Controlling wave propagation through nonlinear engineered granular systems

We study the fundamental dynamic behavior of a special class of ordered granular systems in order to design new, structured materials with unique physical properties. The dynamic properties of granular systems are dictated by the nonlinear, Hertzian, potential in compression and zero tensile strength resulting from the discrete material structure. Engineering the underlying particle arrangement of granular systems allows for unique dynamic properties, not observed in natural, disordered granular media. While extensive studies on 1D granular crystals have suggested their usefulness for a variety of engineering applications, considerably less attention has been given to higher-dimensional systems. The extension of these studies in higher dimensions could enable the discovery of richer physical phenomena not possible in 1D, such as spatial redirection and anisotropic energy trapping. We present experiments, numerical simulation (based on a discrete particle model), and in some cases theoretical predictions for several engineered granular systems, studying the effects of particle arrangement on the highly nonlinear transient wave propagation to develop means for controlling the wave propagation pathways. The first component of this thesis studies the stress wave propagation resulting from a localized impulsive loading for three different 2D particle lattice structures: square, centered square, and hexagonal granular crystals. By varying the lattice structure, we observe a wide range of properties for the propagating stress waves: quasi-1D solitary wave propagation, fully 2D wave propagation with tunable wave front shapes, and 2D pulsed wave propagation. Additionally the effects of weak disorder, inevitably present in real granular systems, are investigated. The second half of this thesis studies the solitary wave propagation through 2D and 3D ordered networks of granular chains, reducing the effective density compared to granular crystals by selectively placing wave guiding chains to control the acoustic wave transmission. The rapid wave front amplitude decay exhibited by these granular networks makes them highly attractive for impact mitigation applications. The agreement between experiments, numerical simulations, and applicable theoretical predictions validates the wave guiding capabilities of these engineered granular crystals and networks and opens a wide range of possibilities for the realization of increasingly complex granular material design.

Tools for the study of dynamical spacetimes

This thesis covers a range of topics in numerical and analytical relativity, centered around introducing tools and methodologies for the study of dynamical spacetimes. The scope of the studies is limited to classical (as opposed to quantum) vacuum spacetimes described by Einstein's general theory of relativity. The numerical works presented here are carried out within the Spectral Einstein Code (SpEC) infrastructure, while analytical calculations extensively utilize Wolfram's Mathematica program.

We begin by examining highly dynamical spacetimes such as binary black hole mergers, which can be investigated using numerical simulations. However, there are difficulties in interpreting the output of such simulations. One difficulty stems from the lack of a canonical coordinate system (henceforth referred to as gauge freedom) and tetrad, against which quantities such as Newman-Penrose Psi_4 (usually interpreted as the gravitational wave part of curvature) should be measured. We tackle this problem in Chapter 2 by introducing a set of geometrically motivated coordinates that are independent of the simulation gauge choice, as well as a quasi-Kinnersley tetrad, also invariant under gauge changes in addition to being optimally suited to the task of gravitational wave extraction.

Another difficulty arises from the need to condense the overwhelming amount of data generated by the numerical simulations. In order to extract physical information in a succinct and transparent manner, one may define a version of gravitational field lines and field strength using spatial projections of the Weyl curvature tensor. Introduction, investigation and utilization of these quantities will constitute the main content in Chapters 3 through 6.

For the last two chapters, we turn to the analytical study of a simpler dynamical spacetime, namely a perturbed Kerr black hole. We will introduce in Chapter 7 a new analytical approximation to the quasi-normal mode (QNM) frequencies, and relate various properties of these modes to wave packets traveling on unstable photon orbits around the black hole. In Chapter 8, we study a bifurcation in the QNM spectrum as the spin of the black hole a approaches extremality.

Experiments with a laser interferometric gravitational wave antenna

Sources and effects of astrophysical gravitational radiation are explained briefly to motivate discussion of the Caltech 40 meter antenna, which employs laser interferometry to monitor proper distances between inertial test masses. Practical considerations in construction of the apparatus are described. Redesign of test mass systems has resulted in a reduction of noise from internal mass vibrations by up to two orders of magnitude at some frequencies. A laser frequency stabilization system was developed which corrects the frequency of an argon ion laser to a residual fluctuation level bounded by the spectral density √s_v(f) ≤ 60µHz/√Hz, at fluctuation frequencies near 1.2 kHz. These and other improvements have contributed to reducing the spectral density of equivalent gravitational wave strain noise to √s_h(f)≈10^(-19)/√ Hz at these frequencies.

