Progressive Travelling Waves: Motion, Energy and Equation

A progressive (travelling) wave is a disturbance that moves through a medium carrying energy from one place to another, without carrying the medium itself along. The shout that reaches a friend across the field, the ripple spreading from a dropped stone, and the light leaving a bulb are all progressive waves: a shape in motion, delivering energy to wherever it arrives.

This is the workhorse wave of physics — every signal, from speech to Wi-Fi to sonar pings, is a progressive wave. Its equation y(x, t) gives every point's displacement at every instant, from which you read speed, frequency, energy flow, and direction.

Key Concepts

• Amplitude A (metres): maximum displacement of a particle from rest — sets how "strong" the wave is.
• Wavelength λ (metres): distance between two identical points (crest to crest); period T (seconds): time for one full oscillation; frequency f = 1/T (hertz).
• k = 2π/λ (rad/m): spatial packing; ω = 2πf (rad/s): temporal oscillation rate.
• Wave speed v = fλ = ω/k (m/s): fixed by the medium (tension, density, elasticity), not by the source.
• Energy transport: power ∛ A²f² — double the amplitude, quadruple the energy flow — while each particle just oscillates in place.

Formulas

• Wave equation of motion: y(x,t) = A sin(kx - ωt + φ). The minus sign means the wave travels toward +x; a plus sign (kx + ωt) means travel toward −x. φ is the phase constant.
• v = fλ = ω/k, with v in m/s, f in Hz, λ in m.
• Maximum particle speed = Aω (how fast the medium jiggles) — this is not the wave speed v.
• Assumptions and limits: uniform non-dispersive medium (v independent of f), small amplitudes, no energy loss. In dispersive media (deep-water waves, glass for light) different frequencies travel at different speeds and pulses spread out.

Worked Example

A wave on a rope is described (SI units) by y = 0.04 sin(πx - 50πt).

1. Compare with A sin(kx - ωt): amplitude A = 0.04 m, k = π rad/m, ω = 50π rad/s. The minus sign → travels toward +x.
2. λ = 2π/k = 2 m; f = ω/2π = 25 Hz; period T = 0.04 s.
3. Wave speed: v = fλ = 25 x 2 = 50 m/s. Check: v = ω/k = 50π/π = 50 m/s. Consistent.
4. Maximum particle speed: Aω = 0.04 x 50π ≈ 6.28 m/s — the rope jiggles at up to 6.28 m/s while the wave shape races at 50 m/s. Two very different speeds!

Common Misconceptions

• "The minus sign in (kx - ωt) means the wave goes backwards." Backwards — the minus sign gives motion toward +x. Think of it this way: to keep the phase constant as time grows, x must grow too.
• "Particle speed equals wave speed." No — wave speed (fλ) is how fast the pattern moves; particle speed (max Aω) is how fast the medium jiggles. In the example above they differ by a factor of 8.
• "A bigger shake makes a faster wave." Amplitude does not set speed — the medium does. Shouting louder doesn't make sound arrive sooner; only a different medium (water vs air) changes v.

Real-World Applications

• Sonar and ultrasound time the round trip of progressive pulses to measure depth or image organs.
• Speech and music are progressive pressure waves carrying information to ears and microphones.
• Fibre optics and Wi-Fi encode data on progressive electromagnetic waves.
• Earthquake early warning uses fast P-waves outrunning the damaging S-waves.

In short: a progressive wave is a travelling shape y = A sin(kx ∓ ωt) ferrying energy at v = fλ while the medium merely oscillates. Read k, ω, and the sign to get wavelength, frequency, speed, direction, and power.