂|. Valid only if |f₁−f₂| ≤ 10 Hz (otherwise not perceptible)

y₁ = A sin(2πf₁t) y₂ = A sin(2πf₂t)

y = y₁+y₂ = 2A cos(π(f₁−f₂)t) · sin(π(f₁+f₂)t)

Amplitude factor: 2A cos(π(f₁−f₂)t) → varies with frequency (f₁−f₂)/2

Intensity ∝ amplitude² → completes max→min→max in 1/(f₁−f₂) seconds

∴ Beat frequency = f₁ − f₂

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Common Mistake Alert

Beat frequency = |f₁ − f₂|, NOT (f₁ + f₂)/2. Also: if a fork of frequency f gives n beats with a fork of f₀, then the unknown fork could be f₀+n OR f₀−n. To determine which, load the fork with wax (decreases f) and observe whether beats increase or decrease. This logic is tested in JEE.

Doppler Effect — The Thinking Approach

The Doppler effect is the apparent change in frequency of a wave when source and/or observer are in relative motion.

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Thinking Step — Build the Formula

Think physically: if observer moves TOWARD source, they encounter wavefronts faster → apparent f increases. If source moves TOWARD observer, wavefronts crowd together → λ decreases → f increases. This reasoning gives you the sign convention automatically.

Doppler Formula

f' = f₀ · (v ± v_o) / (v ∓ v_s)

v = speed of sound | Upper signs: approaching | Lower signs: receding

Observer moving TOWARD source: use + in numerator (v + v_o)

Observer moving AWAY from source: use − in numerator (v − v_o)

Source moving TOWARD observer: use − in denominator (v − v_s)

Source moving AWAY from observer: use + in denominator (v + v_s)

Memory: "Toward = Top + Bottom −" for f increase. "Away = Top − Bottom +" for f decrease

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Common Mistake Alert — The Big One

Source moving and observer moving give DIFFERENT results even if relative velocity is same. Doppler formula is NOT symmetric in source/observer interchange. v_s = v_o = 10 m/s toward each other gives DIFFERENT f' for "source moving" vs "observer moving". This is a JEE Advanced trap.

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Exam Insight

Special case: if both source and observer move in the SAME direction with same speed → no Doppler shift! The relative speed of approach is zero. Also: Doppler applies to light (with relativistic correction at high speeds). NEET level: only sound Doppler needed.

Resonance — When Frequencies Match

Resonance occurs when the frequency of a driving force matches the natural frequency of the system. The amplitude of oscillation becomes maximum.

Resonance Tube Experiment

A tuning fork is held over a tube of variable length. Resonance occurs when tube length satisfies the standing wave condition for a closed pipe.

1st resonance: L₁ = λ/4 − e (e = end correction)

2nd resonance: L₂ = 3λ/4 − e

L₂ − L₁ = λ/2 → λ = 2(L₂−L₁)

End correction: e = (L₂ − 3L₁)/2 (eliminating λ)

Speed of sound: v = f × λ = 2f(L₂−L₁)

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Strategy Tip

In resonance tube experiment, end correction eliminates itself in the difference L₂−L₁. So v = 2f(L₂−L₁) is clean. If only one resonance point is given, end correction matters and e must be known. This is the #1 CBSE numerical in the Waves chapter.

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