Simulation of wave propagation using graph-theoretical algorithm

Abstract

We simulate the wave propagation through various mediums using a graph-theoretical path-finding algorithm. The mediums are discretized to the square lattices, where each node is connected up to its 4th nearest neighbours. The edge connecting any 2 nodes is weighted by the time of flight of the wave between the nodes, which is calculated from the Euclidean distance between the nodes divided by the average velocity at the positions of those nodes. According to Fermat’s principle of least time, wave propagation between 2 nodes will follow the path with minimal weight. We thus use the path-finding algorithm to find such a path. We apply our method to simulate wave propagation from a point source through a homogeneous medium. By defining a wavefront as a contour of nodes with the same time of flight, we obtain a spherical wave as expected. We next investigate the wave propagation through a boundary of 2 mediums with different wave velocities. The result shows wave refraction that exactly follows Snell’s law. Finally, we apply the algorithm to determine the velocity model in a wood sample, where the wave velocity is determined by the angle between the propagation direction and the radial direction from its pith. By comparing the time of flight from our simulation with the measurements, the parameters in the velocity model can be obtained. The advantage of our method is its simplicity and straightforwardness. In all the above simulations, the same simple path-finding code is used, regardless of the complexity of the wave velocity model of the mediums. We expect that our method can be useful in practice when an investigation of wave propagation in a complex medium is needed.