On the Propagation Direction of Traveling Waves

In a number of problems related to the spatial propagation of waves, it is necessary to distinguish between waves moving in one and in the other direction. Examples of such problems include the propagation of waves from a point pulsating source, the problem of spatial optimal disturbances, the problem of determining the absolute or convective nature of instability, etc. In addition, when calculating wave motion in an inhomogeneous medium by marching methods, the solution is projected onto the subspace of waves propagating in one direction for numerical stabilization, which also requires their correct filtering. In the literature, the generally accepted indicators of the direction of wave propagation are the Briggs criterion, which follows from the causality principle, and, in some works, the sign of the group velocity. This paper discusses their interpretations and the relationship between them. Examples are given in which the identification of the wave propagation direction by the sign of the group velocity is erroneous and leads to qualitatively incorrect results. For the first time, a case is considered where the direct application of the Briggs criterion is impossible due to the absorption of the discrete mode representing the wave by the continuous spectrum. A generalization of the Briggs criterion for this case is given and examples of its application are discussed.