Questions
5–6 MCQs per paper
Difficulty
Medium-Hard
Importance
High yield for JEE Main/Advanced and NEET
Overview
Waves describe the transmission of energy and momentum through a medium via periodic disturbances. This topic is fundamental for JEE and NEET as it forms the basis for optics, acoustics, and modern physics, requiring a solid grasp of phase relationships and wave equations.
Wave Motion and Speed
A wave is represented by the function y(x,t) = f(ax ± bt). The velocity of a wave is determined by the properties of the medium, specifically its elasticity and inertia.
- Wave speed v = ω/k
- Phase velocity v = fλ
- Speed in a stretched string v = √(T/μ)
- Speed of sound in gas v = √(γP/ρ)
- Particle velocity vp = -v * (dy/dx)
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Superposition and Interference
The principle of superposition states that the resultant displacement is the algebraic sum of individual displacements. This leads to constructive and destructive interference based on path difference.
- Constructive interference: Path difference Δx = nλ
- Destructive interference: Path difference Δx = (2n-1)λ/2
- Resultant amplitude A = √(A1^2 + A2^2 + 2A1A2 cosφ)
- Intensity I proportional to A^2
Standing Waves
Standing waves form when two identical waves travel in opposite directions in a bounded medium. Energy is localized between nodes and antinodes rather than being transported through space.
- Equation: y = 2A sin(kx) cos(ωt)
- Nodes occur at x = nλ/2
- Antinodes occur at x = (2n-1)λ/4
- Fundamental frequency of string (fixed-fixed): f = v/2L
- Fundamental frequency of organ pipe (closed): f = v/4L
Beats and Doppler Effect
Beats arise from the superposition of waves with slightly different frequencies, creating a periodic rise and fall in intensity. The Doppler effect describes the frequency shift perceived due to relative motion.
- Beat frequency fb = |f1 - f2|
- General Doppler formula: f' = f(v ± vo)/(v ± vs)
- Use sign convention: towards increases frequency, away decreases frequency
- Observer moving towards source increases f'
Formula Sheet
y = A sin(ωt ± kx + φ)
v = √(T/μ)
f' = f(v + vo)/(v - vs)
fb = |f1 - f2|
I = I1 + I2 + 2√(I1I2) cosφ
Exam Tip
Always verify the phase difference between two waves before applying intensity formulas; a simple sign error in path difference calculations is the most common reason for lost marks.
Common Mistakes
- Confusing the phase velocity with the particle velocity (vp is the slope of the wave profile multiplied by wave speed).
- Applying the Doppler effect formula without proper sign conventions for the velocity of the source and observer.
- Ignoring boundary conditions (open vs closed ends) when calculating harmonics for organ pipes.
More Revision Notes
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