Waves

The wave equation v = fλ

For any wave, speed v = fλ, where f is frequency and λ is wavelength. Speed of sound in air ≈ 340 m/s; speed of light in vacuum c = 3×10⁸ m/s. A 256 Hz tuning fork in air has λ = 340/256 ≈ 1.33 m. On a stretched string, v = √(T/μ), where T is tension and μ = mass per unit length. Doubling tension increases speed by √2, not 2.

Transverse vs longitudinal waves

In transverse waves the medium oscillates perpendicular to wave propagation (waves on strings, electromagnetic waves); in longitudinal waves parallel (sound in air, P-waves in earthquakes). Only transverse waves can be polarised — Polaroid sunglasses exploit this. Sound cannot be polarised, a fact used to confirm its longitudinal nature.

Superposition: interference, beats, standing waves

When two waves overlap, displacements add. Constructive interference: path difference = nλ; destructive: (n+½)λ. Two waves of slightly different frequencies f₁ and f₂ produce beats at frequency |f₁ − f₂|. Standing waves on a string fixed at both ends have wavelengths λ_n = 2L/n and frequencies f_n = nv/(2L); a closed organ pipe has only odd harmonics f_n = (2n−1)v/(4L).

Doppler effect

For sound, observed frequency f' = f·(v ± v_o)/(v ∓ v_s), with sign convention: source/observer moving toward each other increase f, away decrease f. An ambulance siren of 800 Hz approaching at 30 m/s is heard at f' = 800·340/(340−30) ≈ 877 Hz. For light, the relativistic Doppler formula f' = f√((1−β)/(1+β)) applies, but for MDCAT non-relativistic approximations suffice.

Worked patterns and pitfalls

A common MCQ: a string of length 0.5 m has v = 200 m/s; its fundamental is f₁ = v/(2L) = 200 Hz. The third harmonic is 600 Hz, not 800 — a closed pipe's third harmonic differs because only odd modes are present. Beat-frequency MCQs trap students into adding instead of subtracting. The key references are HRW Chapters 16-17, Serway Chapter 17, and FSc Punjab Chapter 8 (Waves) and Chapter 9 (Physical Optics) for interference details.