🔬 Wave Lab

JEENEET

Interactive wave visualizers, graph-reading diagnostics, and experimental analysis. This replaces rote learning of wave shapes with visual understanding — the foundation for solving any graph-based problem.

y–x Graph (Snapshot at t = t₀) y = A sin(kx − ωt)

y–t Graph (At fixed x = x₀) Particle motion

Wave Controls

Amplitude A40

Frequency k2

Speed ω3

Wave Speed (v = ω/k)1.5 m/s

Wavelength (λ = 2π/k)3.14 m

Period (T = 2π/ω)2.09 s

Max Particle v (Aω)120

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Thinking Step

y–x graph: snapshot of wave at one instant. Tells you λ and A.
y–t graph: one particle's SHM over time. Tells you T and A.
They look identical but mean COMPLETELY different things. JEE frequently shows one and asks you to identify which graph it is.

How to Read a y–x Graph (Snapshot)

Wavelength λDistance between any two consecutive points at the same displacement moving in the same direction. Crest-to-crest or trough-to-trough.
Amplitude AMaximum displacement from equilibrium. Height of crest (or depth of trough).
Wave directionNot determinable from a single snapshot. Need two snapshots (or y-t graph for a particle).
Particle velocityAt any point: slope = ∂y/∂x. Particle velocity = −v × (∂y/∂x). Positive slope → particle velocity opposite to wave velocity.
Particle at crest∂y/∂x = 0 → particle velocity = 0. Max acceleration (−ω²A). Acceleration opposite to displacement.

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Common Mistake Alert

Students confuse "particle at crest has maximum velocity" — WRONG! Particle at crest has ZERO velocity (it's momentarily at rest, like a pendulum at extreme position). Maximum particle velocity is at mean position (displacement = 0).

Wave 1 + Wave 2 = Resultant

Wave 1 Wave 2 Resultant

Superposition Controls

Amplitude A₁30

Amplitude A₂30

Phase diff φ (×π)0π

Freq ratio f₂/f₁1.0

Resultant Amplitude A—

Interference Type—

I_max / I_min ratio—

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Exam Insight

When A₁ = A₂: I_max = 4I (2A squared), I_min = 0. When A₁ ≠ A₂: I_min > 0. Perfect destructive interference requires EQUAL amplitudes. JEE Advanced often gives unequal amplitudes to trick students.

Phase Difference → Path Difference Conversion

Phase Difference φ	Path Difference Δx	Type	Resultant Amplitude
0, 2π, 4π...	0, λ, 2λ... (nλ)	Constructive	A₁ + A₂
π, 3π, 5π...	λ/2, 3λ/2... ((2n-1)λ/2)	Destructive	\|A₁ − A₂\|
π/2	λ/4	Partial	√(A₁²+A₂²)

Standing Wave + Components

Forward wave Backward wave Standing wave

Standing Wave Controls

Mode (harmonics)n=2

Amplitude30

Number of loops2

Nodes (total)3

Antinodes (total)2

Harmonic label2nd harmonic

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Strategy Tip — Count Loops

In a vibrating string with n loops: number of nodes = n+1, number of antinodes = n. This is the fastest way to verify harmonic number. "3 loops → 4 nodes → 3rd harmonic." Commit this to memory.

Node & Antinode — Key Properties

N — Node

Displacement y = 0 always
Particle velocity = 0 always
Pressure variation = MAXIMUM
Strain = maximum (for sound)
Located at sin(kx) = 0

AN — Antinode

Displacement y = max (±2A)
Max particle velocity = 2Aω
Pressure variation = ZERO
Strain = zero
Located at sin(kx) = ±1

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Common Mistake Alert — Sound Pressure in Standing Wave

In a sound standing wave: node (displacement) = antinode (pressure). Antinode (displacement) = node (pressure). The TWO waves are 90° out of phase spatially. Most NEET students get this backwards and lose 4 marks every year.

Beats — Amplitude Variation

Envelope (Amplitude modulation)

Beat Controls

Frequency f₁ (rel)4

Frequency f₂ (rel)5

Beat frequency1.0 Hz

Average frequency4.5 Hz

Beat period1.0 s

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Thinking Step

Notice: as f₁ and f₂ get farther apart, beats get faster. When they're equal, no beats (single frequency). Waxing (getting louder) = constructive interference. Waning (getting softer) = destructive interference. You hear f_beat complete cycles per second.

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Exam Insight — Graph Questions are Easy Marks

JEE Main gives 1–2 graph-reading questions each year on waves. They look intimidating but follow a fixed pattern. This section gives you the complete diagnostic framework.

📈 Reading y–x Graph: Extract λ, A, v

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Given a y–x graph (snapshot at t=0), identify:

λ: Peak-to-peak distance on x-axis
A: Max y-value on y-axis
k = 2π/λ
v: Needs second graph or given separately
Direction: Need two snapshots or equation

From y–x graph

λ = x₂ − x₁ (same phase)

k = 2π/λ | then v = ω/k (if ω known)

📉 Reading y–t Graph: Extract T, f, ω

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Given a y–t graph for one particle:

T: Period — time for one full oscillation
f = 1/T
ω = 2π/T = 2πf
Initial phase: y-value at t=0

From y–t graph

T from graph → f = 1/T → ω = 2πf

v_particle = −Aω sin(ωt + φ)

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Common Mistake Alert

From y-t graph: if y(0) = A (starts at crest), the equation is y = A cos(ωt). NOT sin. Using sin when it should be cos → wrong phase → wrong particle velocity calculation at t=0.

🔄 Identifying Standing Wave from Graph

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How to confirm a wave is standing (not progressive):

✅ Standing Wave Signs

Some points NEVER move (nodes)
All particles oscillate in phase or opposite phase
Amplitude varies with position
No net energy transport

→ Progressive Wave Signs

All particles eventually oscillate
Same amplitude everywhere
Phase varies with position
Energy transport occurs

🎓 Resonance Tube: Experimental Graph Analysis

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In the resonance tube experiment, a graph of "resonance length L vs frequency f" (or 1/f) is plotted:

Resonance Tube Analysis

L₁ = λ/4 − e L₂ = 3λ/4 − e

L₂ − L₁ = λ/2 → v = 2f(L₂ − L₁)

Graph of L vs n (resonance number): slope = λ/2. Intercept = −e (end correction). This is a CBSE practical question worth 2 marks.

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Strategy Tip

If asked for end correction from graph: extend line to L-axis — the negative intercept is e. If asked for v: v = 2f × slope. One formula, two applications — this is the full analysis.

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