Waves — NEET Notes

Transverse and longitudinal waves

A mechanical wave is a travelling disturbance in an elastic medium. The particles of the medium do not travel with the wave; they oscillate in place while the pattern of disturbance — and the energy it carries — moves outward. The direction in which those particles oscillate, relative to the direction the wave moves, is what separates the two great families of mechanical waves. In a transverse wave, the particles oscillate perpendicular to the direction of wave propagation: a pulse on a stretched string, ripples on water, the electric and magnetic fields of light. In a longitudinal wave, the particles oscillate parallel to the propagation direction, producing alternating regions of compression and rarefaction: sound in air, a slinky pushed end-on, seismic P-waves.

Whether a medium can carry a transverse wave depends on whether it can sustain shear stress. Solids can; fluids cannot. So transverse mechanical waves travel only through solids and along free surfaces of liquids, while longitudinal waves travel through every elastic medium — solid, liquid or gas. In steel, both types coexist with different speeds. In air, only longitudinal sound is possible. Electromagnetic waves are the great exception: they are transverse, but they need no material medium at all, propagating through vacuum at c = 2.998 × 10⁸ m/s.

Displacement relation in a progressive wave

A sinusoidal progressive wave moving in the +x direction is described by the equation

y(x, t) = A sin(kx − ωt + φ)

Displacement equation of a one-dimensional progressive wave

Five quantities live inside that equation, and every NEET question on the wave equation tests at least one of them. A is the amplitude — the maximum displacement of a particle from its equilibrium position. k is the angular wave number, k = 2π/λ, measured in rad/m; it counts how many radians of phase fit into one metre of space. ω is the angular frequency, ω = 2π/T = 2πf, measured in rad/s; it counts how many radians of phase tick by per second. The combination (kx − ωt + φ) is the phase, and φ is the initial phase at x = 0, t = 0. A wave travelling in the −x direction has the form y = A sin(kx + ωt + φ): the relative signs of kx and ωt encode the direction.

Fix t and the equation gives the snapshot of the wave in space — a sine curve. Fix x and the equation gives the time-history of one particle — also a sine curve, the same simple harmonic motion the previous chapter described, except now neighbouring particles execute it slightly out of phase. As time advances, the entire wave pattern translates rigidly at speed v = ω/k. The wavelength λ is the distance between two consecutive points in the same phase — between successive crests or troughs. The period T is the time for one full cycle at a fixed point.

The speed of a travelling wave

Every progressive wave obeys one universal relation that links its speed, frequency and wavelength:

v = f · λ = ω / k

Wave speed — true for every progressive wave, mechanical or electromagnetic

The speed v is determined by the medium, not by the source. The source decides only the frequency at which it shakes the medium. Once v and f are fixed, λ is fixed too: λ = v/f. Across NEET's syllabus, three families of waves matter, and each has its own speed formula. For a transverse wave on a stretched string, the restoring force is the tension T, the inertia is the linear mass density μ (mass per unit length), and the speed is v = √(T/μ). For a longitudinal wave in a fluid, the restoring property is the bulk modulus B and the inertia is the density ρ: v = √(B/ρ). For an electromagnetic wave in vacuum, the constants of free space alone fix the speed: c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s, the universal speed limit.

Sound is around 15 times slower than the speed of a transverse wave in a steel rail and around a million times slower than light. Yet all three are described by the same v = fλ. Newton's original formula for sound in a gas, v = √(P/ρ), assumed the compressions and rarefactions happen isothermally. Laplace corrected this — the cycles are so fast that no heat flows in or out, so they are adiabatic. The bulk modulus that matters is therefore the adiabatic one, B_ad = γP, where γ = Cₚ/Cᵥ. Putting this back gives v = √(γP/ρ), and for air γ = 7/5 raises the predicted speed from 280 m/s to 331 m/s, in excellent agreement with measurement. Sound speed in a gas grows as √T (since P/ρ ∝ T at constant gas composition) and is essentially independent of pressure at constant temperature.

The principle of superposition

What happens when two waves cross each other in the same medium? Each behaves as if the other were not there, and at any instant the net displacement at every point is just the algebraic sum of the displacements due to each wave alone. After the waves have passed through each other, they emerge with their original shapes intact. This is the principle of superposition, the most quietly important idea in the chapter — every interference pattern, every standing wave, every beat phenomenon is one of its consequences.

