Waves: NEET notes

1. Waves carry energy, not matter

NCERT §14.1

• Drop a pebble into a still pond and rings spread outward over the surface. A cork floating on the water only bobs up and down where it is; it does not ride out with the rings. The disturbance moves, the water stays.
• A wave is a pattern of disturbance that travels without the medium as a whole travelling with it. Wind is air moving from place to place; sound is not.
• Waves carry energy, and the pattern can carry information: speech, music, radio and light signals all travel as waves.
• Mechanical waves, such as sound, water waves and seismic waves, obey Newton's laws and need a material medium: water, air, rock.
• Electromagnetic waves, such as visible light, ultraviolet, radio, microwaves and X-rays, need no medium and cross a vacuum. In vacuum they all travel at the same speed c = 299,792,458 m s⁻¹.
• Matter waves describe electrons, protons, neutrons and other particles; the electron microscope is an application of them.
• The physics of waves grew out of the study of springs and pendulums in the seventeenth century; Christiaan Huygens (1629-1695), Robert Hooke and Isaac Newton are among the names linked with it.
• A medium passes a disturbance on because its parts are coupled by elastic forces. Picture a chain of balls joined by springs: pull one end and each spring, as it stretches, tugs the next ball, so the disturbance spreads along the chain.
• Air works the same way. Compressing a small region raises its density and pressure, that region pushes on the next, and a sound wave passes on as a travelling change of density. In a solid the atoms of the lattice behave like masses held by springs.

2. Transverse and longitudinal waves

NCERT §14.2

• Give one end of a stretched string a single up-and-down jerk and a pulse runs along it. A continuous periodic jerk sends a train of pulses: a sinusoidal wave if the end moves in SHM.
• In a transverse wave the particles of the medium oscillate at right angles to the direction the wave travels, as on the string.
• In a longitudinal wave the particles oscillate back and forth along the direction of travel. A piston pushed and pulled at one end of an air-filled pipe sends out regions of high density, compressions, alternating with regions of low density, rarefactions. Sound in air is longitudinal.
• Both are progressive (travelling) waves: the pattern moves from one part of the medium to another.
• A transverse wave needs a medium that resists a change of shape (shear), so transverse mechanical waves travel only in solids and along strings, not through the body of a liquid or gas.
• A longitudinal wave needs only resistance to compression, which solids, liquids and gases all have. A steel bar can carry both kinds; air carries only longitudinal waves.
• Surface waves on water come in two types. Capillary waves are ripples no more than a few centimetres long, held by surface tension. Gravity waves have wavelengths from several metres to several hundred metres, and gravity supplies their restoring force.
• In water waves the particles move both up and down and back and forth, and the motion reaches down to the bottom with falling amplitude. Ocean waves combine transverse and longitudinal motion.
• In the same medium, transverse and longitudinal waves generally travel at different speeds.

3. Displacement relation of a progressive wave

NCERT §14.3

• A wave needs a displacement that depends on position and time together. For a harmonic transverse wave on a string moving along +x: y(x, t) = a sin(kx − ωt + φ).
• A general form A sin(kx − ωt) + B cos(kx − ωt) is the same wave with a = √(A² + B²) and φ = tan⁻¹(B/A).
• Changing the sign to y = a sin(kx + ωt + φ) makes the wave travel towards −x instead.
• Amplitude a is the largest displacement of a particle from its mean position; it is taken as positive. A crest is a point of largest positive displacement, a trough a point of largest negative displacement.
• The phase is the whole argument (kx − ωt + φ); it fixes the displacement at any x and t. φ is the initial phase angle, the phase at x = 0 and t = 0.
• Wavelength λ is the least distance between two points in the same phase, for example crest to crest. The angular wave number (propagation constant) k = 2π/λ, in rad m⁻¹.
• Every particle does SHM of angular frequency ω, so the period is T = 2π/ω and the frequency ν = 1/T = ω/2π.
• A longitudinal wave has the same form, s(x, t) = a sin(kx − ωt + φ), where s is the particle's displacement along the direction of travel.
• Worked example (NCERT): y = 0.005 sin(80.0x − 3.0t) in SI units. a = 0.005 m = 5 mm; k = 80.0 rad m⁻¹, so λ = 2π/80.0 = 7.85 cm; ω = 3.0 rad s⁻¹, so T = 2π/3.0 = 2.09 s and ν = 0.48 Hz.
• At x = 30.0 cm and t = 20 s the phase is 80.0 × 0.3 − 3.0 × 20 = −36 rad, which is the same as −36 + 12π = 1.699 rad (about 97°); the displacement is 0.005 sin(1.699) ≈ 5 mm.

