Wave Speed Equation

The frequency is fixed by the source and does not change when the wave crosses a boundary. Since v = fλ and f is constant, the wavelength must change proportionally to the wave speed. If the wave speeds up (e.g., sound going from air into water), the wavelength increases; if it slows down (e.g., light going from air into glass), the wavelength decreases. This change in wavelength is what causes the wave to bend — the phenomenon of refraction. Different parts of the wavefront enter the new medium at different times and travel at different speeds, so the direction of propagation changes. The change of wavelength is exactly compensated by the change of speed so that f stays constant.

The phase velocity v = ω/k = fλ is the speed at which a single-frequency wave crest moves. The group velocity v_g = dω/dk is the speed at which the envelope of a wave packet — a localized pulse or signal — moves. In a non-dispersive medium, v_g = v and both are constant. In a dispersive medium, v depends on frequency, and v_g differs from v; the wave packet spreads as it propagates because different frequency components travel at different speeds. For light in vacuum, v_g = v = c exactly. For light in glass, v_g is slightly less than v, and the difference is related to the dispersion of the refractive index. For ocean waves, the group velocity is half the phase velocity — which is why surfers ride the group velocity of the swell.

For electromagnetic waves, the wave speed in vacuum is c = 1/√(ε₀μ₀) ≈ 2.998×10⁸ m/s, where ε₀ and μ₀ are the permittivity and permeability of free space. In a material medium, the speed is v = c/n, where n is the refractive index. The wave speed equation v = fλ still applies: f is the frequency of the oscillation (set by the source, such as an electron transition or an antenna), λ is the spatial period of the electromagnetic field oscillation, and v is the speed of light in the medium. For visible light, the frequency is around 4–7.5×10¹⁴ Hz, giving wavelengths in vacuum of about 400–750 nm. When light enters glass, the speed drops to about 2×10⁸ m/s and the wavelength shrinks by the same factor, but the frequency — and therefore the colour — remains unchanged.

Standing waves form when a wave reflects back and forth between two boundaries, and the reflected wave interferes with the incoming wave. The result is a pattern of nodes (points of zero motion) and antinodes (points of maximum motion) that appears stationary. Standing waves only exist at particular frequencies, called resonant frequencies, for which the wavelength fits an integer number of half-wavelengths into the distance between the boundaries: L = nλ/2. Combining this with v = fλ gives the resonant frequencies f_n = nv/(2L) for a string fixed at both ends, or f_n = (2n−1)v/(4L) for a pipe open at one end. This is the physics behind musical instruments, laser cavities, and microwave resonators.

For waves that carry energy and information, no — nothing can travel faster than c. However, certain non-physical quantities can appear to move faster than light. The phase velocity of a wave in a dispersive medium can exceed c (as in the region just below an absorption line, where the refractive index is less than 1), but this does not carry energy or information. The group velocity is always less than or equal to c in a normal dispersive medium. In anomalous dispersion, the group velocity can exceed c or even become negative, but this still does not violate causality because the "front velocity" — the speed at which the leading edge of a pulse first appears — is always exactly c or less. For mechanical waves like sound or ocean waves, the speed is limited by the strength and stiffness of the medium, and is always far less than c.

Angular frequency ω = 2πf measures the phase change per unit time: a wave completes one full cycle of phase (2π radians) in a time T = 1/f, so ω = 2π/T. Wavenumber k = 2π/λ measures the phase change per unit distance: the wave completes one full cycle of phase (2π radians) over a distance λ, so k = 2π/λ. Together they describe the phase of a wave at any point and time: φ = kx − ωt. The wave speed is v = ω/k, which is equivalent to v = fλ. Working in ω and k is convenient because the phase is a simple linear function of position and time, and many wave equations (such as the Schrödinger equation and the Helmholtz equation) are naturally written in terms of these quantities.