Finally, observations made with the antenna in February and March of 1987 are described. Kilohertz-band gravitational waves produced by the remnant of the recent supernova are shown to be theoretically unlikely at the strength required for confident detection in this antenna (then operating at poorer sensitivity than that quoted above). A search for periodic waves in the recorded data, comprising Fourier analysis of four 105-second samples of the antenna strain signal, was used to place new upper limits on periodic gravitational radiation at frequencies between 305 Hz and 5 kHz. In particular, continuous waves of any polarization are ruled out above strain amplitudes of 1.2 x 10^(-18) R.M.S. for waves emanating from the direction of the supernova, and 6.2 x 10^(-19) R.M.S. for waves emanating from the galactic center, between 1.5 and 4 kilohertz. Between 305 Hz and 5kHz no strains greater than 1.2 x 10^(-17) R.M.S. were detected from either direction. Limitations of the analysis and potential improvements are discussed, as are prospects for future searches.

Phase conjugation via four-wave mixing in a resonant medium

This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process.

In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed.

In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile.

Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given.

Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10^-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.

Propagation of the fast magnetosonic wave in a tokamak plasma

The propagation of the fast magnetosonic wave in a tokamak plasma has been investigated at low power, between 10 and 300 watts, as a prelude to future heating experiments.

The attention of the experiments has been focused on the understanding of the coupling between a loop antenna and a plasma-filled cavity. Special emphasis has been given to the measurement of the complex loading impedance of the plasma. The importance of this measurement is that once the complex loading impedance of the plasma is known, a matching network can be designed so that the r.f. generator impedance can be matched to one of the cavity modes, thus delivering maximum power to the plasma. For future heating experiments it will be essential to be able to match the generator impedance to a cavity mode in order to couple the r.f. energy efficiently to the plasma.

As a consequence of the complex impedance measurements, it was discovered that the designs of the transmitting antenna and the impedance matching network are both crucial. The losses in the antenna and the matching network must be kept below the plasma loading in order to be able to detect the complex plasma loading impedance. This is even more important in future heating experiments, because the fundamental basis for efficient heating before any other consideration is to deliver more energy into the plasma than is dissipated in the antenna system.

The characteristics of the magnetosonic cavity modes are confirmed by three different methods. First, the cavity modes are observed as voltage maxima at the output of a six-turn receiving probe. Second, they also appear as maxima in the input resistance of the transmitting antenna. Finally, when the real and imaginary parts of the measured complex input impedance of the antenna are plotted in the complex impedance plane, the resulting curves are approximately circles, indicating a resonance phenomenon.

The observed plasma loading resistances at the various cavity modes are as high as 3 to 4 times the basic antenna resistance (~ .4 Ω). The estimated cavity Q’s were between 400 and 700. This means that efficient energy coupling into the tokamak and low losses in the antenna system are possible.

Damping and saturation of a second order electron plasma wave echo

A description is given of experimental work on the damping of a second order electron plasma wave echo due to velocity space diffusion in a low temperature magnetoplasma. Sufficient precision was obtained to verify the theoretically predicted cubic rather than quadratic or quartic dependence of the damping on exciter separation. Compared to the damping predicted for Coulomb collisions in a thermal plasma in an infinite magnetic field, the magnitude of the damping was approximately as predicted, while the velocity dependence of the damping was weaker than predicted. The discrepancy is consistent with the actual non-Maxwellian electron distribution of the plasma.

In conjunction with the damping work, echo amplitude saturation was measured as a function of the velocity of the electrons contributing to the echo. Good agreement was obtained with the predicted J₁ Bessel function amplitude dependence, as well as a demonstration that saturation did not influence the damping results.

Battery-Powered RF Pre-Ionization System for the Caltech Magnetohydrodynamically-Driven Jet Experiment: RF Discharge Properties and MHD-Driven Jet Dynamics

This thesis describes investigations of two classes of laboratory plasmas with rather different properties: partially ionized low pressure radiofrequency (RF) discharges, and fully ionized high density magnetohydrodynamically (MHD)-driven jets. An RF pre-ionization system was developed to enable neutral gas breakdown at lower pressures and create hotter, faster jets in the Caltech MHD-Driven Jet Experiment. The RF plasma source used a custom pulsed 3 kW 13.56 MHz RF power amplifier that was powered by AA batteries, allowing it to safely float at 4-6 kV with the cathode of the jet experiment. The argon RF discharge equilibrium and transport properties were analyzed, and novel jet dynamics were observed.

