Discussion on “Pseudo-Mach angles for Rayleigh ground waves generated by trains moving at conventional speeds [J. Sound Vib. 570 (2024) 118121]”

The paper of Krylov [1] demonstrates that radiation of plane Rayleigh waves excited by a constant load moving on a beam with discrete, periodic supports that rest on an elastic half-space can take place at speeds of the moving load smaller than the Rayleigh-wave velocity and at angles that are different from the conventional Mach angle. This discussion paper presents an expression for the frequency-dependent radiation angle as well as conditions for the existence of the plane Rayleigh waves, it further clarifies the mechanism of the radiation of the waves, and it demonstrates their physical significance. First, an alternative closed-form representation for the response in the space-frequency domain is presented to reveal expressions for the wavenumbers in horizontal directions (and thus for the angles of the radiated waves) as well as the conditions for existence based on those. Then, by comparing with the wave field excited in the canonical case of an harmonically varying load moving directly on the surface of the half-space (at sub-Rayleigh velocity), it is found that the basic mechanism of plane-wave generation is not fundamentally different. In both cases, the plane Rayleigh waves are radiated due to the time-periodic nature of the loading moving/progressing on the half-space surface. More specifically, all harmonic components representing the set of sequentially exerted sleeper forces continuously radiate Rayleigh waves leading to the occurrence of plane waves, just like the harmonic load moving directly on the half-space surface creates them. It is therefore claimed that the plane Rayleigh waves demonstrated in [1] are not unconventional; they are simply the elementary components of the well-known curved Rayleigh-wave pattern excited by the harmonic load moving directly on the half-space surface, as well as of the very similar Rayleigh-wave pattern excited by the set of sleeper forces. The radiated plane waves do not add up to the classical Mach cone – which is indeed due to the frequency dependence of the radiation angle [1] – but together constitute the curved wave patterns. The destructive interference of the plane Rayleigh waves is therefore only partial and their physical significance is evident.