wavefields

A wavefield is the spatial and temporal distribution of a wave disturbance in a medium or field. It can be scalar or vector valued, representing quantities such as pressure in acoustics, electric and magnetic fields in electromagnetism, or particle probability amplitude in quantum contexts. A wavefield encodes amplitude, phase, and polarization, and evolves according to the governing wave equations.

For a simple acoustic wave in a homogeneous medium, the scalar wavefield u(x,t) satisfies the wave equation d^2u/dt^2 = c^2 ∇^2 u, where c is the wave speed. In the frequency domain, time-harmonic fields satisfy the Helmholtz equation ∇^2 U + k^2 U = 0 with k = ω/c. More generally, Maxwell's equations describe electromagnetic wavefields as vector fields E and B, with polarization and impedance playing roles.

Wavefield analysis often uses superposition, Fourier transforms, and Green's functions to decompose complex motion into simpler components. The Green's function represents the response to a point source, and the total wavefield is the convolution of sources with the Green's function. In inhomogeneous or anisotropic media, wave speeds vary spatially and the wave equation becomes more complex.

Applications span acoustics, optics, seismology, and medical imaging. Wavefields are used to model room acoustics, design optical wavefronts, or reconstruct subsurface structure in seismic imaging via forward and inverse problems. Techniques such as time-reversal, interferometry, and wavefield synthesis build or manipulate wavefields for imaging and focusing.

Numerical methods such as finite-difference time-domain, finite element, or spectral methods simulate wavefields in complex geometries. Measurements of wavefields in experiments provide data for inversion and characterization of materials, structures, and Earth.