Although the RF plasma source was conceived as a wave-heated helicon source, scaling measurements and numerical modeling showed that inductive coupling was the dominant energy input mechanism. A one-dimensional time-dependent fluid model was developed to quantitatively explain the expansion of the pre-ionized plasma into the jet experiment chamber. The plasma transitioned from an ionizing phase with depressed neutral emission to a recombining phase with enhanced emission during the course of the experiment, causing fast camera images to be a poor indicator of the density distribution. Under certain conditions, the total visible and infrared brightness and the downstream ion density both increased after the RF power was turned off. The time-dependent emission patterns were used for an indirect measurement of the neutral gas pressure.

The low-mass jets formed with the aid of the pre-ionization system were extremely narrow and collimated near the electrodes, with peak density exceeding that of jets created without pre-ionization. The initial neutral gas distribution prior to plasma breakdown was found to be critical in determining the ultimate jet structure. The visible radius of the dense central jet column was several times narrower than the axial current channel radius, suggesting that the outer portion of the jet must have been force free, with the current parallel to the magnetic field. The studies of non-equilibrium flows and plasma self-organization being carried out at Caltech are relevant to astrophysical jets and fusion energy research.

Noise Cancellation for Gravitational Wave Detectors

The LIGO gravitational wave detectors are on the brink of making the first direct detections of gravi- tational waves. Noise cancellation techniques are described, in order to simplify the commissioning of these detectors as well as significantly improve their sensitivity to astrophysical sources. Future upgrades to the ground based detectors will require further cancellation of Newtonian gravitational noise in order to make the transition from detectors striving to make the first direct detection of gravitational waves, to observatories extracting physics from many, many detections. Techniques for this noise cancellation are described, as well as the work remaining in this realm.

Wave uplift pressures on horizontal platforms

The major objective of the study has been to investigate in detail the rapidly-varying peak uplift pressure and the slowly-varying positive and negative uplift pressures that are known to be exerted by waves against the underside of a horizontal pier or platform located above the still water level, but not higher than the crests of the incident waves.

In a "two-dimensional" laboratory study conducted in a 100-ft long by 15-in.-wide by 2-ft-deep wave tank with a horizontal smooth bottom, individually generated solitary waves struck a rigid, fixed, horizontal platform extending the width of the tank. Pressure transducers were mounted flush with the smooth soffit, or underside, of the platform. The location of the transducers could be varied.

The problem of a d equate dynamic and spatial response of the transducers was investigated in detail. It was found that unless the radius of the sensitive area of a pressure transducer is smaller than about one-third of the characteristic width of the pressure distribution, the peak pressure and the rise-time will not be recorded accurately. A procedure was devised to correct peak pressures and rise-times for this transducer defect.

The hydrodynamics of the flow beneath the platform are described qualitatively by a si1nple analysis, which relates peak pressure and positive slowly-varying pressure to the celerity of the wave front propagating beneath the platform, and relates negative slowly-varying pressure to the process by which fluid recedes from the platform after the wave has passed. As the wave front propagates beneath the platform, its celerity increases to a maximum, then decreases. The peak pressure similarly increases with distance from the seaward edge of the platform, then decreases.

Measured peak pressure head, always found to be less than five times the incident wave height above still water level, is an order of magnitude less than reported shock pressures due to waves breaking against vertical walls; the product of peak pressure and rise-time, considered as peak impulse, is of the order of 20% of reported shock impulse due to waves breaking against vertical walls. The maximum measured slowly-varying uplift pressure head is approximately equal to the incident wave height less the soffit clearance above still water level. The normalized magnitude and duration of negative pressure appears to depend principally on the ratio of soffit clearance to still water depth and on the ratio of platform length to still water depth.

Electromagnetic wave propagation and scattering in a randomly-inhomogeneous dielectric sphere

Electromagnetic wave propagation and scattering in a sphere composed of an inhomogeneous medium having random variations in its permittivity are studied by utilizing the Born approximation in solving the vector wave equation. The variations in the permittivity are taken to be isotropic and homogeneous, and are spatially characterized by a Gaussian correlation function. Temporal variations in the medium are not considered.

Two particular problems are considered: i) finding the far-zone electric field when an electric or magnetic dipole is situated at the center of the sphere, and ii) finding the electric field at the sphere's center when a linearly polarized plane wave is incident upon it. Expressions are obtained for the mean-square magnitudes of the scattered field components; it is found that the mean of the product of any two transverse components vanishes. The cases where the wavelength is much shorter than correlation distance of the medium and where it is much longer than it are both considered.

