Solution to kinematic equations

The above differential equations will often reoccur in later analysis, so they are very important. Interestingly, these differential equations have a simple solution. Taking the Snell wave to go through the origin at time zero, an expression for the arrival time of the Snell wave at any other location is given by

$$\begin{eqnarray} t_0(x,z) \ \ \ &=&\ \ \ {\sin\,\theta \over v }\ x\ +\ \int_0^z... ...\,x\ +\ \int_0^z\ \sqrt{ {1 \over v ( z ) ^2 }\ -\ p^2 } \ \ d z\end{eqnarray}$$ (12) (13)

The validity of equations (12) and  (13) is readily checked by computing ${\partial t_0}/{\partial x}$ and $\partial t_0 / \partial z$,then comparing with (10) and (11).

An arbitrary waveform f(t) may be carried by the Snell wave. Use (12) and (13) to define the time t₀ for a delayed wave f[t-t₀ (x,z)] at the location (x,z).  

$$\hbox{SnellWave}(t,x,z)\eq f \, \left( \ t\ -\ p\,x\ -\ \int_0^z\ \sqrt{ {1 \over v ( z )^2}\ -\ p^2 } \ \ dz \ \right)$$ (14)

Equation (14) carries an arbitrary signal throughout the whole medium. Interestingly, it does not agree with wave propagation theory or real life because equation (14) does not correctly account for amplitude changes that result from velocity changes and reflections. Thus it is said that Equation (14) is ``kinematically'' correct but ``dynamically'' incorrect. It happens that most industrial data processing only requires things to be kinematically correct, so this expression is a usable one.