"Each wave moves as if the others were not present; the constituents of the medium suffer displacements due to both, and the net displacement is their algebraic sum."

NCERT — the principle of superposition of waves

For two harmonic waves of equal amplitude a and wavelength travelling in the same direction but with a phase difference φ, superposition gives a single resultant wave of the same frequency and wavelength but with amplitude A(φ) = 2a cos(φ/2). When φ = 0 the waves are in phase: A = 2a, the largest possible amplitude — constructive interference. When φ = π the waves are exactly out of phase: A = 0, total cancellation — destructive interference. In between, the amplitude varies smoothly with φ. The principle is linear: it works because the wave equation itself is linear in y. Real media break linearity at high amplitudes — shock waves, sonic booms — but at typical NEET amplitudes the principle holds exactly.

Reflection of waves at boundaries

When a wave hits a boundary, part of it is reflected and part is transmitted into the second medium. At a rigid boundary — a string clamped to a wall, the closed end of a pipe — the reflected wave suffers a phase change of π, equivalent to flipping its sign. The boundary must remain motionless at all times, and the only way the incident and reflected waves can sum to zero displacement at the wall is if they are exactly out of phase. Mathematically, if the incident wave is y = A sin(kx − ωt), the reflected wave is y_r = A sin(kx − ωt + π) = −A sin(kx − ωt).

At an open boundary — a string ending at a free ring on a smooth rod, the open end of an organ pipe — there is no phase change on reflection. The boundary is free to move; the displacement at the boundary doubles rather than vanishing. The reflected wave is y_r = A sin(kx − ωt). The same logic, applied dynamically through Newton's third law, gives the same result: the wall pushes back on the string, generating a wave that differs by π in phase from the incident one.

Standing waves and normal modes

When reflection takes place at two boundaries — a string fixed at both ends, an air column closed at one end — the wave bounces back and forth, and after enough reflections the result is a steady pattern called a standing wave or stationary wave. To see how it arises, superpose an incident wave travelling in +x with its reflection travelling in −x: y₁ = a sin(kx − ωt), y₂ = a sin(kx + ωt). Adding,

y(x, t) = 2a sin(kx) · cos(ωt)

Standing wave — space and time decouple

The key feature: kx and ωt no longer appear in the single combination (kx − ωt). The amplitude 2a sin(kx) depends only on position; the time factor cos(ωt) is the same everywhere. Every particle oscillates at the same frequency ω, but with an amplitude that varies from point to point. Points where sin(kx) = 0 — that is, kx = nπ, so x = nλ/2 — have zero amplitude and never move; these are the nodes. Points where |sin(kx)| = 1, at x = (n + ½)λ/2, oscillate with maximum amplitude 2a; these are the antinodes. The distance between consecutive nodes is λ/2; between a node and the next antinode, λ/4.

String fixed at both ends — all harmonics

A string of length L clamped at both ends must have nodes at x = 0 and x = L. The first condition is automatic from y = 2a sin(kx) cos(ωt). The second forces sin(kL) = 0, that is, kL = nπ, so the allowed wavelengths are λ = 2L/n for n = 1, 2, 3, … and the natural frequencies of the string are

f_n = n · v / 2L = (n/2L) · √(T/μ),   n = 1, 2, 3, …

The lowest of these, f₁ = v/2L, is the fundamental or first harmonic. The next, f₂ = 2f₁, is the second harmonic; f₃ = 3f₁ is the third; and so on. A vibrating string supports all integer multiples of the fundamental — every harmonic is allowed. When you pluck a sitar string, you excite many of these modes simultaneously, and their relative amplitudes determine the timbre. Where you pluck matters: plucking at the centre excites odd harmonics strongly and damps even ones, because the centre is a node for even modes.

Open pipe vs closed pipe

An air column in a pipe behaves like a string in reverse — the open ends are antinodes (displacement maxima, pressure nodes), and any closed end is a node. The boundary conditions determine which harmonics are allowed.

For a pipe open at both ends, each end is an antinode, so L must hold an integer number of half-wavelengths: L = nλ/2. The allowed frequencies are f_n = n · v/2L for n = 1, 2, 3, … All harmonics — first, second, third, fourth — are present, just as on a string. The fundamental is f₁ = v/2L.

