Waves and the travelling wave equation

Drop a stone into a still pond and rings spread out. The water itself does not travel to the bank: a floating leaf only bobs up and down. What travels is the disturbance, and with it energy. That is a wave: each particle of the medium oscillates about its own place and hands the motion on to the next one, a little later.

• Mechanical waves need a medium: water waves, sound, the waves of an earthquake. That is why there is no sound on the moon, which has no air.
• Electromagnetic waves need none: light, radio waves and X-rays cross empty space, all at 3 × 10⁸ m/s.
• To carry a mechanical wave, a medium needs elasticity (to pull each particle back), inertia (mass, to carry it past the middle) and little friction.

Two kinds of mechanical wave:

Every medium can be squeezed, so longitudinal waves travel in solids, liquids and gases. Transverse waves need a medium that resists being sheared, so they travel in solids and along strings, but not through the body of a liquid or a gas.

A travelling harmonic wave moving towards +x is y(x, t) = a sin(kx – ωt + φ):

• a, the amplitude, is the greatest displacement of a particle.
• k = 2π/λ is the angular wave number; λ, the wavelength, is the distance from one crest to the next.
• ω = 2π/T = 2πν is the angular frequency.
• kx – ωt means the wave moves towards +x; kx + ωt means it moves towards –x.

Example: y = 0.02 sin(4πx – 100πt), in metres and seconds. Then a = 2 cm, λ = 2π/4π = 0.5 m, ν = 100π/2π = 50 Hz (T = 0.02 s), and it moves towards +x. Points 12.5 cm apart (a quarter of λ) differ in phase by (2π/λ)Δx = π/2.

Every particle oscillates with the same amplitude and frequency; only the phase changes along the wave. Do not mix up the two speeds: the wave moves steadily, while each particle speeds up and slows down in SHM. In the example, a particle's top speed is aω ≈ 6.3 m/s.