It is natural to begin studies of waves with equations that describe plane waves in a medium of constant velocity.

Figure 7 depicts a ray moving down into the earth at an angle from the vertical.

front
Downgoing ray and wavefront.
Figure 7 |

Perpendicular to the ray is a wavefront.
By elementary geometry the angle between the wave**front**
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
like that shown to be the wavefront in Figure 7 is

(4) |

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:

(5) |

(6) |

(7) |

12/26/2000