An insight into Fermat's principle via acoustic propagation in inhomogeneous air temperature field

Fermat's principle shows that an acoustic or optical eigenray takes a minimal, maximal, or saddle point travel time. Yet, no literature has fully characterized mechanisms on the eigenray extremum, except for a recognized evaluation method by the sign-constancy of the Hessian of a travel-time functional. Aiming at addressing such an issue, acoustic ray propagation in complicated inhomogeneous air temperature field was investigated in a theoretical and numerical way since experimental study can hardly be achieved. First, three numerical approaches, Fermat's variational method, Hamiltonian method, and forward deploying triangle method, were comparatively programed for acoustic ray tracing and gave results agreeing exactly with each other. Based on the essential validation, the numerical method was utilized to study the characteristics of multipath acoustic propagation in complex medium, with defined inhomogeneous air temperature fields as illustration. Next, a delta-neighborhood model was proposed and it turned out to successfully characterize the physics of the acoustic eigenray extrema. As a result, the wave travel-time Fermat functional was shown to be a generalized parabola opening upward, with a single extremum of minimum, or multiple minima together with local maxima or saddle points. In addition, for multipath propagation, the global minimal eigenray resides on the same side with the transmitter-receiver pair relative to the refractive index concave center, while the other stationary rays dwell on the other side. The Fermat's principle is thus further insighted, which governs the mechanism on acoustic/optic wave propagation in complicated medium.