Modes of Bars

Abstract

Abstract The perspectives and techniques developed in the previous chapters will now be applied to calculation of wave propagation in solids. Their application to longitudinal and shear waves will be both familiar and simple. Even more satisfying is the success of those same techniques in finding solutions for waves in a system that does not obey the wave equation and exhibit a phase speed that varies with frequency. Measurement of the frequencies of the normal modes of thin bars will be used to determine the bars’ elastic constants to high precision. The relationship between measured modal frequencies and the elastic moduli is particularly simple because the torsional, flexural, and longitudinal modes of bars can be selectively excited and detected. The technique of resonant ultrasound spectroscopy will allow the extraction of moduli from resonance frequencies even for samples with dimensions that are not as conducive as those of thin bars by a process that is significantly more computationally intensive. The flexural rigidity of wires under tension will be analyzed to determine the normal modes of a “stiff string,” and those effects will be discussed in relation to the tuning of pianos