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When subjected to a small perturbation, an object vibrates in
a characteristic way. These vibrations can be so distinctive
that they allow for the immediate identification of the
object. A familiar example is the sound of musical
instruments: a musical piece played on a viola clearly sounds
different from when played on a violin. In the same way,
vibrations of atoms can be used to identify a material. The
chemical composition of our sun was first deduced in the 19th
century from absorption lines in its spectrum, absorption
lines whose properties are controlled by the elementary
excitations of the constituent molecules.
Furthermore, natural vibrations in response to perturbations are important in dynamical processes. A natural vibrational mode, more commonly called an instability mode, grows over time taking energy from the underlying dynamics and redistributing it into a new form. This can often distort the underlying dynamics so much that a qualitatively new dynamics results. The transition from laminar to turbulent flow is a striking example. The formation of a singularity, a divergence in one or more physical quantities in the mathematical description of an event after a finite amount of time, represents one extreme outcome of a dynamical process. The concept of a singularity is relevant to a variety of processes, from the formation of black holes to the fission of nuclei. It is commonly accepted that the transient dynamics leading up to a singular event should be largely erased. Therefore, experimental and analytical tools for analyzing a singularity focus on characterizing the few universal features (i.e. scaling exponents and invariants) that are independent of initial and boundary conditions and serve to encapsulate the nature of the singularity dynamics. In systems where the phenomena of interest are difficult to observe, for example, star formation or the creation of a galaxy cluster via gravitational implosion, the assumption of a universal dynamics makes the problem tractable and enables analytical progress. Yet, whether such a universal dynamics is attained in practice remains an open question. One way to address this question is to focus on other examples of singularity formation, such as the break-up of a liquid drop, which can be observed directly in experiments and are simpler to understand theoretically. The outcomes have so far largely supported the view that universal dynamics are attained near a singularity. However, our recent experiments on a particularly simple example of singularity formation, the detachment of an air bubble from an underwater nozzle, revealed a very different type of dynamics. The final moments of this process are controlled by an inertial implosion as the potential energy difference between water and air, which initiates the implosion, is entirely converted into kinetic energy and concentrated into a vanishingly small region as the neck of air collapses to a point. This singularity dynamics is not universal; rather it possesses a precise memory of the transient dynamics leading up to the singularity. Small asymmetries in the initial neck shape are not completely erased as the singularity is approached, but instead persist and affect the dynamics even after the singularity occurs. This precise memory defies the tools used previously to characterize the singularity dynamics, since these were developed based on the assumption that the singularity dynamics has little memory of transients. Our work demonstrates that while these more recent and sophisticated tools fail, the older and simpler method of characterizing the natural vibrational modes succeeds in describing the dynamics near the singularity. Analysis of an idealized model of an implosion singularity shows that a precise memory of the transients implies an unusual behavior for the vibrational modes. In general, the amplitudes of a vibration change over time as energy is siphoned out of (or into) the main collapse and into the azimuthal modes, but the precise memory in this singularity forces the amplitudes to become constant as the singularity approaches. As a result, the azimuthal energy distribution, which is simply a composition of the cylindrically symmetric collapse and all the vibration modes, becomes essentially "frozen." Moreover, we have induced a vibrational mode in an experiment and directly measured its time-evolution. The measured vibration evolves in a fashion consistent with that calculated, showing the vibration predicted in the idealized model as a consequence of the precise memory of the singularity is indeed relevant to the actual dynamical evolution. |