Impulse responses for reentrant discontinuities in bores

Abstract

We establish an optical analogy for the processing of spherical wavefronts in conical bore segments. A reentrant discontinuity acts like a converging lens or mirror to give outgoing waves a more positive vergence than incoming waves [M.P. Keating, Am. J. Phys. 43, 766–769 (1975)]. Impulse responses for single interactions with such a discontinuity include exponentially growing wakes, which become unmanageable very quickly in digital convolutions. Geometrical considerations rule out a reentrant discontinuity existing in isolation, and its troublesome behavior is kept under control by multiple reflections involving another discontinuity of stronger, negative power. In closed systems this process shows a “sorcerer's apprentice” type of behavior, in which a temporary solution to the problem turns into the problem for the next cycle. Fortunately that problem becomes more deltalike as time progresses, and there is a tendency for it to be cancelled by a neighboring delta. In open systems the behavior is much tam...