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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 $ \theta $ from the vertical.

 
front
Figure 7
Downgoing ray and wavefront.

front
view

Perpendicular to the ray is a wavefront. By elementary geometry the angle between the wavefront and the earth's surface is also $ \theta $.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 $ v / \sin \theta $, is faster than v. Likewise, the speed of the intercept of the wavefront and the vertical axis is $ v / \cos \theta $.A mathematical expression for a straight line like that shown to be the wavefront in Figure 7 is  
 \begin{displaymath}
z \ \ =\ \ z_0 \ -\ x \ \tan \, \theta\end{displaymath} (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:  
 \begin{displaymath}
z \ \ =\ \ {v \, t \over \cos \, \theta } \ -\ x \ \tan \, \theta\end{displaymath} (5)
Solving for time gives  
 \begin{displaymath}
t(x,z) \ \ =\ \ {z\over v }\ \cos\,\theta \ +\ {x \over v }\ \sin \, \theta\end{displaymath} (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.  
 \begin{displaymath}
\hbox{moving wavefield} \ \ =\ \ 
f\left( \ t\ -\ {x \over v}\ \sin\,\theta \ -\ {z\over v}\ \cos\,\theta\right)\end{displaymath} (7)


next up previous print clean
Next: Snell waves Up: DIPPING WAVES Previous: DIPPING WAVES
Stanford Exploration Project
12/26/2000