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
Highlights
Assume a sufficiently small displacement from equilibrium (i.e., “linear” behavior) and apply a Taylor series expansion to evaluate the net forces acting on the differential element
The bar was cooled below 4.2 K to permit the use of a superconducting quantum interferometer (SQUID) that measured the motion of a simple harmonic oscillator, tuned to f1
The quartz crystal microbalance (QCM) is used to “weigh” very thin films, such as those deposited on microelectronic circuits by vacuum evaporation of metals
Summary
There are three independent types of waves that can be excited in thin bars of solid materials at frequencies that are sufficiently low that the wavelengths of these waves are much greater than the cross-sectional dimensions (i.e., λ ) S1/2): (i) longitudinal waves of compression and expansion, (ii) torsional waves that produce twisting, and (iii) flexural waves that cause the bar to bend. To determine the net longitudinal force, dFx, acting on the differential element of length, dx, we expand Eq (5.2) in a Taylor series about Fx (x) and assume that Young’s modulus is a constant. The fundamental free-free longitudinal mode of a very large aluminum bar, 3 m in length, with a mass of 2300 kg, was used in an attempt to detect gravitational waves.1 [1] Using Young’s modulus and the mass density of aluminum at room temperature, the speed of longitudinal waves cL 1⁄4 5510 m/s, making f1 ffi 860 Hz. In operation, the bar was cooled below 4.2 K to permit the use of a superconducting quantum interferometer (SQUID) that measured the motion of a simple harmonic oscillator, tuned to f1. 2n À 1 πcB ωn 1⁄4 2 L for n 1⁄4 1, 2, 3,
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
