SHM & Waves

Concept overview

Simple harmonic motion is any motion whose restoring force is proportional to displacement: F = −kx. The resulting differential equation d²x/dt² = −ω²x has sinusoidal solutions x(t) = A cos(ωt + φ) with angular frequency ω = √(k/m). Every SHM system has this same equation — spring-mass, simple pendulum, compound pendulum, torsional pendulum, LC circuit.

For a spring-mass system, ω = √(k/m) and period T = 2π√(m/k). For a simple pendulum with small oscillations, ω = √(g/L) and T = 2π√(L/g). Energy in SHM oscillates between kinetic ½mv² and potential ½kx² with total E = ½kA² constant.

Waves carry energy without transporting matter. A wave on a string with tension T and mass per unit length μ travels at v = √(T/μ). Sound in air travels at v = √(γP/ρ). The wave equation ∂²y/∂t² = v² ∂²y/∂x² has travelling-wave solutions y(x, t) = A sin(kx − ωt), where k = 2π/λ and ω = 2πf. The dispersion relation ω = vk connects them.

Superposition explains interference and beats. Two coherent waves of similar frequencies produce beats at Δf = |f₁ − f₂|. The Doppler effect shifts the observed frequency for a source and observer in relative motion: f' = f(v ± v_o)/(v ∓ v_s), where the top signs are for approach and bottom for recession.

Key formulas & concepts

• SHM differential equation: d²x/dt² = −ω²x; solution x = A cos(ωt + φ)
• Spring-mass ω: ω = √(k/m); T = 2π√(m/k)
• Simple pendulum (small θ): T = 2π√(L/g)
• Energy in SHM: E = ½kA² (constant total)
• Wave on string: v = √(T/μ)
• Sound in air: v = √(γP/ρ) ≈ 340 m/s at STP
• Beat frequency: f_beat = |f₁ − f₂|
• Doppler (source moving): f' = f × v / (v ∓ v_s)

Common mistakes to avoid

• Applying T = 2π√(L/g) to large-amplitude pendulum oscillations — the formula assumes small angle.
• Confusing angular frequency ω with linear frequency f — ω = 2πf, and formulas usually take ω.
• Ignoring phase difference when adding two SHMs — resultant amplitude depends on the phase.
• Using the wrong sign convention in the Doppler formula — always start from the base f' = f × v_apparent / v.

JEE Main relevance

JEE Main asks 2 questions every paper — usually one SHM period/frequency and one wave-related question (superposition, Doppler, or beats). Formula-driven with predictable setups.

JEE Advanced relevance

Advanced problems combine SHM with damped/driven oscillations, or waves with reflection/transmission at interfaces. Doppler problems with both source and observer moving are also standard.

Previous-year questions (PYQ)

SHM period problems and Doppler applications dominate the PYQ archetype pool. Practise both in chapter-wise sessions.