Revisiting Standing Waves on a Circular Path

Abstract

All textbooks on general physics and on wave motion cover the subject of standing waves to some degree. However, standing waves on circular paths barely get remarked on in physics textbooks. The most common allusion to such standing waves in the textbooks is done to provide a reason why the quantization of angular momentum applies to the Bohr model of the atom,1-5 and without which this quantization condition is just a (unexplained) postulate. Less frequently mentioned are macroscopic cases, solid state physics examples, and optical "corrals."6 In contrast, searching the internet for articles dealing with "standing waves in a circle" or "standing waves on circular paths" tends only to bring up references to the short papers by Meiners,7 Vijay,8 and Bloom & Bloom.9 These three papers consider (non-equivalent mechanical) apparatuses designed to produce standing waves in (macroscopic) vibrating wire loops for the purpose of demonstrating standing wave patterns. (Videos of mechanically driven standing waves in circular objects are accessible from the internet. 10,11) Nevertheless, the available physics literature does not appear to provide a mathematical description of standing waves on a circular path suitable for use in undergraduate teaching and this situation requires attention.