Oscillations and Waves is really two chapters that share a mathematical backbone — simple harmonic motion describes something that repeats in place, and a wave is what happens when that same repeating motion propagates outward through a medium.
SHM is defined by one precise condition: restoring force proportional to displacement and directed opposite to it. Every SHM system — a spring, a simple pendulum, a physical pendulum — is really the same differential equation with a different expression for the restoring constant, which is exactly why their time period formulae all share the same underlying structure. Waves then take this oscillation and let it travel, and the wave equation, superposition principle, and Doppler effect all follow from tracking how that travelling disturbance behaves.
SHM and wave formulae
| Quantity | Formula |
|---|---|
| Displacement in SHM | x = A sin(ωt + φ) |
| Angular frequency | ω = 2π/T = 2πf |
| Time period of a spring-mass system | T = 2π√(m/k) |
| Time period of a simple pendulum | T = 2π√(L/g) |
| Maximum velocity in SHM | vmax = Aω |
| Maximum acceleration in SHM | amax = Aω² |
| Wave speed | v = fλ |
| Wave equation (transverse) | y = A sin(ωt − kx) |
| Doppler effect (source moving toward stationary observer) | f' = f·v/(v − vs) |
Topics covered
- Periodic motion and simple harmonic motion — the defining condition of SHM, and how to recognise it in a physical setup
- Displacement, velocity and acceleration in SHM — their sinusoidal relationships, and their maximum values
- Energy in SHM — how kinetic and potential energy exchange while total mechanical energy stays constant
- Spring-mass and simple pendulum systems — deriving each one's time period from Newton's second law
- Damped and forced oscillations, resonance — what happens when energy loss or an external driving force is introduced
- Wave motion and the wave equation — transverse and longitudinal waves, and the mathematical description of a travelling disturbance
- Superposition of waves — constructive and destructive interference, and standing (stationary) waves
- Beats — the audible result of superposing two waves of slightly different frequency
- The Doppler effect — the frequency shift caused by relative motion between source and observer, in sound and light
Why this chapter matters
In NEET and JEE, SHM energy and velocity-at-a-given-displacement numericals are extremely common, as are spring-pendulum time period comparisons. Doppler effect numericals, particularly with a moving source and stationary observer, are a near-guaranteed inclusion, and standing wave and beats questions frequently combine multiple concepts from this chapter into one problem.
For Class 11 boards, expect a derivation of the simple pendulum or spring-mass time period and a numerical on wave superposition or the Doppler effect.
Mistakes students make in this chapter
- Assuming every periodic motion is SHM. SHM specifically requires restoring force proportional to displacement — a bouncing ball or a swinging pendulum at large amplitude are periodic but not exactly SHM.
- Confusing angular frequency ω with the actual angular velocity of a rotating object. In SHM, ω is a mathematical rate parameter (ω = 2π/T), not necessarily connected to any object physically rotating.
- Forgetting kinetic and potential energy exchange but total energy stays constant in SHM. Total mechanical energy at any instant equals ½kA² regardless of position, though the split between kinetic and potential varies continuously.
- Mixing up transverse and longitudinal wave descriptions. In a transverse wave, particles oscillate perpendicular to wave travel (like a string); in a longitudinal wave, they oscillate parallel to it (like sound) — the wave equation looks similar for both but the physical picture is different.
- Using the wrong sign convention in the Doppler effect formula. Whether the source or observer moves toward or away from the other changes the sign in the denominator or numerator — getting this backwards flips whether frequency increases or decreases.
Worked example: a complete SHM analysis
- Find the angular frequency first. Every other quantity in SHM is built from ω, so this is always step one:ω = 2π/T = 2π/2 = π ≈ 3.14 rad/s
- Find the maximum velocity. It occurs at the mean position, where all the energy is kinetic. Convert the amplitude to metres:v_max = Aω = 0.05 × π ≈ 0.157 m/s
- Find the maximum acceleration. It occurs at the extreme positions, where displacement — and therefore the restoring force — is greatest. Note this is where velocity is zero, not maximum:a_max = Aω² = 0.05 × 9.87 ≈ 0.493 m/s²
- Find the speed at x = 3 cm using the general velocity relation:v = ω√(A² − x²) = π√(0.05² − 0.03²)v = π√0.0016 = π × 0.04 ≈ 0.126 m/s
- Set up the energy condition. Write both energies in terms of displacement and set them equal:KE = ½mω²(A² − x²) · PE = ½mω²x²A² − x² = x² → x² = A²/2
- Solve for x:x = A/√2 = 5/1.414 ≈ 3.54 cm
What the notes contain
The PDF derives the time period of both the spring-mass system and the simple pendulum from Newton's second law, works through SHM energy conservation with solved numericals, and covers standing waves, beats and the Doppler effect with worked examples for each specific source-observer motion case.
How to use this: build the concept from these notes first, then do one quick NCERT pass before the exam for exact wording and diagram labels, and spend whatever time is left on previous year questions.
Questions
What condition defines simple harmonic motion?
The restoring force (or acceleration) must be directly proportional to displacement from equilibrium and always directed opposite to it — F = −kx. Any motion satisfying this exact relationship is SHM, and the PDF works through several examples testing this condition.
Why does a pendulum's time period not depend on its mass?
Because both the restoring force (gravity component) and the inertia (mass) scale together and cancel out in the derivation — the full derivation showing exactly why mass cancels is worked through in the PDF.
How is the Doppler effect different for a moving source versus a moving observer?
Both produce a frequency shift, but the mathematical formulae differ slightly because the physical mechanism differs — a moving source compresses or stretches the wavelength itself, while a moving observer experiences a different wave speed relative to themselves. Both cases are worked through separately.
Does this cover beats and standing waves with worked numericals?
Yes, including the relationship between beat frequency and the difference between two source frequencies, and the node-antinode pattern in standing waves on a string.
All other chapters
Every chapter of Class 11 and Class 12 Physics, Chemistry and Biology — each with its own notes and PDF.