(928) |

is there a qualitative change. This time marks the arrival of a second precursor known as the

In order to generalize the result (918) to deal with a stationary phase point at , it is necessary to expand about this point, keeping terms up to . Thus,

(929) |

where

(930) |

for the simple dispersion relation (871). The amplitude (910) is therefore given approximately by

(931) |

This expression reduces to

where

(933) |

and

(934) |

The positive (negative) sign in the integrand is taken for ( ).

The integral in Equation (933) is known as an *Airy integral*. It can
be expressed in terms of Bessel functions of order
, as follows:

(935) |

and

(936) |

From the well-known properties of Bessel functions, the precursor can be seen to have a growing exponential character for times earlier than , and an oscillating character for . The amplitude in the neighborhood of is plotted in Figure 13.

The initial oscillation period of the Brillouin precursor is crudely estimated (by setting ) as

(937) |

The amplitude of the Brillouin precursor is approximately

(938) |

Let us adopt the ordering

(939) |

which corresponds to the majority of physical situations involving the propagation of electromagnetic radiation through dielectric media. It follows, from the previous results, plus the results of Section 7.11, that

(940) |

and

(941) |

Furthermore,

(942) |

and

(943) |

Thus, it is clear that the Sommerfeld precursor is essentially a low amplitude, high frequency signal, whereas the Brillouin precursor is a high amplitude, low frequency signal. Note that the amplitude of the Brillouin precursor, despite being significantly higher than that of the Sommerfeld precursor, is still much less than that of the incident wave.