4. Wave speed on a stretched string

NCERT §14.4, §14.4.1

• Follow a point of fixed phase, such as a crest. For kx − ωt to stay constant as t grows, x must grow at the rate ω/k, so the wave speed is v = ω/k.
• Using ω = 2πν and k = 2π/λ, v = λν = λ/T. In one period of any particle the pattern moves forward by one wavelength. This holds for every progressive wave.
• The speed of a mechanical wave is set by the medium: an elastic property that supplies the restoring force, and an inertial property, the density.
• On a stretched string the restoring force is the tension T and the inertia is the linear mass density µ = m/L, mass per unit length.
• Dimensional analysis: T/µ has dimensions [MLT⁻²]/[ML⁻¹] = [L²T⁻²], a speed squared, so v = C√(T/µ). Dimensions cannot fix C; the exact derivation gives C = 1, so v = √(T/µ).
• The speed depends only on T and µ, not on the wavelength or frequency. The source sets the frequency, and λ = v/ν then follows.
• Worked example (NCERT): a steel wire 0.72 m long of mass 5.0 × 10⁻³ kg under a tension of 60 N. µ = 5.0 × 10⁻³/0.72 = 6.9 × 10⁻³ kg m⁻¹, so v = √(60/6.9 × 10⁻³) = 93 m s⁻¹.
• Tighten a string and waves on it speed up; use a heavier string and they slow down.

5. Speed of sound: Newton and Laplace

NCERT §14.4.2

• Sound squeezes and relaxes small volumes of the medium, so the elastic property that matters is the bulk modulus B = −ΔP/(ΔV/V), measured in pascal like pressure.
• B/ρ has the dimensions of a speed squared, and the exact result is v = √(B/ρ) for longitudinal waves in any medium.
• In a thin solid bar the sideways bulging is negligible, so Young's modulus replaces B: v = √(Y/ρ).
• Sound is generally faster in liquids and solids than in gases. They are denser, but their bulk moduli are larger by a much bigger factor, because they are far harder to compress.
• Newton assumed the compressions happen at constant temperature. For an ideal gas PV = constant then gives B = P, so v = √(P/ρ): Newton's formula.
• Worked example (NCERT): air of molar mass 29.0 × 10⁻³ kg occupies 22.4 × 10⁻³ m³ at STP, so ρ = 1.29 kg m⁻³. Newton's formula then gives about 280 m s⁻¹, roughly 15% below the measured 331 m s⁻¹.
• Laplace's correction: the pressure changes in sound are too fast for heat to flow in or out, so they are adiabatic. PV^γ = constant gives B = γP and v = √(γP/ρ), where γ = Cp/Cv.
• For air γ = 7/5, and the Laplace formula gives 331.3 m s⁻¹ at STP, in agreement with the measured speed.
• The speed of sound in a gas does not change with pressure at constant temperature, since P/ρ stays fixed; it rises with temperature and with humidity.

6. Principle of superposition

NCERT §14.5

• Two pulses sent towards each other along a string pass straight through each other and come out unchanged.
• While they overlap, the displacement at each point is the algebraic sum of the displacements each pulse alone would cause. This is the principle of superposition: y = y₁ + y₂ + … = Σ fᵢ(x − vt).
• Two equal pulses, one up and one down, cancel for an instant when they exactly overlap and the string looks flat; the string's energy is then all kinetic, and the pulses reappear and move on.
• Two harmonic waves of the same amplitude a, frequency and wavelength, moving the same way but differing in phase by φ, add to y = 2a cos(φ/2) sin(kx − ωt + φ/2).
• The sum is again a harmonic wave in the same direction, with the same ω and k, but its amplitude is A(φ) = 2a cos(φ/2).
• If φ = 0 the waves are in phase and the amplitude is 2a: constructive interference.
• If φ = π, one wave is exactly opposite to the other, so they cancel at every point for all time: destructive interference.

