Wave Equation in 3D

PDEs · Axiom Academy

Exploring wave propagation in three-dimensional space: spherical waves, Kirchhoff's formula, and Huygens' principle The wave equation in three dimensions generalizes the 1D case by replacing the single spatial derivative with the Laplacian operator in three variables: Here, the Laplacian ∇²u represents the sum of second partial derivatives in all three spatial directions. The constant c represents the wave speed (speed of light, speed of sound, etc.). One of the most fundamental solutions to the 3D wave equation is the spherical wave. Unlike plane waves, spherical waves emanate from a point source and decrease in amplitude as they spread out. The key feature is the 1/r decay: as the wave spreads over a larger sphere, its energy is distributed over area 4πr², causing amplitude to decay as 1/r. The function f(r - ct) represents the wave profile moving outward at speed c. 3. Spherically Symmetric Solutions When solutions depend only on the radial distance r = √(x² + y² + z²), the 3D wave equation simplifies dramatically. In spherical coordinates: By substituting v = ru, this transforms to the 1D wave equation for v! The general solution becomes u = f(r-ct)/r + g(r+ct)/r, representing outgoing and incoming spherical waves. Kirchhoff's formula provides the general solution to the 3D wave equation with initial conditions. It states that the solution at point x and time t depends only on initial data on a sphere of radius ct:

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