Waves

A mechanical wave is a disturbance that transfers energy through a medium without any net transport of matter; only the disturbance (and energy) travels while individual particles oscillate about their mean positions.

Waves are broadly classified by the relationship between the direction of particle vibration and the direction of wave propagation.

• Transverse wave: particles of the medium vibrate perpendicular to the direction of propagation of the wave. Crests (maximum positive displacement) and troughs (maximum negative displacement) are formed. Example: waves on a stretched string, waves on the surface of water, and electromagnetic waves (which need no medium). Transverse mechanical waves can travel only through media possessing shear elasticity (rigidity), i.e. solids and stretched strings, because sideways restoring forces are required.
• Longitudinal wave: particles of the medium vibrate parallel to (along) the direction of propagation. Regions of crowded particles are called compressions and regions of spread-out particles are called rarefactions. Example: sound waves in air, waves in a spring pushed and pulled along its length. Longitudinal waves need only volume elasticity and can therefore travel through solids, liquids and gases.

For a longitudinal wave, the distance between two consecutive compressions (or two consecutive rarefactions) equals one wavelength, λ, exactly as the distance between two consecutive crests equals one wavelength for a transverse wave.

Worked example.

Given: a sound wave travels through air with speed v = 340 m/s. The distance measured between two successive compressions is 1.7 m.

Formula: wavelength λ = distance between successive compressions; wave speed v = ν λ, so frequency ν = v / λ.

Substitution: λ = 1.7 m, so ν = 340 / 1.7

Result: ν = 200 Hz.

A one-dimensional sinusoidal (harmonic) progressive wave travelling along the positive x-direction is represented as:

y(x, t) = a sin(kx − ωt)

where each symbol has a precise meaning:

• Amplitude (a): the maximum displacement of a particle of the medium from its equilibrium position.
• Wavelength (λ): the distance between two consecutive points that are in the same phase (e.g. two successive crests); related to the propagation constant k by k = 2π/λ.
• Period (T): the time taken by a particle to complete one full oscillation; related to angular frequency ω by ω = 2π/T.
• Frequency (ν): the number of oscillations a particle completes per second, ν = 1/T.
• Wave speed (v): the speed at which the phase (e.g. a crest) advances through the medium, v = ω/k = ν λ.

The quantity (kx − ωt) is called the phase of the wave; every point on the medium having the same phase at a given instant is displaced by the same amount.

Worked example.

Given: a transverse wave travelling on a string is described by y(x,t) = 0.005 sin(80x − 3t), with x, y in metres and t in seconds.

Formula: compare with y = a sin(kx − ωt) to identify a, k, ω; then λ = 2π/k, T = 2π/ω, ν = 1/T, v = ω/k.

Substitution: a = 0.005 m, k = 80 rad/m, ω = 3 rad/s.

• λ = 2π/80 = 0.0785 m ≈ 7.85 × 10⁻² m
• T = 2π/3 = 2.09 s
• ν = 1/2.09 = 0.48 Hz
• v = 3/80 = 0.0375 m/s = 3.75 cm/s

Result: amplitude 5 mm, wavelength ≈ 7.85 cm, period ≈ 2.09 s, frequency ≈ 0.48 Hz, wave speed = 3.75 cm/s (all along +x).

The speed of a mechanical wave is fixed entirely by the properties of the medium (its elasticity and inertia) — it does not depend on the frequency or amplitude of the wave.

• Transverse wave on a stretched string: v = √(T/μ), where T is the tension in the string and μ is the mass per unit length (linear mass density).
• Longitudinal wave in a medium (general): v = √(E/ρ), where E is the relevant modulus of elasticity (Young's modulus Y for a solid rod, bulk modulus B for a liquid) and ρ is the density.
• Longitudinal wave (sound) in a gas — Newton's formula: Newton proposed v = √(P/ρ), assuming compressions and rarefactions occur isothermally. This underestimates the measured speed of sound in air by about 15%.
• Laplace's correction: Laplace pointed out that compressions and rarefactions occur too fast for heat exchange, so the process is adiabatic. The corrected formula is v = √(γP/ρ), where γ = C_p/C_v is the ratio of specific heats of the gas.

Worked example (Laplace's correction for sound in air).

Given: atmospheric pressure P = 1.01 × 10⁵ Pa, density of air ρ = 1.29 kg/m³, γ (air) = 1.4.

Formula: v = √(γP/ρ)

Substitution: v = √[(1.4 × 1.01 × 10⁵) / 1.29] = √(1.096 × 10⁵)

Result: v ≈ 331 m/s, matching the experimentally measured speed of sound in air at 0 °C (Newton's uncorrected formula gives only about 280 m/s, about 15% lower).

