Waves— Notes, Formulas & Revision

Transverse wave: medium particles oscillate PERPENDICULAR to direction of wave propagation.

Examples: waves on a string, light (EM waves), water surface ripples.

Equation: y(x,t) = A·sin(kx − ωt), where A = amplitude, k = wave number, ω = angular frequency.

Polarisation is possible only in transverse waves — direction of oscillation can be in any plane perpendicular to propagation.

Speed: v = ω/k = fλ. Determined by medium properties.

For waves on a string of tension T and linear mass density μ: v = √(T/μ).

Cannot propagate in fluids (gases, ideal liquids) — no shear resistance.

Carry energy and momentum without transferring matter.

Longitudinal wave: medium particles oscillate PARALLEL to direction of wave propagation.

Examples: sound in air/water, P-waves (compression seismic waves), waves along a Slinky.

Compressions and rarefactions alternate along the wave direction.

Speed in a fluid: v = √(B/ρ), where B = bulk modulus, ρ = density.

Sound in air at 20°C: v ≈ 343 m/s. In water: v ≈ 1500 m/s. In steel: v ≈ 5000 m/s.

CANNOT be polarised — oscillation direction is set by propagation.

Can propagate in solids, liquids, gases — wherever pressure can transmit force.

Same wave equation y(x,t) = A·sin(kx − ωt) but y is now particle displacement along x.

Wave propagation: disturbance travels through a medium, transferring energy WITHOUT transferring matter.

Particles oscillate locally; the WAVE PATTERN moves at v = ω/k = fλ.

Wave nature is preserved: same A, f, ω as it propagates (in non-dispersive, lossless medium).

Reflection at boundary: fixed end → inverted reflection. Free end → erect reflection.

Refraction: wave speed changes when entering a new medium ⇒ wavelength changes (frequency stays).

Diffraction: bending around obstacles or through apertures. Significant when aperture size ≈ wavelength.

Superposition: when two waves meet, displacements ADD (algebraic). Each then continues independently.

Wave equation: ∂²y/∂t² = v²·∂²y/∂x² — satisfied by any traveling-wave solution.

Fundamental wave relation: v = f·λ, true for ALL waves (mechanical and EM).

Wavelength λ: spatial period — distance between consecutive crests (or any equivalent points).

Frequency f: temporal period — number of full waves passing per second. Period T = 1/f.

Angular frequency ω = 2πf. Wave number k = 2π/λ.

Speed: v = ω/k = fλ. Depends on the medium (for mechanical waves) or fundamental constants (for EM in vacuum).

In a given medium, f is set by the source; λ adjusts.

When the wave crosses media, v changes, λ changes, f stays the same.

Audible sound: 20 Hz - 20 kHz, λ = 17 m down to 1.7 cm in air.

Wave speed on a stretched string: v = √(T/μ), where T = tension (N), μ = mass per unit length (kg/m).

Depends ONLY on the medium (string), not on the frequency or amplitude.

Increasing tension ⇒ faster wave. Increasing μ (heavier string) ⇒ slower wave.

Wavelength λ adjusts to keep v = fλ correct in each medium.

Pluck a guitar string: travel up and back to form a standing wave; tuning is done by adjusting T (peg) or μ (different strings).

Two strings with same T but different μ have different speeds — basis of musical-instrument variety.

Power transmitted by a sinusoidal wave: P = ½μω²A²v.

If string crosses a boundary (e.g., light to heavy section), partial reflection + transmission occur.

Interference: superposition of two or more waves resulting in a new pattern of amplitudes.

CONSTRUCTIVE (in phase, Δφ = 2nπ): amplitudes ADD ⇒ A_total = A₁ + A₂.

DESTRUCTIVE (out of phase by π, Δφ = (2n+1)π): amplitudes SUBTRACT ⇒ A_total = |A₁ − A₂|.

Generally: A_total = √(A₁² + A₂² + 2A₁A₂·cos Δφ).

Intensity: I ∝ A² ⇒ I_max = (A₁ + A₂)², I_min = (A₁ − A₂)² for coherent sources.

COHERENT sources required: constant phase relation and same frequency.

Examples: Young's double slit (light), two loudspeakers (sound), two-source interference patterns.

Path difference Δx → phase difference Δφ = 2π·Δx/λ. Bright fringe: Δx = nλ. Dark fringe: Δx = (n+½)λ.

Beats: when two waves of slightly different frequencies superpose, the resultant amplitude OSCILLATES at a low 'beat' frequency.

