The dynamics and tuning of orchestral crotales

Crotales are center-weighted, tuned cymbals that are found in the percussion section of most orchestras. They are arranged like a keyboard in octave sets and are commercially available in two octaves, from C6 to C8. Little information about the physics of crotales has been reported in the literature, despite their having the interesting property of producing a particularly pleasing sound. In this study, the acoustic and vibrational properties of crotales from C6 to C8 are theoretically and experimentally investigated. Interferograms of typical vibrational modes are presented, and the frequencies of the acoustically important modes of crotales are identified and reported. The acoustic spectra of the crotales are compared to theoretical predictions for thin circular plates and annular plates clamped at the center. These models are found to be insufficient for predicting the normal modes of the crotales. An accurate model is developed using finite element analysis, and this model is used to investigate the effects of subtle changes in the magnitude and size of the center mass on the acoustic spectrum. This investigation serves as a basis for suggestions for improvement of the crotales by modifying the center mass. Finally, the physical parameters for a set of clamped annular plates are derived such that the set has similar acoustic properties to a set of crotales, but with more accurate tuning. The validity of these parameters is confirmed using finite element analysis

Non-degenerate normal-mode doublets in vibrating flat circular plates

The vibrations of flat circular plates have been studied for hundreds of years, and they are well understood by the scientific community. Unfortunately, when vibrating circular plates are discussed in textbooks, the relationship between pairs of spatially orthogonal vibrational patterns that occur at each of the normal-mode frequencies is often ignored. Usually these orthogonal solutions are presented to the student as being degenerate in frequency; however, in practice the degeneracy of the doublet is often broken and the two spatially orthogonal solutions are separated in frequency. We show theoretically and experimentally that the degeneracy can be broken by a small asymmetry in the plate, and we derive a formula for predicting the magnitude of the frequency-splitting. We have used electronic speckle pattern interferometry to investigate the phenomena of doublet splitting and have confirmed the validity of the theory

Design of Force Fields from Data at Finite Temperature

We investigate the problem of how to obtain the force field between atoms of an experimentally determined structure. We show how this problem can be efficiently solved, even at finite temperature, where the position of the atoms differs substantially from the ground state. We apply our method to systems modeling proteins and demonstrate that the correct potentials can be recovered even in the presence of thermal noise.Comment: 10 pages, 1 postcript figure, Late

A note on entropic uncertainty relations of position and momentum

We consider two entropic uncertainty relations of position and momentum recently discussed in literature. By a suitable rescaling of one of them, we obtain a smooth interpolation of both for high-resolution and low-resolution measurements respectively. Because our interpolation has never been mentioned in literature before, we propose it as a candidate for an improved entropic uncertainty relation of position and momentum. Up to now, the author has neither been able to falsify nor prove the new inequality. In our opinion it is a challenge to do either one.Comment: 2 pages, 2 figures, 2 references adde

Steric constraints in model proteins

A simple lattice model for proteins that allows for distinct sizes of the amino acids is presented. The model is found to lead to a significant number of conformations that are the unique ground state of one or more sequences or encodable. Furthermore, several of the encodable structures are highly designable and are the non-degenerate ground state of several sequences. Even though the native state conformations are typically compact, not all compact conformations are encodable. The incorporation of the hydrophobic and polar nature of amino acids further enhances the attractive features of the model.Comment: RevTex, 5 pages, 3 postscript figure

Quantum Inequalities and Singular Energy Densities

There has been much recent work on quantum inequalities to constrain negative energy. These are uncertainty principle-type restrictions on the magnitude and duration of negative energy densities or fluxes. We consider several examples of apparent failures of the quantum inequalities, which involve passage of an observer through regions where the negative energy density becomes singular. We argue that this type of situation requires one to formulate quantum inequalities using sampling functions with compact support. We discuss such inequalities, and argue that they remain valid even in the presence of singular energy densities.Comment: 18 pages, LaTex, 2 figures, uses eps

A closer look at the uncertainty relation of position and momentum

We consider particles prepared by the von Neumann-L\"uders projection. For those particles the standard deviation of the momentum is discussed. We show that infinite standard deviations are not exceptions but rather typical. A necessary and sufficient condition for finite standard deviations is given. Finally, a new uncertainty relation is derived and it is shown that the latter cannot be improved.Comment: 3 pages, introduction shortened, content unchange