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The simplest waves are expanding circles.
An equation for a circle expanding with velocity *v*
is
| |
(15) |

Considering *t* to be a constant,
i.e. taking a snapshot, equation (15) is that of a circle.
Considering *z* to be a constant,
it is an equation in the (*x* , *t*)-plane for a hyperbola.
Considered in the (*t* , *x* , *z*)-volume,
equation (15) is that of a cone.
Slices at various values of *t* show circles of various sizes.
Slices of various values of *z* show various hyperbolas.
Converting equation (15)
to traveltime depth we get

| |
(16) |

| (17) |

The earth's velocity typically increases
by more than a factor of two between the earth's surface,
and reflectors of interest.
Thus we might expect that equation (17) would have little
practical use.
Luckily, this simple equation will solve many problems for us
if we know how to interpret the velocity as an average velocity.

** Next:** Root-mean-square velocity
** Up:** Waves in strata
** Previous:** Solution to kinematic equations
Stanford Exploration Project

12/26/2000