Beat frequency f_beat = |f₁ − f₂|. Always non-negative.

Used in musical tuning — listen for beats between two strings; tune until beats disappear (f₁ = f₂).

Mathematically: y₁ + y₂ = 2A·cos(2π·(f₁−f₂)/2·t)·cos(2π·(f₁+f₂)/2·t). Slow envelope × fast carrier.

Envelope frequency = (f₁−f₂)/2; we hear beats at TWICE this rate because |envelope| has period halved.

Beat phenomenon is a TIME-DOMAIN manifestation of constructive/destructive interference.

Audible beat range: f_beat ~ 0-15 Hz. Higher beats are perceived as roughness or two distinct tones.

Lasers: heterodyne detection uses beat frequencies between two slightly detuned lasers.

Standing wave: pattern formed by superposition of two equal-amplitude waves traveling in opposite directions.

y(x,t) = 2A·sin(kx)·cos(ωt) — separates space and time dependence.

Has fixed NODES (zero amplitude) and ANTINODES (max amplitude). Particles oscillate but pattern doesn't propagate.

Distance between consecutive nodes: λ/2. Same for consecutive antinodes.

Distance from node to nearest antinode: λ/4.

Found on: vibrating strings (musical instruments), open/closed pipes, microwave ovens, laser cavities.

All points between two nodes oscillate IN PHASE; on opposite sides of a node, π out of phase.

Energy transport: ZERO in a perfect standing wave (no net flow).

NODES: points of ZERO displacement in a standing wave — always at rest.

ANTINODES: points of MAXIMUM displacement — oscillate with amplitude 2A.

Nodes and antinodes ALTERNATE along the wave; spacing between consecutive nodes (or antinodes) = λ/2.

Distance from node to NEAREST antinode = λ/4.

For a string FIXED at both ends: ends are nodes. Allowed λ = 2L/n (n = 1, 2, 3, …).

For a tube CLOSED at one end (open at other): closed end is node, open is antinode. Allowed λ = 4L/(2n−1).

For a tube OPEN at both ends: both ends are antinodes. Allowed λ = 2L/n.

Number of nodes & antinodes determines the HARMONIC number.

Sound is a LONGITUDINAL pressure wave — alternating compressions and rarefactions of the medium.

Speed in air at 20°C: v ≈ 343 m/s. In water: ~1500 m/s. In steel: ~5000 m/s.

Sound requires a MEDIUM — cannot propagate in vacuum (unlike EM waves).

Speed in fluid: v = √(B/ρ), with B = bulk modulus, ρ = density.

In gas: v = √(γRT/M) — depends on T (not P or ρ separately).

Speed of sound rises ~0.6 m/s per °C in air. Doubling T (Kelvin) increases v by √2.

Audible range: 20 Hz - 20 kHz. Below: infrasound. Above: ultrasound.

Sound intensity: power per unit area, measured in W/m² or in decibels (logarithmic).

Sound speed depends on the MEDIUM. Generally: solid > liquid > gas.

Air at 20°C: ~343 m/s. Water: ~1500 m/s. Steel: ~5000 m/s. Glass: ~5000 m/s. Rubber: ~60 m/s.

Speed in a medium: v = √(B/ρ) for fluids, √(Y/ρ) for thin rod in solid, √(γP/ρ) for ideal gas.

Sound in solids can be longitudinal OR transverse (shear). Sound in fluids only longitudinal.

In gas: v ∝ √T (Kelvin), independent of pressure. Air at 0°C: 331 m/s.

Diatomic gas (air ~80% N₂): γ = 1.4. Monatomic (helium): γ = 5/3.

Sound speed in helium is 3× that in air at same T (lighter gas, lower M) — high-pitched 'Donald Duck' voice.

Underwater communication uses sound because water's slow attenuation (vs radio waves).

Doppler effect: apparent frequency of a wave changes when SOURCE moves relative to OBSERVER.

Source moving TOWARD stationary observer: apparent f INCREASES (waves compressed in front).

Source moving AWAY from stationary observer: apparent f DECREASES (waves stretched behind).

Formula (source moves, observer stationary): f' = f × v/(v ∓ v_s). Minus when source approaches.

v_s = source speed; v = wave speed (in medium).

Higher harmonics shift more than fundamental (in relative terms).

If v_s ≥ v: produces a sonic boom (cone of compressed waves at Mach angle θ = sin⁻¹(v/v_s)).