For a pipe closed at one end (open at the other), the closed end is a node and the open end is an antinode. The distance from a node to the next antinode is λ/4, so L must equal an odd number of quarter-wavelengths: L = (2n − 1)λ/4 for n = 1, 2, 3, …, which gives f_n = (2n − 1) · v/4L. The fundamental is f₁ = v/4L — exactly half the open-pipe fundamental. The next allowed mode is f₃ = 3v/4L; then f₅ = 5v/4L. Only odd harmonics are present. A closed pipe of the same length sounds an octave lower than an open one, and richer in odd partials.

Beats — when close frequencies interfere

When two harmonic sound waves of nearly equal frequency are heard together, the ear picks up a single average pitch — but also a slow waxing and waning of loudness. That throbbing is a beat. To see how it arises, take two waves at a fixed location: s₁ = a cos ω₁t and s₂ = a cos ω₂t. By superposition,

s = a cos ω₁t + a cos ω₂t = [2a cos ω_bt] · cos ω_at

where ω_a = (ω₁ + ω₂)/2 is the average angular frequency and ω_b = (ω₁ − ω₂)/2 is half the difference. The factor cos(ω_at) is the rapid oscillation the ear hears as a tone. The bracket [2a cos ω_bt] is a slowly varying envelope — its amplitude is maximum every half-period of ω_b, so the loudness peaks twice per cycle of cos(ω_bt). The beat frequency — the rate at which loudness peaks — is therefore

f_beat = |f₁ − f₂|

Beat frequency — the difference of the two close frequencies

The ear can resolve beats up to about 10 Hz; beyond that, the throbbing blurs into a low buzz. Musicians use beats to tune instruments: two strings sound exactly in tune when no beats remain. NEET 2020 used this in a clever way — two guitar strings produce 6 Hz beats; when one string's tension is reduced its frequency drops, and if the beats increase to 7 Hz, the lowered string must have been on the lower side of the other to begin with. So if string A is 530 Hz and beats with B at 6 Hz, B could be 524 Hz or 536 Hz. Decreasing B's tension reduces its frequency. If B were 536 Hz it would move toward 530 Hz (beats decrease); if B were 524 Hz it would move away from 530 Hz (beats increase). The observed increase identifies B as 524 Hz.

Doppler effect in sound

When source and observer move relative to each other, the frequency the observer hears differs from the frequency the source emits. Stand on a railway platform and listen to the engine's whistle: the pitch is higher as the train approaches, lower as it recedes. This is the Doppler effect, discovered by Christian Doppler in 1842. It works for every kind of wave — sound, light, radar, the cosmic redshift that tells us the universe is expanding — but the formulas for sound (where the medium matters) and light (where only relative motion matters) differ slightly. Here we treat sound.

Three quantities enter: the speed of sound in the medium, v; the velocity of the source along the source-to-observer line, v_s; and the velocity of the observer along the same line, v_o. The trick is to derive the result in two steps — first allow only the source to move, then only the observer, then combine.

The signs of v_s and v_o follow one rule: positive in the direction from source to observer. If the source moves toward the observer, v_s is positive — and the denominator (v − v_s) shrinks, pushing f′ above f. If the observer moves toward the source, v_o is positive (because the convention is along source→observer, but the observer's motion toward source means against this, so by NCERT's convention v_o is taken positive when motion is along source→observer; carefully, the standard form is f′ = f · (v + v_o)/(v − v_s) with v_o, v_s positive when motion brings source and observer closer). Recede instead of approach and the signs flip. NEET 2017 asked the case where both cars move toward each other at 22 m/s and 16.5 m/s, with v = 340 m/s and f = 400 Hz: f′ = 400 × (340 + 16.5)/(340 − 22) ≈ 448 Hz. NEET 2016 set a clever echo problem — siren moves at 15 m/s toward a cliff, observer at rest behind the siren. The cliff acts first as a stationary observer (hears 800 × 330/(330 − 15) = 838 Hz) then as a stationary source emitting that 838 Hz back. The observer behind the siren hears 838 Hz, with the second leg adding no shift because both observer and reflector are at rest.