What is a wave? Transverse and longitudinal waves
A wave is a disturbance that moves through a medium and carries energy, while each particle of the medium only moves a little about its own place.
- Transverse wave: particles move at right angles to the direction of the wave. It has crests (tops) and troughs (bottoms). Examples: a wave on a string, ripples on water (nearly), light.
- Longitudinal wave: particles move along the direction of the wave. It has compressions (crowded, high pressure) and rarefactions (spread out, low pressure). Example: sound in air.
Transverse mechanical waves need a medium that resists shape change, so they travel in solids and on strings, not through the inside of a gas. Longitudinal waves can travel in solids, liquids and gases.
A progressive (travelling) wave keeps moving forward, carrying energy along.
Wavelength, frequency and wave speed
- Wavelength λ: distance between two nearest points in the same phase, for example crest to crest.
- Frequency f: waves passing a point each second (= frequency of the source).
- Period T = 1/f.
In one period the wave moves forward by one wavelength, so v = λ/T = fλ.
Speed depends on the medium
- String under tension T with mass per unit length μ: v = √(T/μ).
- Sound in a solid rod: v = √(Y/ρ); in a fluid: v = √(B/ρ), where Y and B are elastic moduli and ρ the density.
- Sound in a gas: Newton assumed the change is isothermal, v = √(P/ρ) ≈ 280 m/s for air, which is too low. Laplace’s correction: the squeezing is so fast that heat has no time to flow (adiabatic), so v = √(γP/ρ) ≈ 331 m/s at 0 °C (γ = 1.4 for air).
- Speed of sound rises with temperature (v ∝ √T in kelvin), and does not change with pressure at fixed temperature. Humid air carries sound slightly faster.
Displacement relation for a progressive wave
For a wave moving along +x, the displacement of the particle at x at time t is
y(x, t) = A sin(kx − ωt + φ)
- A = amplitude, the biggest displacement of any particle.
- k = 2π/λ = angular wave number (rad/m).
- ω = 2π/T = 2πf = angular frequency (rad/s).
- (kx − ωt + φ) is the phase.
Wave speed from the equation: v = ω/k. A wave moving along −x is written y = A sin(kx + ωt + φ).
Phase difference between two points Δx apart at the same time: Δφ = (2π/λ) Δx. Each particle does SHM; points further along simply lag.
Try it: a rope and a slinky
Tie one end of a long rope (or dupatta) to a door handle. Flick your hand up once: a hump runs along it (transverse). Pull the rope tighter and flick again: the hump runs faster. If you have a slinky, lay it on the floor and push one end: a crowded region runs along it (longitudinal). In the 3D, step 5 lets you do both and read v and λ.
Key formulas and definitions
- v = fλ = λ/T
- y = A sin(kx − ωt + φ), k = 2π/λ, ω = 2πf, v = ω/k
- String: v = √(T/μ)
- Sound in gas: v = √(γP/ρ) (Laplace); Newton: √(P/ρ)
- Phase difference Δφ = (2π/λ) Δx
Worked examples
1. A wave has f = 500 Hz and λ = 0.68 m. Find its speed.
v = fλ = 500 × 0.68 = 340 m/s.
2. A radio station broadcasts at 100 MHz. Radio waves travel at 3 × 10⁸ m/s. Find λ.
λ = v/f = 3 × 10⁸ / 10⁸ = 3 m.
3. A string of mass 20 g and length 2 m is stretched with 80 N. Find the wave speed on it.
μ = 0.02/2 = 0.01 kg/m. v = √(T/μ) = √(80/0.01) = √8000 ≈ 89.4 m/s.
4. y = 0.02 sin(4πx − 400πt) (SI). Find A, λ, f and v.
A = 0.02 m. k = 4π → λ = 2π/k = 0.5 m. ω = 400π → f = 200 Hz. v = ω/k = 400π/4π = 100 m/s (along +x).
5. Two points on a wave of λ = 2 m are 0.5 m apart. Find the phase difference between them.
Δφ = (2π/λ)Δx = (2π/2) × 0.5 = π/2.
6. Speed of sound in air at 27 °C is 347 m/s. Find it at 127 °C.
v ∝ √T. T₁ = 300 K, T₂ = 400 K. v₂ = 347 × √(400/300) = 347 × 1.155 ≈ 401 m/s.
7. Use Newton’s and Laplace’s formulas for air (P = 1.01 × 10⁵ Pa, ρ = 1.29 kg/m³, γ = 1.4).
Newton: v = √(P/ρ) = √(78300) ≈ 280 m/s. Laplace: v = √(1.4 × 78300) ≈ 331 m/s, which matches experiment.
8. If the tension in a string is made 4 times, what happens to the wave speed and, for the same frequency, the wavelength?
v ∝ √T, so v doubles. λ = v/f, so λ also doubles.
Common mistakes
- Thinking the particles travel with the wave. They only move about their own positions; energy travels.
- Mixing k and λ. k = 2π/λ, so a wave with k = 4π rad/m has λ = 0.5 m, not 4π.
- Believing a louder (bigger amplitude) sound travels faster. Speed depends on the medium, not on amplitude or frequency.
- Using Newton’s formula √(P/ρ) for sound in air. The correct one is Laplace’s √(γP/ρ).