Generalizing The Mach Cone For Sources Traveling Along Arbitrary Trajectories

The concept of Mach cone arises when a wave source travels along a linear trajectory through a given medium with speed larger than the wave speed in that medium, and a shock wave is created. It is commonly associated to mechanical waves (specially sound), although also appears in electromagnetic waves as in the case of Cerenkov radiation. In all cases, the conical shape (or arrow tip shape in 2D media) is a consequence of the linear trajectory of the source. Here, we generalize this situation and determine the shape of the Mach wave front for sources traveling along arbitrary trajectories. To do so, instead of waves, we consider that the source emits rays as in geometrical optics, and determine the equations describing the points in the space that are reached simultaneously by the rays, which form the Mach wave front. To show the applicability of our general results, we obtain explicitly the Mach wave fronts for three different trajectories: linear, circular and sinusoidal.