raypaths

Raypaths are the trajectories followed by waves as they propagate through a medium whose properties vary in space. In uniform media, rays travel in straight lines; in media with spatial variations of refractive index or wave speed, rays bend according to the local gradient of those properties.

In optics, raypaths are determined by Fermat's principle, which states that the optical path length between two points is extremal. The ray equations can be written as d/ds (n dr/ds) = ∇n, where n is the refractive index and s is the arc length along the ray. At interfaces between media with different refractive indices, Snell's law constrains the incidence and refraction angles. In gradient-index (GRIN) materials, the refractive index changes continuously, producing smooth bending without distinct boundaries.

In seismology, raypaths describe the paths of seismic waves through the Earth's interior. The velocity structure of the crust and mantle causes refraction, reflection, and turning rays, shaping travel-time curves used in imaging subsurface structures. Raypath analysis supports seismic tomography and interpretation of geophysical data.

Computationally, raypaths are obtained by solving the ray equations numerically, often using the eikonal approximation or more advanced ray-tracing techniques. Methods include direct integration, shooting, and fast marching approaches. The accuracy of raypath models depends on the wavelength relative to medium heterogeneities and on the smoothness of the velocity or refractive-index field.

Applications span optics design, atmospheric and underwater acoustics, radar and lidar propagation, and geophysical imaging. Raypaths provide a practical and intuitive framework for predicting wave behavior in complex media and for extracting structural information from wave observations.