Volume 81, Issue 4, April 2013
 PAPERS


Towards the Kelvin wake and beyond
View Description Hide DescriptionThe difference between wave propagation in dispersive and nondispersive media can be effectively demonstrated by observing the wave patterns invoked by uniformly moving surface disturbances. Although the dispersion relation of surface waves on water is complicated, there are some frequency intervals where the phase velocity of the waves reduces to the simple power law behavior . Among these cases are gravity waves short compared to the depth of the water ( ), short capillary waves ( ), and long waves in shallow water ( ). Making use of this powerlaw behavior, we vary the exponent and visualize the smooth transition from a nondispersive to a dispersive medium.

Elucidating the stop bands of structurally colored systems through recursion
View Description Hide DescriptionInterference is the source of some of the spectacular colors of animals and plants in nature. In some of these systems, the physical structure consists of an ordered array of layers with alternating high and low refractive indices. This periodicity leads to an optical band structure that is analogous to the electronic band structure encountered in semiconductor physics: specific bands of wavelengths (the stop bands) are perfectly reflected. Here, we present a minimal model for optical band structure in a periodic multilayer structure and solve it using recursion relations. The stop bands emerge in the limit of an infinite number of layers by finding the fixed point of the recursion. We compare to experimental data for various beetles, whose optical structure resembles the proposed model. Thus, using only the phenomenon of interference and the idea of recursion, we are able to elucidate the concept of band structure in the context of the experimentally observed high reflectance and iridescent appearance of structurally colored beetles.

The pilotwave perspective on quantum scattering and tunneling
View Description Hide DescriptionThe de BroglieBohm “pilotwave” theory replaces the paradoxical waveparticle duality of ordinary quantum theory with a more mundane and literal kind of duality: each individual photon or electron comprises a quantum wave (evolving in accordance with the usual quantum mechanical wave equation) and a particle that, under the influence of the wave, traces out a definite trajectory. The definite particle trajectory allows the theory to account for the results of experiments without the usual recourse to additional dynamical axioms about measurements. Instead, one need simply assume that particle detectors click when particles arrive at them. This alternative understanding of quantum phenomena is illustrated here for two elementary textbook examples of onedimensional scattering and tunneling. We introduce a novel approach to reconcile standard textbook calculations (made using unphysical planewave states) with the need to treat such phenomena in terms of normalizable wave packets. This approach allows for a simple but illuminating analysis of the pilotwave theory's particle trajectories and an explicit demonstration of the equivalence of the pilotwave theory predictions with those of ordinary quantum theory.

Coupled secondquantized oscillators
View Description Hide DescriptionSecond quantization is a powerful technique for describing quantum mechanical processes in which the number of excitations of a single particle is not conserved. A textbook example of second quantization is the presentation of the simple harmonic oscillator in terms of creation and annihilation operators, which, respectively, represent addition or removal of quanta of energy from the oscillator. Our aim in this article is to bolster this textbook example. Accordingly, we explore the physics of coupled secondquantized oscillators. These explorations are phrased as exactly solvable eigenvalue problems, the mathematical structure providing a framework for the physical understanding. The examples we present can be used to enhance the discussion of secondquantized harmonic oscillators in the classroom, to make a connection to the classical physics of coupled oscillators, and to acquaint students with systems employed at the frontiers of contemporary physics research.

A classroom demonstration of reciprocal space
View Description Hide DescriptionAn array of nanowires and a laser pointer are used for a simple visualization of twodimensional reciprocal space. The experiment can be performed without any preparation and in any classroom. It aids the teaching of scattering experiments, and illustrates the underlying principles of electron, xray, and neutron scattering. A detailed study of the diffraction pattern was performed by mounting the sample with nanowires on a stage designed for xray scattering. The setup is well suited for undergraduate students, who get training in sample alignment in a small lab instead of at a largescale facility. The exact positions of the diffraction spots are calculated and monitored experimentally for a rotation of the sample. By fitting to this set of images, it is possible to determine the lattice vectors of the artificial crystal with an uncertainty of less than 1%.

The rise and fall of spinning tops
View Description Hide DescriptionThe motion of four different spinning tops was filmed with a highspeed video camera. Unlike pointed tops, tops with a rounded peg precess initially about a vertical axis that lies well outside the top, and then spiral inward until the precession axis passes through a point close to the centerofmass. The centerofmass of a top with a rounded peg can rise as a result of rolling rather than sliding friction, contrary to the explanation normally given for the rise of spinning tops. A tippe top was also filmed and was observed to jump vertically off a horizontal surface several times while the centerofmass was rising, contrary to the usual assumption that the normal reaction force on a tippe top remains approximately equal to its weight. It was found that the centerofmass of a tippe top rises as a result of rolling friction at low spin frequencies and as a result of sliding friction at high spin frequencies. It was also found that, at low spin frequencies, a tippe top can precess at two different frequencies simultaneously.

Synchronization of a thermoacoustic oscillator by an external sound source
View Description Hide DescriptionSince the pioneering work of Christiaan Huygens on the sympathy of pendulum clocks, synchronization phenomena have been widely observed in nature and science. In this paper, we describe a simple experiment, with a thermoacoustic oscillator driven by a loudspeaker, which exhibits several aspects of synchronization. Both the synchronization region of leading order around the oscillator's natural frequency f _{0} and regions of higher order (around f _{0}∕2 and f _{0}∕3) are measured as functions of the loudspeaker voltage and frequency. We also show that increasing the coupling between the loudspeaker and the oscillator gives rise under some circumstances to the death of selfsustained oscillations (quenching). Moreover, two additional set of experiments are performed: the first investigates a feedback loop in which the signal captured by the microphone is delivered to the loudspeaker through a phaseshifter; the second investigates the nontrivial interaction between the loudspeaker and the oscillator when the latter acts as a relaxation oscillator (spontaneous and periodic onset/damping of selfsustained oscillations). The experiment is easy to build and highly demonstrative; it might be of interest for classroom demonstrations or an instructional lab dealing with nonlinear dynamics.

A missing magnetic energy paradox
View Description Hide DescriptionWhile the interaction forces between two electric and two magnetic dipoles are formally identical, their interaction energies differ because in addition to mechanical work, the magnetic energy includes the electrical work needed to keep the dipole moments unaltered. This energy difference appears to contradict a calculation based on the integrals of the squares of the electric and magnetic fields since the electric and magnetic dipole fields have precisely the same geometry.