7. Reflection and standing waves

NCERT §14.6

• A wave that meets a boundary is reflected, as an echo is. At a boundary between two media part of it is reflected and part transmitted; the transmitted part obeys the laws of refraction and the reflected part the laws of reflection.
• Rigid boundary (a string tied to a wall): the end cannot move, so the wall pulls the string the opposite way (Newton's third law). The pulse comes back inverted, a phase change of π. If yi = a sin(kx − ωt), then yr = a sin(kx − ωt + π) = −a sin(kx − ωt).
• Free boundary (a string tied to a ring that slides freely on a rod, or the open end of an organ pipe): the pulse comes back upright, with no phase change, and the displacement at the end becomes twice the pulse's.
• A wave reflected at a boundary overlaps the incoming wave. Two identical waves moving in opposite directions, a sin(kx − ωt) and a sin(kx + ωt), add to y = 2a sin kx cos ωt.
• This is a standing wave: kx and ωt appear separately, so the pattern does not travel. Each point does SHM with its own amplitude 2a sin kx.
• Nodes, where sin kx = 0, never move: x = nλ/2 for n = 0, 1, 2, …
• Antinodes, where |sin kx| = 1, swing with the largest amplitude 2a: x = (n + ½)λ/2.
• Neighbouring nodes are λ/2 apart, and so are neighbouring antinodes; a node and the next antinode are λ/4 apart.
• All particles between two neighbouring nodes move in phase, with different amplitudes; particles on opposite sides of a node are in opposite phase.

8. Harmonics of strings and pipes

NCERT §14.6.1

• A string fixed at both ends must have a node at each end. Only standing waves with a whole number of half-wavelengths fit: L = nλ/2, so λ = 2L/n.
• Its natural frequencies, the normal modes, are ν = nv/2L, n = 1, 2, 3, …, with v = √(T/µ).
• The lowest, n = 1, ν₁ = v/2L, is the fundamental mode or first harmonic. n = 2 is the second harmonic, n = 3 the third, and so on; all whole multiples of ν₁ are present.
• A string usually vibrates in a mix of several modes at once. On a sitar or violin, the point where the string is plucked or bowed decides which modes come out stronger.
• Closed pipe (a glass tube partly filled with water): the closed end is a displacement node, where the pressure change is largest; the open end is a displacement antinode, where the pressure change is least.
• So L = (n + ½)λ/2 and ν = (n + ½)v/2L for n = 0, 1, 2, … The fundamental is v/4L, and the others are 3v/4L, 5v/4L, …: only odd harmonics.
• Open pipe: both ends are antinodes, so L = nλ/2 and ν = nv/2L, n = 1, 2, 3, … Every harmonic is present and the fundamental v/2L is twice that of a closed pipe of the same length.
• If a string or air column is driven at a frequency close to one of its natural frequencies, it resonates. A tabla membrane, clamped all round its rim, has normal modes fixed by the rule that no point on the rim moves.
• Worked example (NCERT): a pipe 30.0 cm long with v = 330 m s⁻¹. Open at both ends, ν₁ = 330/0.6 = 550 Hz and the modes are 550n Hz, so a 1.1 kHz source resonates the second harmonic. Closed at one end, the fundamental is 330/1.2 = 275 Hz and only odd multiples occur; 1.1 kHz is four times 275 Hz, an even multiple, so there is no resonance.

