A laboratory experiment to test the limits of Bernoulli-Euler theory for flexural waves in bars

At Brigham Young University, one of the acoustics courses taught in the Physics and Astronomy Department focuses on resonance topics. This course is a graduate level course that uses a differential equation approach to wave motion and resonances on strings, bars, membranes, and plates. This paper discusses the theory of flexural, or bending, waves in free-free rods and describes a laboratory experiment to test the rod thickness limits of the Bernoulli-Euler theory for rods. It is found that the fundamental frequency departs from Bernoulli-Euler theory as the thickness exceeds 6% of a flexural wavelength, however the mode shape above this limit remains unchanged (at least for the rods tested which included thicknesses up until the thickness equals 16% of a wavelength).