CONCLUSION

The solution to each partial differential equation, given the amplitude values along an initial curve, will generate a surface. The surface represents the correct distribution of amplitudes across the 2-D operator. The problem of finding the characteristic curves is formulated in an overdetermined form. I expected to find both families of curves as solutions to the partial differential equation. However each PDE offers a single family of curves which coincides with the required characteristic, while the other associated family of curves is different from the form presented in Figure 1. The answer to this problem could be found in either using a higher order of differential equations to have as solutions the two families of curves, or to require a single family of curves to be described by the characteristics associated with the PDE. Using a single family of curves will not generate the cutoff in the DMO ellipse. While the approach to find the partial differential equations was an inductive approach, a deductive analysis of the amplitudes of the operator could generate a better understanding of the problem.