Reflections and the high-frequency limit

It is well known that the contact between two different materials can cause acoustic reflections. A material contact is defined to be a place where either K or $\rho$ changes by a spatial step function. In one dimension either $\partial K / \partial x$ or $\partial \rho / \partial x$ or both would be infinite at a point, and we know that either can cause a reflection. So it is perhaps a little surprising that while the density derivative is explicitly found in (17), the incompressibility derivative is not. This means that dropping the density gradients in (17) will not eliminate all possible reflections. Dropping the terms will slightly simplify further analysis, however, and since constant density is a reasonable case, the terms are often dropped.

There are also some well-known mathematical circumstances under which the first-order terms may be ignored. Fix your attention on a wave going in any particular direction. Then $\omega$, k_x, and k_z have some prescribed ratio. In the limiting case that frequency goes to infinity, the P_tt, P_xx, and P_zz terms in (18) all tend to the second power of infinity. Suppose two media gradually blend into one another so that $\partial \rho / \partial x$ is less than infinity. The terms neglected in going from (17) to (18) are of the form $\rho_x \, P_x$ and $\rho_z \, P_z$.As frequency tends to infinity, these terms only tend to the first power of infinity. Thus, in that limit they can be neglected.

These terms are usually included in theoretical seismology where the goal is to calculate synthetic seismograms. But where the goal is to create earth models from seismic field data--as in this book--these terms are generally neglected. Earth imaging is more difficult than calculating synthetic seismograms. But often the reason for neglecting the terms is simply to reduce the clutter. These terms may be neglected for the same reason that equations are often written in two dimensions instead of three: the extension is usually possible but not often required. Further, these terms are often ignored to facilitate Fourier solution techniques. Practical situations might arise for which these terms need to be included. With the finite-difference method, they are not difficult to include. But any effort to include them in data processing should also take into account other factors of similar significance, such as the assumption that the acoustic equation approximates the elastic world.

EXERCISES:

1. Soil is typically saturated with water below a certain depth, which is known as the water table. Experience with hammer-seismograph systems shows that seismic velocity typically jumps up to water velocity ($V_{{} -2 { H_2 O}}\,=$ $1500 \ m/s$)at the water table. Say that in a certain location, the ground roll is observed to be greater than the reflected waves, so a decision has been made to bury geophones. The troublesome ground roll is observed to travel at six-tenths the speed of a water wave. How deep must the geophones be buried below the water table to attenuate the ground roll by a factor of ten? Assume the data contains all frequencies from 10 to 100 Hz. (Hints: $\ln 10 \approx 2$, $2 \pi \approx 6$, etc.)
2. Consider the function  

   $$P ( z , t ) \ \ = \ \ P_0 \ {1 \over \sqrt { Y ( z ) } } \ ... ... \omega \, \sqrt { { \rho ( \xi ) \over K ( \xi ) } } \ d \xi }$$ (22)

   where as a trial solution for the one-dimensional wave equation:  

   $$\left( \ { \partial^2 \over \partial t^2 } \ - \ { K ( z ) ... ...rtial \rho \over \partial z } \ { \partial P \over \partial z }$$ (23)

   Substitute the trial solution (22) into the wave equation (23). Discuss the trade-off between changes in material properties and the validity of your solution for different wavelengths.