About the direction of travel of traveling waves

In a number of problems involving spatial wave propagation, it is necessary to distinguish between waves traveling in one direction and in the other. Examples of such problems are the propagation of waves from a point the problem of pulsating source; the problem of spatial optimal perturbations; the problem of determining the absolute or convective character of instability, etc. In addition, when calculating the wave motion in the inhomogeneous medium by marching methods for numerical stabilization, the projection of the solution onto the space of waves propagating in the same direction is used, which also requires their correct screening. Commonly accepted in the literature indicators of the direction of wave motion are the Briggs criterion derived from the causality principle and, in some papers, the sign of the group velocity. This paper discusses their interpretations and the relationship between them. Examples are given where the identification of the wave direction by the sign of the group velocity is erroneous and leads to qualitatively incorrect results. The case when direct application of the Briggs criterion is impossible due to absorption of the discrete mode describing the wave by a continuous spectrum is considered for the first time. A generalization of the Briggs criterion to this case is given and examples of its application are given.