The principle of superposition states that when two or more waves travel through the same region of a medium simultaneously, the resultant displacement of any particle at any instant is the algebraic sum of the displacements each wave would have produced individually:

y(x,t) = y₁(x,t) + y₂(x,t) + …

This holds because the wave equation is linear in y, and it explains phenomena such as interference, beats, and standing waves.

Consider two waves of the same frequency and amplitude a, arriving at a point with a phase difference φ between them. Using the superposition principle, the resultant is again a simple harmonic oscillation of amplitude:

R = 2a cos(φ/2)

• When φ = 0, 2π, 4π, … (in phase), R = 2a — constructive interference (maximum amplitude).
• When φ = π, 3π, … (out of phase), R = 0 — destructive interference (waves cancel).

Worked example.

Given: two waves of equal amplitude a₁ = a₂ = 0.02 m and the same frequency arrive at a point with a phase difference φ = 60° = π/3 rad.

Formula: resultant amplitude R = √(a₁² + a₂² + 2a₁a₂ cos φ), which for a₁ = a₂ = a reduces to R = 2a cos(φ/2).

Substitution: R = 2 × 0.02 × cos(30°) = 0.04 × 0.866

Result: R ≈ 0.0346 m = 3.46 cm.

Reflection of waves. When a travelling wave meets a boundary, part or all of it is reflected. The nature of reflection depends on the boundary:

• At a rigid (fixed) boundary — e.g. a string end tied to a wall — the reflected wave undergoes a phase reversal of π (180°); a crest returns as a trough. The fixed point remains a node (zero displacement).
• At a free (open) boundary — e.g. the open end of an air column — the wave is reflected without any phase change, and that end becomes an antinode (maximum displacement).

Standing (stationary) waves. When two identical waves travelling in opposite directions (the incident and reflected waves) superpose, the result is a standing wave — the waveform no longer travels; certain points (nodes) never move, while points midway between them (antinodes) oscillate with maximum amplitude. Adjacent nodes (or adjacent antinodes) are separated by λ/2, and a node and its neighbouring antinode are separated by λ/4.

Normal modes of a string fixed at both ends (e.g. a sonometer wire or a guitar string of length L): both ends must be nodes, so only wavelengths satisfying L = nλ/2 (n = 1, 2, 3, …) are allowed. The normal-mode (natural) frequencies are:

ν_n = n v / (2L), n = 1, 2, 3, …

All harmonics (integer multiples of the fundamental) are possible.

Organ pipes. In a pipe closed at one end, the closed end must be a node and the open end an antinode, allowing only odd harmonics:

ν_n = (2n − 1) v / (4L), n = 1, 2, 3, …

In a pipe open at both ends, both ends are (approximately) antinodes, so, as for the stretched string, all harmonics are allowed:

ν_n = n v / (2L), n = 1, 2, 3, …

Worked example.

Given: a string of length L = 1.0 m, fixed at both ends, along which transverse waves travel with speed v = 200 m/s.

Formula: ν_n = n v / (2L)

Substitution: fundamental, n = 1: ν₁ = (1 × 200)/(2 × 1); first overtone, n = 2: ν₂ = (2 × 200)/(2 × 1)

Result: fundamental frequency ν₁ = 100 Hz; first overtone (second harmonic) ν₂ = 200 Hz.

When two sound waves of nearly equal (but not identical) frequencies ν₁ and ν₂ and equal amplitude a travel together, their superposition produces a resultant whose amplitude rises and falls periodically. This periodic waxing and waning of loudness is called beats.

If y₁ = a sin(2πν₁t) and y₂ = a sin(2πν₂t), superposition gives:

y = y₁ + y₂ = [2a cos(2π Δν t)] sin(2π ν̄ t)

where Δν = (ν₁ − ν₂)/2 and ν̄ = (ν₁ + ν₂)/2. This is interpreted as a wave of frequency ν̄ (close to both original frequencies) whose amplitude is modulated by the slowly varying term 2a cos(2πΔνt).

Loudness is maximum whenever cos(2πΔνt) = ±1; since both +1 and −1 give a maximum, the number of loud beats heard per second (the beat frequency) works out to be simply the difference of the two original frequencies:

ν_beat = |ν₁ − ν₂|

Beats are commonly used to tune musical instruments: two notes are adjusted until the beat frequency falls to zero, indicating equal frequencies.

Worked example.

Given: two tuning forks of frequencies ν₁ = 256 Hz and ν₂ = 260 Hz are sounded together.

Formula: ν_beat = |ν₁ − ν₂|; beat period T_beat = 1/ν_beat.

Substitution: ν_beat = |256 − 260| = 4; T_beat = 1/4

Result: 4 beats are heard every second (beat frequency = 4 Hz), with a beat period of 0.25 s.