On acoustic cavitation of slightly subcritical bubbles

The classical Blake threshold indicates the onset of quasistatic evolution leading to cavitation for gas bubbles in liquids. When the mean pressure in the liquid is reduced to a value below the vapor pressure, the Blake analysis identifies a critical radius which separates quasistatically stable bubbles from those which would cavitate. In this work, we analyze the cavitation threshold for radially symmetric bubbles whose radii are slightly less than the Blake critical radius, in the presence of time-periodic acoustic pressure fields. A distinguished limit equation is derived that predicts the threshold for cavitation for a wide range of liquid viscosities and forcing frequencies. This equation also yields frequency-amplitude response curves. Moreover, for fixed liquid viscosity, our study identifies the frequency that yields the minimal forcing amplitude sufficient to initiate cavitation. Numerical simulations of the full Rayleigh-Plesset equation confirm the accuracy of these predictions. Finally, the implications of these findings for acoustic pressure fields that consist of two frequencies will be discussed.Comment: 14 pages, Presented at APS/DFD conference in Philadelphia 199

Transport in Transitory Dynamical Systems

We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the "action" of relevant heteroclinic orbits at the intersection of invariant manifolds of "forward" and "backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure

Deformations of Gabor Frames

The quantum mechanical harmonic oscillator Hamiltonian generates a one-parameter unitary group W(\theta) in L^2(R) which rotates the time-frequency plane. In particular, W(\pi/2) is the Fourier transform. When W(\theta) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family of frames F_\theta generated by the non-compactly supported windows g_\theta=W(theta)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When \theta=\pi/2, we obtain the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family F_\theta therefore interpolates these two familiar examples.Comment: 8 pages in Plain Te

Search for the magnetic field of the O7.5 III star xi Persei

Cyclical wind variability is an ubiquitous but as yet unexplained feature among OB stars. The O7.5 III(n)((f)) star xi Persei is the brightest representative of this class on the Northern hemisphere. As its prominent cyclical wind properties vary on a rotational time scale (2 or 4 days) the star has been already for a long time a serious magnetic candidate. As the cause of this enigmatic behavior non-radial pulsations and/or a surface magnetic field are suggested. We present a preliminary report on our attempts to detect a magnetic field in this star with high-resolution measurements obtained with the spectropolarimeter Narval at TBL, France during 2 observing runs of 5 nights in 2006 and 5 nights in 2007. Only upper limits could be obtained, even with the longest possible exposure times. If the star hosts a magnetic field, its surface strength should be less than about 300 G. This would still be enough to disturb the stellar wind significantly. From our new data it seems that the amplitude of the known non-radial pulsations has changed within less than a year, which needs further investigation.Comment: 2 pages, 6 figures, contributed poster at IAU Symposium 259 "Cosmic Magnetic Fields: from Planets, to Stars and Galaxies", Tenerife, Spain, November 3-7, 200