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📚 Official Learning Libraries & Channels

🌊 Waves – JEE Advanced Objective Library

Complete Mechanical Waves objective questions from basic to toughest JEE Advanced level.

👉 Open Waves Library

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🌊 WAVES (Mechanical Waves)

Stage 1 – Page 1 : Foundation & Core Concepts (JEE / Intermediate Level)

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1️⃣ What is a Mechanical Wave?

A mechanical wave is a disturbance that travels through a material medium, transferring energy and momentum without transporting matter.

• Medium is compulsory
• Energy propagates, particles only oscillate

Examples: Sound waves, waves on a string, water surface waves

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2️⃣ What Actually Moves in a Wave?

In wave motion:

• Particles do not travel with the wave
• Each particle oscillates about its mean position
• Net displacement of particle after one cycle is zero

JEE Key Insight:
Waves transfer energy, not mass.

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3️⃣ Types of Mechanical Waves

(A) Transverse Waves

• Particle motion ⟂ wave direction
• Have crests and troughs
• Require shear restoring force

Examples: String waves, surface water waves

(B) Longitudinal Waves

• Particle motion ∥ wave direction
• Have compressions and rarefactions
• Pressure and density variations occur

Examples: Sound waves in air, waves in solids

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4️⃣ Important Wave Terminology

Quantity	Symbol	Meaning
Wavelength	λ	Distance between successive crests/compressions
Frequency	f	Oscillations per second
Time Period	T	Time for one oscillation
Wave Speed	v	Speed of wave propagation
Angular Frequency	ω	2πf
Wave Number	k	2π/λ

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5️⃣ Fundamental Wave Relation

v = fλ

Changing frequency does not change wave speed. Wave speed depends only on the medium.

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6️⃣ Speed of Mechanical Waves

(A) Wave on a Stretched String

v = √(T / μ)

• T → Tension in string
• μ → Mass per unit length

Increasing tension increases speed.
Increasing mass density decreases speed.

(B) Longitudinal Waves

v = √(Elastic Property / Inertia)

In solids → Young’s modulus
In fluids → Bulk modulus

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7️⃣ Principle of Superposition

When two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements.

y_net = y₁ + y₂

This principle forms the basis of interference, standing waves and beats.

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8️⃣ Mathematical Representation of a Wave

y(x,t) = A sin(kx − ωt + φ)

• A → Amplitude
• k → Wave number
• ω → Angular frequency
• φ → Phase constant

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

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9️⃣ Energy in Mechanical Waves

Energy carried by a wave is proportional to the square of amplitude.

JEE Trap:
Amplitude affects energy, not wave speed.

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🔟 Phase & Phase Difference

Phase describes the state of oscillation of a particle.

Δφ = (2π / λ) × Δx

Used extensively in interference and path-difference problems.

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🧠 Stage-1 Exam Thinking Rule

• Identify wave type
• Check medium property controlling speed
• Separate particle motion from wave motion
• Remember: frequency remains constant across media

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Stage 1 – Page 1 Completed Successfully ✅