INTRODUCTION

First we will review the calculus of step functions and power functions and their Fourier transforms leading up to the half-order differential operator. With this background we can answer the question, what is the difference between a line and the sequence of short dashes that make up the line? Likewise, what is the difference between a plane and a dense array of tiny patches that make up the plane? Is each tiny patch the same as a point source? These questions will be answered from the viewpoint of wave propagation where it is conceptually and computationally convenient to make plane waves by superposing an array of spherical waves. We will find that a plane wave with an impulsive waveform can be made from a superposition of circles (or hyperbolas) or spheres (or hyperboloids of revolution) but that in three dimensions the hyperbolid carries not an impulsive waveform, but a time derivative, d/dt, and in two dimensions the hyperbola carries the half-order derivative waveform. These waveforms are generally clear on synthetic data but on field data they are usually obscured by waveforms from other sources.