Comment on "Observation of the inverse Doppler effect".

Seddon and Bearpark present a creative and exciting observation of a reversed Doppler effect when an electromagnetic shock propagates through a transmission line (1). We find that the physical origin of this anomalous effect is fundamentally different from the one suggested by Seddon and Bearpark (that v phase v group Ͻ 0), but that the experimental results can be properly validated with the correct theory. The system studied by Sed-don and Bearpark falls into the general class of systems that involve a propagating shocklike excitation in a periodic medium, for which we have predicted reversed Doppler effects using a different theoretical framework (2). For this system, an extended Brillouin zone (BZ) scheme should be used, rather than the periodic BZ scheme considered by Seddon and Bearpark (3). In their analysis, a phase-matching condition v shock ϭ v phase leads to the conclusion that radiation emitted by the shock has a ⌿ 0 (wave vector) value in the second BZ, where v phase v group Ͻ 0. Although the condition v shock ϭ v phase predicts the correct emission frequency ␻ 0 , the suggestion that this emitted radiation has a ⌿ value in the second BZ is not founded. The discretized nature of this system precludes unique measurement of v phase (assignment of ⌿ to a particular BZ) by measuring voltages or other quantities at points that are spatially periodically related. Radiation well characterized by plane waves in the first band of periodic systems is poorly characterized by plane waves with wave-vector values outside the first BZ. Imposing a periodicity on the vacuum dispersion reveals a region similar to that of the Seddon and Bearpark transmission-line system , where v phase v group Ͻ 0, that is clearly unphysical (Fig. 1). Applied to vacuum, the analysis of Seddon and Bearpark [equation 1 in (1)] incorrectly predicts that a reversed Doppler shift can occur in that system. Away from the cutoff frequency, physical values of wave vector ⌿ in the experiment of Seddon and Bearpark fall within the first BZ, where v phase v group Ͼ 0. We have shown (2) that the phase of the reflection coefficient of the shock front is time dependent, unlike that of a normal moving reflecting surface assumed by Seddon and Bearpark in equation 1. This key feature is the actual origin of the inverse Doppler effect and explains how it can be observed in …