IIT JEE Physics Notes: Waves & Oscillations Set 5

Introduction

Waves and Oscillations is one of the most important and frequently tested topics in IIT JEE — both Main and Advanced. Every year, at least 3 to 5 questions appear directly from this chapter. The good news is that this topic is highly conceptual and formula-driven, which means once you understand the core ideas clearly, you can solve most questions quickly and confidently. These notes cover everything you need to master for the practice paper and beyond.

---

Part 1: Simple Harmonic Motion (SHM) — Core Concepts

SHM is the foundation of this entire chapter. A particle executes SHM when its acceleration is always directed toward the mean position and is proportional to its displacement from it. Mathematically: a = −ω²x

The general equation of motion is: x = A sin(ωt + φ)

where A is amplitude, ω is angular frequency, and φ is the initial phase.

Key quantities you must memorize:

• Velocity at displacement x: v = ω√(A² − x²)
• Maximum velocity (at mean position): v_max = Aω
• Maximum acceleration (at extreme position): a_max = Aω²
• Ratio of max acceleration to max velocity = ω (This is a direct JEE question type)
• Time period: T = 2π/ω

Energy in SHM is a critical concept. Total mechanical energy is constant throughout the motion: E = ½mω²A². Kinetic energy is maximum at the mean position and zero at extremes. Potential energy is zero at the mean position and maximum at the extremes. KE equals PE when displacement x = A/√2 — this is a very common JEE question.

Phase difference between displacement and velocity is always π/2 (velocity leads displacement by 90°). Between displacement and acceleration, it is π.

---

Part 2: Spring-Mass Systems and Pendulums

For a spring of constant k and mass m: T = 2π√(m/k)

Important trick: If a spring is cut into n equal pieces, the spring constant of each piece becomes nk (spring constant is inversely proportional to length). So if one piece is used with the same mass, the new time period is T/√n.

For springs in parallel: k_eff = k₁ + k₂, so ω = √((k₁+k₂)/m) For springs in series: 1/k_eff = 1/k₁ + 1/k₂

For a simple pendulum: T = 2π√(L/g). Time period is independent of mass and amplitude (for small angles). If gravity becomes g/4, the time period doubles. On the Moon, the pendulum swings slower because g is smaller.

Pendulum clock error: In summer, thermal expansion increases effective length L, which increases T, so the clock runs slow. In winter, L decreases, T decreases, and the clock runs fast. This is a frequently tested conceptual question.

---

Part 3: Superposition of SHM

When two SHMs with the same frequency but different phases are superimposed, the resultant amplitude is: A_R = √(A₁² + A₂² + 2A₁A₂cosδ)

where δ is the phase difference between them.

Special cases:

• δ = 0 (same phase): A_R = A₁ + A₂ (maximum)
• δ = π (opposite phase): A_R = |A₁ − A₂| (minimum)
• δ = π/3 and A₁ = A₂ = A: A_R = √3 · A (direct JEE question)
• δ = π/2: A_R = √(A₁² + A₂²)

---

Part 4: Wave Motion — Progressive Waves

A progressive wave carries energy from one point to another without transfer of matter. General equation: y = A sin(kx − ωt)

Key relations:

• Wave speed: v = ω/k = fλ
• k = 2π/λ is the wave number
• ω = 2πf is angular frequency

Speed of transverse wave on a string: v = √(T/μ), where T is tension and μ is linear mass density (mass per unit length). Neither amplitude nor frequency affects wave speed — only tension and linear mass density do. This is tested very frequently.

Power transmitted: P = ½μA²ω²v, which means P ∝ A²ω². Remember this proportionality for MCQs.

Reflection of waves: When a wave travels from a denser medium (lower speed) to a rarer medium (higher speed), the reflected wave has NO phase change. When it reflects from a denser medium (lower speed on the other side), there is a phase change of π. Students often get this reversed — be careful.

Stationary (Standing) Waves: Formed by superposition of two identical waves traveling in opposite directions. Equation: y = 2A cos(kx) sin(ωt). Distance between two adjacent nodes = λ/2. Distance between a node and the nearest antinode = λ/4.

---

Part 5: Sound Waves, Organs, and Doppler Effect

Speed of sound: v = √(γP/ρ) = √(γRT/M). Since v ∝ √T (absolute temperature), doubling temperature makes speed √2 times, not 2 times.

Organ pipes — most tested formulas:

• Open pipe harmonics (all): f_n = nv/2L. Fundamental = v/2L.
• Closed pipe harmonics (odd only): f_n = (2n−1)v/4L. Fundamental = v/4L.
• Ratio of fundamental frequencies of closed to open pipe of same length = 1:2
• Ratio of fundamental to second overtone in closed pipe = 1:5 (since second overtone is the 3rd harmonic: 5v/4L)

Beats: When two sources of slightly different frequencies f₁ and f₂ are sounded together, beats are heard at frequency = |f₁ − f₂|. For 256 Hz and 260 Hz: beats = 4 per second.

Sound intensity and decibels: L = 10 log₁₀(I/I₀). If intensity increases 100 times, sound level increases by 20 dB. A 40 dB sound becoming 100I gives 60 dB.

Displacement node vs. pressure node: A displacement node is a pressure antinode, and vice versa. This is conceptually important and frequently asked.

Doppler Effect: When source moves toward stationary observer: f = f₀ · v/(v − v_s). When source moves away: f = f₀ · v/(v + v_s). When observer moves toward source: f = f₀ · (v + v_o)/v.

For Melde’s experiment on a vibrating string: f ∝ √T. To double the frequency, tension must be quadrupled.

---

Quick Revision Checklist Before Exam

• KE = PE at x = A/√2
• Max acceleration / Max velocity = ω
• Spring cut into n parts: each has constant nk
• Closed pipe supports only odd harmonics; ratio with open pipe = 1:2
• v_sound ∝ √T (not T)
• Reflected wave from rarer medium: no phase change
• Beats = |f₁ − f₂|
• Displacement node = Pressure antinode
• Doppler: source approaching → frequency increases

---

These notes cover every concept tested in the practice paper. Revise each formula with its derivation at least once, and solve the paper again after 2 days without looking at answers to test your real retention. Consistency in practice is what separates JEE qualifiers from the rest.