Including lateral velocity variation

Lateral velocity variation v=v(x) has not been included in the program, but it is not difficult to install. It enters in two places. It enters first in equation (60). If the wavefield is such that k_x is small enough, then equation (60) is the only place it is needed. Second, it enters in the tridiagonal coefficients through the v in equation (56). The so-called thin-lens approximation of optics seems to amount to including the equation (60) part only. An example of lateral velocity variation is in Figure 8.

 

Mlateralvel Figure 8 Make changes to the program to include a thin-lens term with a lateral velocity change of 40% across the frame produced by a constant slowness gradient. Identify other parts of the program which are affected by lateral velocity variation. You need not make these other changes. Why are they expected to be small?

 

Mfortyfive Figure 9 Incorporate the 45$^\circ$ term, $\partial_{xxz}$, for the collapsing spherical wave. Use zero-slope sides. Compare your result with the 15$^\circ$ result obtained via the program wavemovie. Mark an X at the theoretical focus location.