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## Rays and fronts

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 Figure 7 Downgoing ray and wavefront.

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 like that shown to be the wavefront in Figure 7 is
 (4)

In this expression z0 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)
Solving for time gives
 (6)
Equation (6) tells the time that the wavefront will pass any particular location (x , z). The expression for a shifted waveform of arbitrary shape is f(t - t0 ). Using (6) to define the time shift t0 gives an expression for a wavefield that is some waveform moving on a ray.
 (7)

Next: Snell waves Up: DIPPING WAVES Previous: DIPPING WAVES
Stanford Exploration Project
12/26/2000