wave number — from formula anatomyThe wave number is a measure of spatial frequency: it tells you how many complete wave cycles fit into a unit of length. In physics, it appears in the mathematical description of waves, from light and sound to quantum mechanical wavefunctions. It is the spatial analogue of temporal frequency — just as frequency counts oscillations per second, wave number counts oscillations per meter.
The concept emerged in the 19th century with the mathematical treatment of wave phenomena, though the term 'wave number' became standard in spectroscopy and quantum theory in the early 20th century.
For a wave of wavelength λ, the wave number k is defined as k = 2π/λ. This definition makes it a natural companion to angular frequency ω = 2πf, so that a traveling wave can be written compactly as A·sin(kx − ωt). In quantum mechanics, the wave number is directly proportional to momentum via the de Broglie relation p = ħk, making it a bridge between wave and particle descriptions.
Wave numbers are used in spectroscopy to label spectral lines, in crystallography to index diffraction peaks, and in oceanography to describe surface wave fields.
In spectroscopy, wave numbers are often quoted in reciprocal centimeters (cm⁻¹), so a visible red photon at 700 nm has a wave number of about 14,300 cm⁻¹.
| \psi | متغير | wave function — الدالة الموجية |
| x | متغير | position coordinate — موضع metre |
| k | متغير | wave number — عدد الموجة per metre |
| V(x) | متغير | potential function — الطاقة الكامنة (Potential Energy) joule |
| \frac{d^2}{dx^2} | مؤثر | second derivative with respect to position — المشتقة الثانية |
| \tilde{\kappa} | متغير | Fourier transform of convergence — تقارب |
| \tilde{\kappa}_B | متغير | Fourier transform of B-mode convergence — b mode convergence |
| \tilde{\gamma}_1 | متغير | Fourier transform of first shear component — shear component |
| \tilde{\gamma}_2 | متغير | Fourier transform of second shear component — shear component |
| k_1 | متغير | First component of wave vector — wave vector component radian per metre |
| k_2 | متغير | Second component of wave vector — wave vector component radian per metre |
| k | متغير | Magnitude of wave vector — عدد الموجة radian per metre |
| ω | متغير | angular frequency — تردد زاوي radian per second |
| k | متغير | wave number — عدد الموجة radian per metre |
| c | متغير | propagation speed — wave speed metre per second |
| | | | مؤثر | returns the magnitude of a real number — absolute value operator |
| ω | متغير | Angular frequency of the wave — تردد زاوي radian per second |
| k | متغير | Wave number (spatial frequency) — عدد الموجة radian per metre |