Scalar diffraction from a circular aperture is a ubiquitous problem that arises in a variety of disciplines, such as optics (lenses), acoustics (speakers), electromagnetics (dish antennas), and ultrasonics (piston transducers). The problem endures despite centuries of research because each new generation of researchers rediscovers it and adds some novel insight or new result to the existing literature. This text promises a few new results and several novel insights, particularly with regard to spatial averaging. Although the text emphasizes ultrasonic diffraction, the results and insights developed are general and may be applied to the many practical problems involving scalar diffraction from a circular aperture. The material is based on a synthesis of mathematics, physical optics, linear systems theory, and scalar diffraction theory.