Kugelwelle

The Kugelwelle, or spherical wave, is a wave with spherical symmetry that propagates outward from a localized source. It is the three-dimensional analogue of a plane wave, producing curved wavefronts that are spheres centered on the source. In idealized terms, Kugelwellen describe isotropic radiation from a point source in a homogeneous medium.

Mathematically, a Kugelwelle is a solution to the wave equation ∇^2ψ − (1/c^2)∂^2ψ/∂t^2 with spherical symmetry. For a monochromatic field, the spatial part takes the form ψ(r, t) = (A/r) e^{i(kr − ωt)} with k = ω/c, so the amplitude decays as 1/r and the phase advances with distance. The general time-domain form is ψ(r, t) = (1/r) F(t − r/c).

Key properties include wavefronts that are spheres r = constant, and an intensity that falls off as 1/r^2 due to geometric spreading. In the far field, the field behaves approximately as a Kugelwelle, while the near field contains more complex patterns determined by the source geometry and boundary conditions.

Applications and limitations: the Kugelwelle serves as a model for point-like or isotropic sources in acoustics, electromagnetism, and quantum mechanics. Real sources are finite and often emit non-isotropic fields; media can be inhomogeneous, reflecting or refracting waves and causing deviations from ideal spherical propagation. The model is most accurate when the source size is small compared to the wavelength and the observation distance is large compared with the source.