Signal Arrival

Let us now try to establish at which time, $t_2$ , a signal first arrives at depth $x$ inside the dielectric medium whose amplitude is comparable with that of the wave incident at time $t=0$ on the surface of the medium ($x=0$ ). Let us term this event the ``arrival'' of the signal. It is plausible, from the discussion in Section 7.12 regarding the stationary phase approximation, that signal arrival corresponds to the situation at which the point of stationary phase in $\omega$ -space corresponds to a pole of the function $F(\omega)$ . In other words, when $\omega_s$ approaches the frequency $2\pi/\tau$ of the incident signal. It is certainly the case that the stationary phase approximation yields a particularly large amplitude signal when $\omega_s\rightarrow 2\pi/\tau$ . Unfortunately, as has already been discussed, the method of stationary phase becomes inaccurate under these circumstances. However, calculations involving the more robust method of steepest decent confirm that, in most cases, the signal amplitude first becomes significant when $\omega_s=2\pi/\tau$ . Thus, the signal arrival time is

$$t_2 = \frac{x}{v_g(2\pi/\tau)},$$ (944)

where $v_g(2\pi/\tau)$ is the group velocity calculated using the frequency of the incident signal. It is clear from Figure 12 that

$$t_0 < t_1 < t_2.$$ (945)

Thus, the main signal arrives later than the Sommerfeld and Brillouin precursors.