Wave Speed
v = fλ — how frequency, wavelength, and the medium determine how fast a wave travels.
The Wave Speed Formula
Every periodic wave has three related quantities: speed v, frequency f, and wavelength λ. They are linked by:
v = fλ
The wavelength λ is the distance between successive identical points on the wave (e.g., crest to crest). The frequency f is the number of complete cycles per second (Hz). The product gives the distance the wave advances per second.
This relationship is kinematic — it applies to all wave types regardless of the physical mechanism. It follows directly from the definition: in time T = 1/f, one wavelength passes, so v = λ/T = λf.
The full wave equation governing a wave's spatial and temporal behaviour is: ∂²y/∂t² = v² ∂²y/∂x², where solutions take the form y = A cos(kx − ωt), with wave number k = 2π/λ and angular frequency ω = 2πf. The speed v = ω/k = fλ.
Variable Table
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Wave speed | m/s |
| f | Frequency | Hz (= s⁻¹) |
| λ | Wavelength | m |
| T | Period (= 1/f) | s |
| k | Wave number (= 2π/λ) | rad/m |
| ω | Angular frequency (= 2πf) | rad/s |
Speed in Different Media
Sound in Air
At 0°C: v_sound ≈ 331 m/s. At 20°C: v_sound ≈ 343 m/s. The temperature dependence: v ≈ 331√(T/273) m/s, where T is in kelvin. In water: ~1,480 m/s. In steel: ~5,960 m/s.
Transverse Waves on a String
v = √(T/μ), where T is the tension (N) and μ is the linear mass density (kg/m). A tighter, lighter string carries waves faster.
Electromagnetic Waves
In vacuum: c = 299,792,458 m/s ≈ 3 × 10⁸ m/s (exact by definition). In a medium with refractive index n: v = c/n. For visible light in glass (n ≈ 1.5): v ≈ 2 × 10⁸ m/s.
Worked Examples
Example 1 — Finding wavelength
A sound wave in air (20°C) has frequency 440 Hz. Find its wavelength.
λ = v/f = 343/440 ≈ 0.780 m (78 cm — concert A note).
Example 2 — Wave speed on a string
A guitar string has tension 80 N and linear density 5 × 10⁻³ kg/m. Find the wave speed.
v = √(T/μ) = √(80 / 5×10⁻³) = √16,000 ≈ 126 m/s.
Example 3 — Frequency from speed and wavelength
A water wave has wavelength 2 m and travels at 3 m/s. Find its frequency and period.
f = v/λ = 3/2 = 1.5 Hz. T = 1/f = 1/1.5 ≈ 0.67 s.
Dispersion
A wave is non-dispersive if all frequencies travel at the same speed (e.g., sound in air, light in vacuum). It is dispersive if speed depends on frequency. In dispersive media, a wavepacket (superposition of different frequencies) spreads out over time.
The phase velocity v_p = ω/k describes individual sinusoids. The group velocity v_g = dω/dk describes the speed of energy and information. For non-dispersive media, v_p = v_g. In dispersive media they differ — a glass prism separates white light into colours because different wavelengths travel at different speeds.
Common Mistakes
- Confusing period and wavelength. Period T is a time (seconds); wavelength λ is a length (metres). f = 1/T but λ ≠ 1/f — λ = v/f.
- Assuming frequency changes when a wave enters a new medium. Frequency is set by the source and doesn't change at a boundary. Speed and wavelength change; frequency doesn't.
- Mixing up phase and group velocity. Phase velocity can exceed c in dispersive media — this doesn't violate special relativity because information travels at group velocity.
Speed of Sound: Temperature Dependence
For an ideal gas, the speed of sound is v = √(γP/ρ) = √(γRT/M), where γ is the adiabatic index, R is the gas constant, T is absolute temperature, and M is molar mass. For dry air at temperature T (kelvin): v ≈ 331√(T/273) m/s, or equivalently v ≈ 331 + 0.6θ m/s where θ is temperature in °C. At 20°C: v ≈ 343 m/s; at 100°C: v ≈ 387 m/s.
Humidity also slightly increases the speed of sound in air because water vapour (M = 18 g/mol) is lighter than dry air (M ≈ 29 g/mol), reducing the effective molar mass.
Wave Speed and the Refractive Index
When light crosses a boundary between media with different refractive indices n₁ and n₂, the speed changes from c/n₁ to c/n₂ but the frequency remains constant. Snell's law n₁ sin θ₁ = n₂ sin θ₂ follows from the requirement that wavefronts match at the boundary. Total internal reflection occurs when n₁ sin θ₁ = n₂ requires sin θ₂ > 1 — which has no real solution, so no transmitted wave exists. Optical fibres exploit total internal reflection, trapping light inside a high-index core to transmit signals with minimal loss over thousands of kilometres.
Related Topics
Frequently Asked Questions
Wave speed formula?
v = fλ. Wave speed equals frequency times wavelength. This applies to sound, light, water, and all other wave types.
What determines wave speed?
The medium, not the source. Sound in air: ~343 m/s. Light in vacuum: 3×10⁸ m/s. These don't change with frequency or amplitude.
Does frequency change when a wave enters a new medium?
No. Frequency is set by the source. Speed and wavelength both change at a boundary; frequency stays the same.
What is dispersion?
Speed varies with frequency in the medium. White light disperses in glass because different wavelengths travel at different speeds, producing a spectrum.
References
- French, A. P. (1971). Vibrations and Waves. MIT Introductory Physics Series. Chapter 5.
- Griffiths, D. J. (2017). Introduction to Electrodynamics (4th ed.). Cambridge. Chapter 9.
- Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (2000). Fundamentals of Acoustics (4th ed.). Wiley.