The explanation of the observed Doppler shift (1) hinges on the existence, or otherwise, of waves in the second Brillouin zone (BZ) of the transmission line system. We agree with Reed et al. (2) that the second BZ is unphysical in the case of the vacuum. However, we believe that the transmission line used in the Doppler experiment has characteristics that allow the generation of waves in the second BZ of this system.
Waves (space harmonics) may be excited in the second BZ of suitable periodic systems by injecting a pump signal with a propagation velocity that is synchronous with the phase velocity of the required space harmonic. This technique is well established in the field of microwave tube technology, in which an electron beam provides the pump energy (3). For example, backward wave oscillators (BWO), which exhibit normal dispersion in the first BZ and anomalous dispersion in the second BZ, produce microwave output by excitation of a wave (spatial harmonic) in the second BZ (3). As the velocity of the electron beam is reduced, the phase velocity of the excited wave is reduced continuously across the BZ boundary to give a continuous transition from generation of waves with vgroupvphase > 0 (first BZ) to generation of waves with vgroupvphase > 0 (second BZ). The BWO clearly demonstrates that the generation of waves in the second BZ of periodic systems is not unphysical when a pumping pulse is used to generate the synchronous wave.
Formation of a wave with νgroupνphase > 0in the second BZ of the transmission line used in the inverse Doppler experiment in (1) has been analyzed previously (4). The system used in the Doppler experiment has the same essential features as the BWO; spatial dispersion is provided by cross-link capacitors in the transmission line and by a corrugated waveguide in the BWO. An electrical pump pulse in the transmission line experiment performs the same function as the electron beam in the BWO, which is generation of a synchronous wave. The transmission line is analyzed as a discrete periodic system, whereas the BWO is usually analyzed as a continuous periodic system. The analysis of purely discrete systems leads to mathematical uncertainty as to the phase velocity of waves that are measured at periodically related points, as pointed out by Reed et al. However, practical experience shows that generation of waves in the second BZ of pumped periodic systems is physical, as described above.
Both our analysis and that of Reed et al. agree with the experimental observations that both the oscillation frequency and the group velocity of the generated wave vary continuously as the pump-pulse velocity is reduced. However, stipulation that the generated wave can exist only in the first BZ requires that the generated wave undergo a discontinuous change of phase velocity at the BZ boundary, from +νp to -νp. We maintain that that assertion is not physically correct and that the phase velocity of the generated wave changes continuously to generate a wave in the second BZ, as demonstrated in the BWO and described in previous analyses of the transmission-line system (4).