9. Beats

NCERT §14.7

• Two sounds of nearly equal frequency heard together give a sound at their average frequency whose loudness swells and fades regularly. This waxing and waning is called beats.
• For two waves of equal amplitude, s = s₁ + s₂ = [2a cos ω_b t] cos ω_a t, with ω_b = (ω₁ − ω₂)/2 and ω_a = (ω₁ + ω₂)/2.
• When |ω₁ − ω₂| is small compared with ω₁ + ω₂, the term cos ω_a t is a fast oscillation and 2a cos ω_b t is a slowly changing amplitude around it.
• The loudness peaks whenever cos ω_b t reaches +1 or −1, which happens twice in each period of cos ω_b t. So the beat frequency is ν_beat = ν₁ − ν₂.
• Example: waves of 11 Hz and 9 Hz give a resultant at 10 Hz that swells and fades 2 times a second.
• Musicians use beats to tune instruments: they adjust one against a reference until the beats slow down and vanish.
• Worked example (NCERT): sitar string A plays Dha at 427 Hz, and 5 beats per second are heard with string B, so B is 422 Hz or 432 Hz. Tightening B raises its frequency and the beats drop to 3 per second; only 422 Hz moves towards 427 Hz, so ν_B = 422 Hz.
• Musical pillars: the Nellaiappar temple in Tamil Nadu, built by the Pandyan dynasty in the 7th century, has stone pillars that give musical notes (Sa Re Ga Ma Pa Dha Ni Sa) when tapped, grouped as Shruti, Gana Thoongal and Laya Thoongal pillars. Such pillars are also found at Hampi, Kanyakumari and Thiruvananthapuram.

Must-know facts

1. A wave carries energy and information; the particles of the medium only oscillate about their mean positions.
2. Mechanical waves need a medium; electromagnetic waves do not and travel at c = 299,792,458 m s⁻¹ in vacuum.
3. Transverse: particle motion ⊥ travel, needs shear, only in solids and strings. Longitudinal: particle motion ∥ travel, in solids, liquids and gases.
4. y = a sin(kx − ωt + φ) travels towards +x; a sin(kx + ωt + φ) travels towards −x.
5. k = 2π/λ, ω = 2π/T = 2πν.
6. v = ω/k = λν = λ/T.
7. String: v = √(T/µ), independent of λ and ν. The source fixes ν; the medium fixes v.
8. Longitudinal: v = √(B/ρ); thin bar v = √(Y/ρ).
9. Newton v = √(P/ρ) ≈ 280 m s⁻¹ in air at STP; Laplace v = √(γP/ρ) = 331.3 m s⁻¹ with γ = 7/5.
10. Superposition of two equal waves with phase difference φ: amplitude 2a cos(φ/2).
11. Rigid end: reflection with phase change π. Free end: no phase change.
12. Standing wave y = 2a sin kx cos ωt: nodes at nλ/2, antinodes at (n + ½)λ/2, spacing λ/2.
13. String fixed at both ends and open pipe: ν = nv/2L, all harmonics.
14. Closed pipe: ν = (n + ½)v/2L; fundamental v/4L, only odd harmonics.
15. Beat frequency = ν₁ − ν₂; 11 Hz and 9 Hz give 2 beats per second.
16. Ex 14.3: 0.72 m, 5.0 g wire at 60 N gives 93 m s⁻¹. Ex 14.6: 427 Hz, 5 then 3 beats, so 422 Hz.

Common traps

Formulas

Key terms

Wave

A disturbance that travels through a medium, or through space, carrying energy without carrying matter along.

Mechanical wave

A wave that needs a material medium and is governed by Newton's laws, such as sound or a wave on a string.

Transverse wave

A wave in which particles oscillate perpendicular to the direction of travel.

Longitudinal wave

A wave in which particles oscillate along the direction of travel.

Compression and rarefaction

Regions of higher and of lower density than normal in a longitudinal wave.

Progressive wave

A wave whose pattern travels from one part of the medium to another.

Crest and trough

Points of largest positive and largest negative displacement.

Wavelength

The least distance between two points in the same phase.

Angular wave number

k = 2π/λ, the phase change per metre.

Linear mass density

Mass per unit length of a string, µ.

Bulk modulus

B = −ΔP/(ΔV/V), the resistance of a medium to compression.

Principle of superposition

Where waves overlap, each point's displacement is the sum, with signs, of what each wave alone would give it.

Standing wave

The non-travelling pattern formed by two identical waves moving in opposite directions.

Node and antinode

Points of a standing wave with zero amplitude and with the largest amplitude.

Normal mode

A natural frequency at which a bounded system such as a string or air column can oscillate.

Fundamental and harmonics

The lowest normal mode and the modes at whole-number multiples of it.

Beats

Regular rise and fall of loudness when two sounds of nearly equal frequency are heard together.