Rays and fronts

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

 

front Figure 4 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 $\theta$.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 would be  

$$z \ \ =\ \ z_0 \ -\ x \ \tan \, \theta$$ (1)

like that shown to be the wavefront in Figure 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:  

$$z \ \ =\ \ v \, {t \over \cos \, \theta } \ -\ x \ \tan \, \theta$$ (2)

Solving for time gives  

$$t(x,z) \ \ =\ \ {z\over v }\ \cos\,\theta \ +\ {x \over v }\ \sin \, \theta$$ (3)

Equation (3) tells the time that the wavefront will pass any particular location (x , z). The expression for a shifted waveform of arbitrary shape is $f(t \ -\ t_0 )$.Using (3) to define the time shift t₀ gives an expression for a wavefield that is some waveform moving on a ray.  

$$\hbox{moving wavefield} \ \ =\ \ f\left( \ t\ -\ {x \over v}\ \sin\,\theta \ -\ {z\over v}\ \cos\,\theta\right)$$ (4)