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🌊 Waves – JEE Advanced Objective Library

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🌊 WAVES (Mechanical Waves)

Stage 1 – Page 3 : Mathematical Description of Waves (JEE Core)

1️⃣ Standard Wave Equation

The most important equation in wave mechanics is the progressive wave equation.

y(x,t) = A sin (kx − ωt + φ)

Symbol	Meaning
A	Amplitude
k	Wave number = 2π/λ
ω	Angular frequency = 2πf
φ	Initial phase

2️⃣ Wave Velocity from Equation

Comparing phase:

kx − ωt = constant

Wave velocity:

v = ω / k = fλ

JEE Warning:
This velocity is wave speed, not particle speed.

3️⃣ Types of Wave Equations

y = A sin(kx − ωt) → Wave moving in +x direction
y = A sin(kx + ωt) → Wave moving in −x direction

Shortcut:
Minus sign → forward wave
Plus sign → backward wave

4️⃣ Displacement vs Position Graph (y vs x)

If time is fixed (t = constant):

Graph shows wave shape
Distance between two crests = wavelength (λ)
Amplitude = maximum height

This graph tells:

λ directly
No information about time or frequency

5️⃣ Displacement vs Time Graph (y vs t)

If position is fixed (x = constant):

Graph shows SHM of particle
Time period = T
Frequency = 1/T

This graph tells:

Particle motion
No information about wavelength

6️⃣ Phase Concept (Very Important)

Phase of a wave at any point:

Phase = (kx − ωt + φ)

Same phase → particles move identically
Phase difference decides interference

Phase difference between two points:

Δφ = kΔx = (2π/λ)Δx

7️⃣ Phase Difference – Special Cases

Δφ = 0 → Same phase
Δφ = π → Opposite phase
Δφ = π/2 → Quarter cycle apart

IIT Tip:
Phase problems are easier than time-based problems.

8️⃣ Differentiation of Wave Equation

Velocity of particle:

v = ∂y/∂t = −Aω cos(kx − ωt)

Acceleration of particle:

a = −ω²y

Acceleration always towards mean position
Particle executes SHM

9️⃣ Important JEE Traps

Confusing y–x graph with y–t graph
Finding λ from y–t graph ❌
Using wave speed as particle speed ❌
Ignoring initial phase φ

🧠 Exam Thinking Method

Fix x → SHM
Fix t → wave shape
Differentiate only for particle quantities

Stage 1 – Page 3 Completed ✅