Figure 4 depicts a ray moving down into the earth at an angle from the vertical.

front
Downgoing ray and wavefront.
Figure 4 |

Perpendicular to the ray is a wavefront.
By elementary geometry the angle between the wavefront and the earth's surface
is also .The ray increases its length at a speed *v*.
The speed that is observable on the earth's surface is the intercept
of the wavefront with the earth's surface.
This speed, namely , is faster than *v*.
Likewise, the speed of the intercept of the wavefront and
the vertical axis is .A mathematical expression for a straight line would be

(1) |

In this expression *z _{0}* is the intercept between the wavefront
and the vertical axis.
To make the intercept move downward, replace it by the
appropriate velocity times time:

(2) |

(3) |

(4) |

10/31/1997