Equilibration of a liquid droplet-vapor-gas mixture downstream of a shock wave

The equations of motion for the flow of a mixture of liquid droplets, their vapor, and an inert gas through a normal shock wave are derived. A set of equations is obtained which is solved numerically for the equilibrium conditions far downstream of the shock. The equations describing the process of reaching equilibrium are also obtained. This is a set of first-order nonlinear differential equations and must also be solved numerically. The detailed equilibration process is obtained for several cases and the results are discussed.

An Optically recombined laser interferometer for gravitational wave detection

Kilometer scale interferometers for the detection of gravitational waves are currently under construction by the LIGO (Laser Interferometer Gravitational-wave Observatory) and VIRGO projects. These interferometers will consist of two Fabry-Perot cavities illuminated by a laser beam which is split in half by a beam splitter. A recycling mirror between the laser and the beam splitter will reflect the light returning from the beam splitter towards the laser back into the interferometer. The positions of the optical components in these interferometers must be controlled to a small fraction of a wavelength of the laser light. Schemes to extract signals necessary to control these optical components have been developed and demonstrated on the tabletop. In the large scale gravitational wave detectors the optical components must be suspended from vibration isolation platforms to achieve the necessary isolation from seismic motion. These suspended components present a new class of problems in controlling the interferometer, but also provide more exacting test of interferometer signal and noise models.

This thesis discusses the first operation of a suspended-mass Fabry-Perot-Michelson interferometer, in which signals carried by the optically recombined beams are used to detect and control all important mirror displacements. This interferometer uses an optical configuration and signal extraction scheme that is planned for the full scale LIGO interferometers with the simplification of the removal of the recycling mirror. A theoretical analysis of the performance that is expected from such an interferometer is presented and the experimental results are shown to be in generally good agreement.

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.

The resolution of the thermodynamic paradox and the theory of guided wave propagation in anisotropic media

The resolution of the so-called thermodynamic paradox is presented in this paper. It is shown, in direct contradiction to the results of several previously published papers, that the cutoff modes (evanescent modes having complex propagation constants) can carry power in a waveguide containing ferrite. The errors in all previous “proofs” which purport to show that the cutoff modes cannot carry power are uncovered. The boundary value problem underlying the paradox is studied in detail; it is shown that, although the solution is somewhat complicated, there is nothing paradoxical about it.

The general problem of electromagnetic wave propagation through rectangular guides filled inhomogeneously in cross-section with transversely magnetized ferrite is also studied. Application of the standard waveguide techniques reduces the TM part to the well-known self-adjoint Sturm Liouville eigenvalue equation. The TE part, however, leads in general to a non-self-adjoint eigenvalue equation. This equation and the associated expansion problem are studied in detail. Expansion coefficients and actual fields are determined for a particular problem.

Steady surface wave pattern in a shear flow

The subject under investigation concerns the steady surface wave patterns created by small concentrated disturbances acting on a non-uniform flow of a heavy fluid. The initial value problem of a point disturbance in a primary flow having an arbitrary velocity distribution (U(y), 0, 0) in a direction parallel to the undisturbed free surface is formulated. A geometric optics method and the classical integral transformation method are employed as two different methods of solution for this problem. Whenever necessary, the special case of linear shear (i.e. U(y) = 1+ϵy)) is chosen for the purpose of facilitating the final integration of the solution.

The asymptotic form of the solution obtained by the method of integral transforms agrees with the leading terms of the solution obtained by geometric optics when the latter is expanded in powers of small ϵ r.

The overall effect of the shear is to confine the wave field on the downstream side of the disturbance to a region which is smaller than the wave region in the case of uniform flows. If U(y) vanishes, and changes sign at a critical plane y = y_cr (e.g. ϵy_cr = -1 for the case of linear shear), then the boundary of this asymmetric wave field approaches this critical vertical plane. On this boundary the wave crests are all perpendicular to the x-axis, indicating that waves are reflected at this boundary.

Inside the wave field, as in the case of a point disturbance in a uniform primary flow, there exist two wave systems. The loci of constant phases (such as the crests or troughs) of these wave systems are not symmetric with respect to the x-axis. The geometric optics method and the integral transform method yield the same result of these loci for the special case of U(y) = U_o(1 + ϵy) and for large Kr (ϵr ˂˂ 1 ˂˂ Kr).

An expression for the variation of the amplitude of the waves in the wave field is obtained by the integral transform method. This is in the form of an expansion in small ϵr. The zeroth order is identical to the expression for the uniform stream case and is thus not applicable near the boundary of the wave region because it becomes infinite in that neighborhood. Throughout this investigation the viscous terms in the equations of motion are neglected, a reasonable assumption which can be justified when the wavelengths of the resulting waves are sufficiently large.