How waves travel: transverse and longitudinal waves and the wave equation
A wave carries energy, not matter: the particles only vibrate about their mean positions. For every wave, v = νλ.
What a wave is
Drop a stone into a still pond. Rings of crests and troughs spread out from the point where it fell. A piece of straw on the water only bobs up and down at its place. So the water particles do not travel outward; only the disturbance does. A wave is a disturbance that carries energy from one place to another without carrying matter. A tsunami shows how much energy a wave can carry. Waves that need a medium, such as sound and waves on a string, are mechanical waves. The medium must have mass (inertia) and elasticity. Light and radio waves are electromagnetic waves and can travel through vacuum.
Transverse and longitudinal waves
| Feature | Transverse wave | Longitudinal wave |
|---|---|---|
| How particles move | At right angles to the direction of travel | Along the direction of travel |
| What you see | Crests and troughs | Compressions (crowded particles) and rarefactions (spread-out particles) |
| Where they travel | Solids and the surface of liquids | Solids, liquids and gases |
| Elasticity needed | Rigidity (resistance to change of shape) | Volume elasticity (resistance to change of volume) |
| Examples | Wave on a stretched string; ripples on water; light | Sound in air; a push along a slinky spring |
Words that describe a wave
The amplitude a is the largest displacement of a particle from its mean position. The wavelength λ is the distance between two nearest particles vibrating in the same phase, for example from crest to crest. The time period T is the time for one vibration, and the frequency ν = 1/T is the number of vibrations per second (unit hertz, Hz). In one period the wave moves forward by one wavelength, so v = λ/T = νλ. Two more terms are useful. The angular frequency ω = 2π/T = 2πν is the phase change per second. The propagation constant k = 2π/λ is the phase change per metre. Dividing them gives v = ω/k.
Derivation: equation of a simple harmonic wave
A transverse harmonic wave travels along OX. The particles vibrate along the y-axis.
- 1The particle at the origin O vibrates as y = a sin ωt.
- 2The wave reaches a point P at distance x a little later. So P lags behind O by a phase φ: y = a sin(ωt − φ).
- 3A distance of λ means a phase change of 2π. So the phase change per metre is k = 2π/λ, and φ = kx.
- 4★So the displacement at P is y = a sin(ωt − kx).
- 5Putting ω = 2π/T and k = 2π/λ gives y = a sin 2π(t/T − x/λ).
- 6Using v = λ/T gives y = a sin (2π/λ)(vt − x). If O starts with a phase φ₀, then y = a sin(ωt − kx + φ₀).
- 7So, y = a sin(ωt − kx) describes a harmonic wave moving along +x. The phase difference between two points Δx apart is Δφ = (2π/λ)Δx.
★ key step
Worked example: reading a wave equation
A wave is given by y = 0.02 sin(200πt − 0.5πx), with x and y in metres and t in seconds. Find (a) the amplitude, (b) the frequency, (c) the wavelength, (d) the speed, (e) the phase difference between two points 0.5 m apart.
- Compare with y = a sin(ωt − kx).
- (a) a = 0.02 m.
- (b) ω = 200π rad s⁻¹, so ν = ω/2π = 200π ÷ 2π = 100 Hz.
- (c) k = 0.5π m⁻¹, so λ = 2π/k = 2π ÷ 0.5π = 4 m.
- (d) v = νλ = 100 Hz × 4 m = 400 m s⁻¹. Check: ω/k = 200π ÷ 0.5π = 400 m s⁻¹.
- (e) Δφ = (2π/λ)Δx = (2π ÷ 4 m) × 0.5 m = π/4 rad.
(a) 0.02 m; (b) 100 Hz; (c) 4 m; (d) 400 m s⁻¹; (e) π/4 rad
How to draw a transverse and a longitudinal wave
- Transverse: draw a horizontal line and label it x (direction of travel), with an arrow pointing right.
- Draw a smooth sine curve along it. Label a top point crest and a bottom point trough.
- Mark λ between two neighbouring crests, and a from the line up to a crest.
- Draw small up and down arrows on a few points to show that particles move at right angles to x.
- Longitudinal: draw a row of short vertical lines. Put them close together in some places and label these C (compression); spread them out in between and label these R (rarefaction).
- Mark λ from the middle of one compression to the middle of the next, and draw an arrow along the row for the direction of travel.
The 2004 tsunami that struck Tamil Nadu and the Andaman Islands carried huge energy across the ocean. Yet the deep sea water itself moved very little. Earthquake stations in India record both longitudinal (P) and transverse (S) waves to locate an earthquake.
2 marks: “State two differences between transverse and longitudinal waves.” Transverse: particles vibrate at right angles to the direction of travel; crests and troughs form; need rigidity, so they cannot travel through gases. Longitudinal: particles vibrate along the direction of travel; compressions and rarefactions form; can travel in solids, liquids and gases.
1 marks: “How is the phase difference between two points on a wave related to the path difference between them?” Δφ = (2π/λ) × Δx, where Δx is the path difference.
3 marks: “A wave is described by y = 0.05 sin(100πt − 0.25πx) in SI units. Find its frequency, wavelength and speed.” ω = 100π, so ν = 50 Hz; k = 0.25π, so λ = 2π ÷ 0.25π = 8 m; v = νλ = 50 × 8 = 400 m s⁻¹ (check ω/k = 400 m s⁻¹).
The particles dance on the spot; the energy travels on. Speed = frequency × wavelength.