Waves

The previous chapter studied a single particle oscillating in place. This one lets the oscillation travel. A wave carries energy and momentum from one point to another without carrying the medium with it — the cork bobs, the ripple moves on — and that one sentence explains everything from a plucked string to a radio signal. The chapter separates transverse waves, where the medium oscillates across the direction of travel, from longitudinal waves, where it oscillates along it, and explains why sound in air must be the second kind. The travelling wave is then written down once and used everywhere: $$y(x,t) = a\sin(kx - \omega t + \phi)$$ with $a$ the amplitude, $k = \frac{2\pi}{\lambda}$ the angular wave number, $\omega = 2\pi\nu$ the angular frequency, and the sign between $kx$ and $\omega t$ deciding which way the wave goes. Holding the phase constant gives the wave speed $v = \nu\lambda = \frac{\omega}{k}$, a speed set by the medium and not by the source — so carrying a note into water changes its wavelength but never its frequency. Two speeds are then derived: $v = \sqrt{\frac{T}{\mu}}$ for a transverse wave on a stretched string, and $v = \sqrt{\frac{\gamma P}{\rho}}$ for sound in a gas, where Newton's formula assumed isothermal compressions and came out 15% low until the Laplace correction made them adiabatic. The principle of superposition — displacements add algebraically and the waves then carry on unchanged — produces interference, and, once a wave meets a boundary and reflects, the standing wave $y = 2a\sin kx\cos\omega t$, which does not travel at all. Boundary conditions then quantise everything: a string fixed at both ends supports $\nu_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$ and all harmonics; an open pipe supports all of them too; and a closed pipe supports only the odd ones, so its first overtone is its third harmonic — the most reliable trap in the chapter. Two close frequencies give beats at $\lvert\nu_1-\nu_2\rvert$, and relative motion between source and observer gives the Doppler effect, whose whole difficulty is a sign convention. Topics the rationalised syllabus trimmed — the Doppler effect in full, with the reflecting-surface and moving-medium cases and the sonic boom, and the overtone naming that exam questions turn on — are restored here, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.