Kugelwellen

Kugelwellen, or spherical waves, are wave fields whose wavefronts are concentric spheres centered on a source. They arise when energy radiates from a localized region into a homogeneous medium and exhibit radial symmetry in three dimensions. In free space, an ideal outward-propagating spherical wave has an amplitude that decays with distance as 1/r, with the phase proportional to kr − ωt, where r is the distance from the source, k is the wavenumber, and ω is the angular frequency.

Formally, for a scalar field u(r,t) satisfying the wave equation ∇^2u − (1/c^2)∂^2u/∂t^2 = 0, a time-harmonic solution u(r,t) = Re[Φ(r)e^(−iωt)] satisfies the Helmholtz equation ∇^2Φ + k^2Φ = 0 with k = ω/c. The outgoing spherical wave solution is Φ(r) ∝ e^(ikr)/r, and a standing spherical wave is ∝ sin(kr)/r. The 1/r factor reflects the geometric spreading of energy in three dimensions, so the intensity falls off roughly as 1/r^2.

Generation and occurrence: in acoustics, a small loudspeaker or transducer can approximate a spherical wave in free space; in optics, a point-like light source produces spherical wavefronts near the source. In quantum mechanics, radial solutions to the Schrödinger equation for a free particle also take spherical form. Far from the source, the wave becomes locally planar over small regions.

Applications and limitations: spherical waves are a basic model in sonar, ultrasound imaging, astronomy, and optical teaching. Real media introduce attenuation, scattering, and reflections from boundaries, which modify the ideal 1/r behavior and the shape of the wavefronts, especially near the source or in inhomogeneous media.