Used in: weather radar (raindrop velocity), police radar (car speed), medical Doppler ultrasound (blood flow), astronomy (redshift of galaxies).

Observer moving, source stationary: apparent frequency changes.

Observer moving TOWARD source: encounters more waves per second ⇒ f' > f.

Observer moving AWAY from source: fewer waves per second ⇒ f' < f.

Formula: f' = f × (v ± v_o)/v. Plus when observer approaches.

v_o = observer speed; v = wave speed.

Same overall direction of shift as moving source, but DIFFERENT formula.

Magnitude depends on RATIO v_o/v.

Combined source + observer motion uses both numerator AND denominator changes.

When BOTH source and observer move, use the COMBINED Doppler formula.

f' = f·(v ± v_o)/(v ∓ v_s). Top sign: observer approaches source. Bottom sign: source approaches observer.

Approach geometry: + in top, − in bottom ⇒ shift UP.

Receding geometry: − in top, + in bottom ⇒ shift DOWN.

Sign convention is the trickiest part — drawing a diagram helps.

Special case: if source and observer move at same velocity, f' = f (no relative motion).

Applications: aircraft tracking radar (both moving), astronomy (both rotating), military sonar (both ships moving).

Relativistic effects matter when speeds approach c — modified formulas for light.

String fixed at both ends: standing-wave modes have wavelengths λ_n = 2L/n, frequencies f_n = nf₁ (n = 1, 2, 3, …).

Fundamental (n = 1): one antinode in middle, nodes at both ends. Lowest frequency.

n-th harmonic: n antinodes, n+1 nodes.

All integer multiples of f₁ are PRESENT — string can produce 1st, 2nd, 3rd, 4th, … harmonics. (Difference from closed pipe.)

Frequency: f_n = nv/(2L) = (n/2L)·√(T/μ).

Higher harmonics give a 'richer' sound (timbre depends on harmonic mix).

Plucking position determines WHICH harmonics get excited. Plucking center excites odd harmonics; plucking off-center excites both.

Tuning: change T (peg) or L (finger) or μ (different string).

Organ pipe: a tube with sound standing waves. Two main types: OPEN at both ends, or CLOSED at one end.

OPEN pipe: both ends antinodes (air can move freely). λ_n = 2L/n. ALL harmonics: f₁, 2f₁, 3f₁, …

CLOSED pipe: closed end node (air can't move), open end antinode. λ_n = 4L/(2n−1). Only ODD harmonics: f₁, 3f₁, 5f₁, …

Open pipe fundamental: f₁ = v/(2L). Closed pipe fundamental: f₁ = v/(4L) — HALF the open pipe.

Same length L: open pipe sounds an OCTAVE higher than closed pipe.

Pipe organs use both types for different sounds; reed instruments (clarinet) behave like closed pipes; flute behaves like open pipe.

End correction: actual antinode is slightly OUTSIDE the open end — small correction to L.

Sound speed in pipe: same as in surrounding air (v depends only on temperature, not pipe geometry).

Wave transports ENERGY without transporting matter. Particles oscillate locally; energy moves at wave speed.

Power transmitted by a sinusoidal wave on a string: P = ½μω²A²v.

Quadratic in BOTH amplitude (A) and frequency (ω). Doubling either ⇒ quadruple power.

Intensity I = P/area. For 3D sound waves: I = ½ρvω²A² (W/m²).

Inverse-square law: from a point source, I drops as 1/r² (spreading over sphere of area 4πr²).

Sound levels: decibel scale is logarithmic; 10 dB increase = 10× intensity = √10 ≈ 3.16× amplitude.

Damping/absorption: real waves lose energy as they travel — light absorbed by glass, sound absorbed by walls.

Energy in standing waves: bounces back and forth, no net transport.

Phase difference Δφ: how much one oscillation leads or lags another (in radians or degrees).

Two waves in PHASE (Δφ = 0, 2π, 4π…): crests align, displacements add (constructive).

Two waves OUT OF PHASE by π (180°): crest of one aligns with trough of other, cancel.

Phase difference from path difference: Δφ = (2π/λ)·Δx.

Constructive interference: Δφ = 2nπ (path difference = nλ).

Destructive: Δφ = (2n+1)π (path difference = (n+½)λ).

On phasor diagram: two oscillations are vectors with angle Δφ between them.

Phase difference can be due to: PATH difference, initial PHASE offset, REFRACTION through different medium, REFLECTION at boundary (which can add π).