Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session Q1: Memory and Focusing in Catastrophic Deformations |
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Sponsoring Units: DCMP Chair: Michael Brenner, Harvard University Room: Spirit of Pittsburgh Ballroom A |
Wednesday, March 18, 2009 11:15AM - 11:51AM |
Q1.00001: Tsunami Asymptotics Invited Speaker: Optical analogies, and some singularity theory, give new information about tsunamis. For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth act as lenses and can focus tsunamis onto cusped caustics, and this ``singularity on a singularity'' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude. [Preview Abstract] |
Wednesday, March 18, 2009 11:51AM - 12:27PM |
Q1.00002: Memory encoding vibrations in a disconnecting air bubble Invited Speaker: The implosion that disconnects a submerged air bubble into several bubbles provides a simple example of energy focusing. The most efficient disconnection is an entirely symmetric one terminating in a finite-time singularity. At the final moment, the potential energy at the start of the disconnection is entirely condensed into the kinetic energy of a vanishingly small amount of liquid rushing inwards to disconnect the bubble. In reality, however, the initial shape always possesses slight imperfections. We show that a memory of the imperfection remains and controls the final fate of the focusing. Linear stability reveals that even an infinitesimal perturbation is remembered. A slight initial asymmetry excites vibrations in the cross-section shape of the bubble neck. The vibrations persist over time. Near the singularity, their amplitudes freeze, locking onto constant values, while their frequencies chirp, increasing more and more rapidly. The net effect is that the singularity remembers exactly half of the information about the initial imperfection, the half encoded by the vibration amplitudes. We check this scenario in an experiment by releasing an air bubble from a nozzle with an oblong cross-section. This excites an elongation-compression vibrational mode. We measure the vibration excited and find quantitative agreement with linear stability. When the initial distortion has a small, but finite, size, the saturation of the vibration amplitude causes the symmetric singularity to be pre-empted by an asymmetric contact between two distant points on the interface. Numerics reveal that the contact is typically smooth, corresponding to two inward-curving portions of the bubble surface colliding at finite speed. Both the contact speed and curvature vary non-monotonically with the initial distortion size, with abrupt jumps at specific values. This is because the vibration causes contact to occur at different values of the phase. A contact produced when the shape distortion is pronounced requires a smaller initial amplitude than a contact produced when the vibration is out of phase. (Joint work with Nathan C. Keim, Lipeng Lai, Laura E. Schmidt, Konstantin Turitsyn and Sidney R. Nagel.) [Preview Abstract] |
Wednesday, March 18, 2009 12:27PM - 1:03PM |
Q1.00003: Pattern Transformation Triggered by Deformation Invited Speaker: Periodic elastomeric cellular solids are subjected to uniaxial compression and a novel uniform transformation of the structure is found above a critical value of applied load. The results of a numerical investigation reveal that the pattern switch is triggered by a reversible elastic instability. The mechanism has proved to be useful for controlled imprinting of complex patterns in phononic and photonic crystals. The material also provides an example of a simple, tunable and robust negative Poisson ratio foam. More recently, the inverse problem of an appropriate array of elastic particles has been shown to provide another example of an intriguing pattern switch. [Preview Abstract] |
Wednesday, March 18, 2009 1:03PM - 1:39PM |
Q1.00004: Resonant generation of internal waves on a model continental slope Invited Speaker: Away from shallow, well-mixed surface regions, the density of sea water increases with depth due to variation in salinity and temperature. This continuous density stratification supports \textit{internal} gravity waves, which are the counterpart within the fluid interior of \textit{surface} gravity waves. Internal gravity waves are import for many oceanic processes, such as sediment transportation and ocean mixing. We study internal wave generation in a laboratory model of oscillating tidal flow on a continental margin. Waves are found to be generated only in a near-critical region where the slope of the bottom topography matches that of internal waves. Fluid motion with a velocity an order of magnitude larger than that of the forcing occurs within a thin boundary layer above the bottom surface. The resonant wave is unstable because of strong shear; Kelvin-Helmholtz billows precede wave breaking. We construct a model to extrapolate our results to oceanic conditions. This work [1] provides a new explanation for the intense boundary flows on continental slopes. \\[4pt] [1] H. P. Zhang, B. King and Harry L. Swinney, Phys. Rev. Lett. 100, 244504 (2008). [Preview Abstract] |
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