The speed of waves

In one period a wave moves forward one wavelength, so its speed is v = λ/T = νλ = ω/k. A 440 Hz note in air at 343 m/s has a wavelength of 343/440 ≈ 0.78 m.

The frequency belongs to the source. When a wave passes from one medium into another, its frequency stays the same; its speed and wavelength change together.

The speed depends on the medium, not on the wave: an elastic property (how hard it springs back) divided by an inertia property (how heavy it is).

• Stretched string: v = √(T/μ), where T is the tension and μ the mass per unit length. A guitar string 0.65 m long weighing 2.6 g (μ = 0.004 kg m⁻¹) under 100 N carries waves at √(100/0.004) ≈ 158 m/s.
• Sound in a liquid or gas: v = √(B/ρ), with B the bulk modulus. For water, B ≈ 2.2 × 10⁹ Pa, so v ≈ 1480 m/s.
• Sound along a solid rod: v = √(Y/ρ). For steel, √(2 × 10¹¹/7800) ≈ 5100 m/s.

So sound is fastest in solids and slowest in gases. Strike a long steel rail with a hammer, and a friend with an ear to the rail far away hears two bangs: first through the rail, then through the air.

Newton's mistake. For air, Newton assumed the temperature stays fixed as sound passes (an isothermal change), which gives B = P and v = √(P/ρ) ≈ 280 m/s. The measured value is about 331 m/s at 0 °C.

Laplace's correction, about a century later: the squeezing and stretching happen so fast that heat has no time to flow in or out. The change is adiabatic, so B = γP and v = √(γP/ρ). Air is mostly two-atom gases, γ = 1.4, and 280 × √1.4 ≈ 331 m/s. Even Newton slipped; what matters is checking against measurement.

What changes the speed of sound in a gas:

• Pressure: no effect at fixed temperature, because doubling P also doubles ρ, and P/ρ stays the same.
• Temperature: v ∝ √T (in kelvin). At 30 °C, v = 331 × √(303/273) ≈ 349 m/s: about 0.6 m/s more for each degree.
• Humidity: water vapour is lighter than air, so humid air is less dense and sound goes faster.
• Wind blowing the same way adds its speed; against, it subtracts.
• Frequency: no effect. High and low notes from an orchestra reach you together.