Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincaré procedure is valid to the order considered; weakly resonant, where the Poincaré procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincaré result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter.

Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

Experiments in Neutrino Mass and Mixing: Part I - New Limits on Heavy Neutrino Emission in Nuclear Beta Decay. Part II - Fast Neutron Backgrounds for the San Onofre Neutrino Detector

In Part I of this thesis, a new magnetic spectrometer experiment which measured the β spectrum of ^(35)S is described. New limits on heavy neutrino emission in nuclear β decay were set, for a heavy neutrino mass range between 12 and 22 keV. In particular, this measurement rejects the hypothesis that a 17 keV neutrino is emitted, with sin^2 θ = 0.0085, at the 6δ statistical level. In addition, an auxiliary experiment was performed, in which an artificial kink was induced in the β spectrum by means of an absorber foil which masked a fraction of the source area. In this measurement, the sensitivity of the magnetic spectrometer to the spectral features of heavy neutrino emission was demonstrated.

In Part II, a measurement of the neutron spallation yield and multiplicity by the Cosmic-ray Underground Background Experiment is described. The production of fast neutrons by muons was investigated at an underground depth of 20 meters water equivalent, with a 200 liter detector filled with 0.09% Gd-loaded liquid scintillator. We measured a neutron production yield of (3.4 ± 0.7) x 10^(-5) neutrons per muon-g/cm^2, in agreement with other experiments. A single-to-double neutron multiplicity ratio of 4:1 was observed. In addition, stopped π^+ decays to µ^+ and then e^+ were observed as was the associated production of pions and neutrons, by the muon spallation interaction. It was seen that practically all of the π^+ produced by muons were also accompanied by at least one neutron. These measurements serve as the basis for neutron background estimates for the San Onofre neutrino detector.

Discovery of non-zero neutrino mixing angle θ_13 using Daya Bay antineutrino detectors

The Daya Bay Reactor Antineutrino Experiment observed the disappearance of reactor $\bar{\nu}_e$ from six $2.9~GW_{th}$ reactor cores in Daya Bay, China. The Experiment consists of six functionally identical $\bar{\nu}_e$ detectors, which detect $\bar{\nu}_e$ by inverse beta decay using a total of about 120 metric tons of Gd-loaded liquid scintillator as the target volume. These $\bar{\nu}_e$ detectors were installed in three underground experimental halls, two near halls and one far hall, under the mountains near Daya Bay, with overburdens of 250 m.w.e, 265 m.w.e and 860 m.w.e. and flux-weighted baselines of 470 m, 576 m and 1648 m. A total of 90179 $\bar{\nu}_e$ candidates were observed in the six detectors over a period of 55 days, 57549 at the Daya Bay near site, 22169 at the Ling Ao near site and 10461 at the far site. By performing a rate-only analysis, the value of $sin^2 2\theta_{13}$ was determined to be $0.092 \pm 0.017$.

Red shift and broadening of backward harmonic radiation from electron oscillations driven by femtosecond laser pulse

The characteristics of backward harmonic radiation due to electron oscillations driven by a linearly polarized fs laser pulse are analysed considering a single electron model. The spectral distributions of the electron's backward harmonic radiation are investigated in detail for different parameters of the driver laser pulse. Higher order harmonic radiations are possible for a sufficiently intense driving laser pulse. We have shown that for a realistic pulsed photon beam, the spectrum of the radiation is red shifted as well as broadened because of changes in the longitudinal velocity of the electrons during the laser pulse. These effects are more pronounced at higher laser intensities giving rise to higher order harmonics that eventually leads to a continuous spectrum. Numerical simulations have further shown that by increasing the laser pulse width the broadening of the high harmonic radiations can be controlled.

Magnetic trapping of atomic tritium for neutrino mass measurement

Improved measurement of the neutrino mass via β decay spectroscopy requires the development of new energy measurement techniques and a new β decay source. A promising proposal is to measure the β energy by the frequency of the cyclotron radiation emitted in a magnetic field and to use a high purity atomic tritium source. This thesis examines the feasibility of using a magnetic trap to create and maintain such a source. We demonstrate that the loss rate due to β decay heating is not a limiting factor for the design. We also calculate the loss rate due to evaporative cooling and propose that the tritium can be cooled sufficiently during trap loading as to render this negligible. We further demonstrate a design for the magnetic field which produces a highly uniform field over a large fraction of the trap volume as needed for cyclotron frequency spectroscopy while still providing effective trapping.

Wave induced oscillations in harbors of arbitrary shape

Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.

A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇²f + k²f = 0, is formulated as an integral equation; an approximate method is employed to solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the solutions for the regions outside and inside the harbor.

Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical check on the arbitrary shaped harbor theory.

Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea condition.

Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach Harbor located in Long Beach, California.

Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.

It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective interior boundaries.