Waves in strata

Rays and fronts

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

Figure 3.7 depicts a ray moving down into the earth at an angle $\theta$ 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 $\theta$. 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 3.7 is

$$z = z_0 - x \tan \theta$$ (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:

$$z = {v t \over \cos \theta } - x \tan \theta$$ (5)

Solving for time gives

$$t(x,z) = {z\over v } \cos \theta + {x \over v } \sin \theta$$ (6)

Equation (3.6) tells the time that the wavefront will pass any particular location $(x,z)$. The expression for a shifted waveform of arbitrary shape is $f(t - t_0 )$. Using (3.6) to define the time shift $t_0$ gives an expression for a wavefield that is some waveform moving on a ray.

$$\hbox{moving wavefield} = \ f\left( t - {x \over v} \sin \theta - {z\over v} \cos \theta\right)$$ (7)

Waves in strata