Resonance ultrasound spectroscopy: Highlights of the mathematics.

Abstract

We highlight the key mathematics associated with resonance ultrasound spectroscopy (RUS). From the physics viewpoint, RUS is part of the general problem of elastic vibrations in continuous media. A solid can transmit three waves: one quasilongitudinal and two quasitransverse. At resonance, we see standing waves, which we can represent by an eight‐summation Fourier‐series wave function. We begin by reviewing briefly the 1‐D vibrating‐string problem that arose in the music of many early civilizations (about 4000 BC). Pythagoras of Samos (582–507 BC) first treated this problem mathematically. In the early beginnings of modern science, both Galileo (1564–1642) and Mersenne (1588–1648) considered vibrating strings, which have a perfect analog in axial‐bar vibration. Lamb and Lame contributed in the 1800s. Probably the first inverse RUS measurement came from Cole and Frazer (1964), who studied the sphere‐vibration problem using graphical methods. Most of our talk will highlight the remarkable mathematical contr...