Modifcations to gravitational waves due to matter shells

As detections of gravitational waves (GWs) mount, the need to investigate various effects on the propagation of these waves from the time of emission until detection also grows. We investigate how a thin low density dust shell surrounding a gravitational wave source affects the propagation of GWs. The Bondi-Sachs (BS) formalism for the Einstein equations is used for the problem of a gravitational wave (GW) source surrounded by a spherical dust shell. Using linearised perturbation theory, we and the geometry of the regions exterior to, interior to and within the shell. We and that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust. This finding is novel. In the context of cosmology, apart from the gravitational redshift, the effects are too small to be measurable; but the effect would be measurable if a GW event were to occur with a source surrounded by a massive shell and with the radius of the shell and the wavelength of the GWs of the same order. We extended our investigation to astrophysical scenarios such as binary black hole (BBH) mergers, binary neutron star (BNS) mergers, and core collapse supernovae (CCSNe). In these scenarios, instead of a monochromatic GW source, as we used in our initial investigation, we consider burst-like GW sources. The thin density shell approach is modified to include thick shells by considering concentric thin shells and integrating. Solutions are then found for these burst-like GW sources using Fourier transforms. We show that GW echoes that are claimed to be present in the Laser Interferometer Gravitational-Wave Observatory (LIGO) data of certain events, could not have been caused by a matter shell. We do and, however, that matter shells surrounding BBH mergers, BNS mergers, and CCSNe could make modifications of order a few percent to a GW signal. These modifications are expected to be measurable in GW data with current detectors if the event is close enough and at a detectable frequency; or in future detectors with increased frequency range and amplitude sensitivity. Substantial use is made of computer algebra in these investigations. In setting the scene for our investigations, we trace the evolution of general relativity (GR) from Einstein's postulation in 1915 to vindication of his theory with the confirmation of the existence of GWs a century later. We discuss the implications of our results to current and future considerations. Calculations of GWs, both analytical and numerical, have normally assumed their propagation from source to a detector on Earth in a vacuum spacetime, and so discounted the effect of intervening matter. As we enter an era of precision GW measurements, it becomes important to quantify any effects due to propagation of GWs through a non-vacuum spacetime Observational confirmation of the modification effect that we and in astrophysical scenarios involving black holes (BHs), neutron stars (NSs) and CCSNe, would also enhance our understanding of the details of the physics of these bodies.

Existence of Steady States of the Massless Einstein–Vlasov System Surrounding a Schwarzschild Black Hole

We show that there exist steady states of the spherically symmetric massless Einstein–Vlasov system which surround a Schwarzschild black hole. The steady states are (thick) shells with finite mass and compact support. Furthermore we prove that an arbitrary number of shells, necessarily well separated, can surround the black hole. To our knowledge this is the first result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numerical investigation about the properties of the shells.

Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin–Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an expression of the shear correction factor variable (αn) in terms of the thickness (h) and the radius of curvature (R). The main advantage of the proposed model is that it has the possibility to analyze thin, medium and thick tunnel ventilation shafts. As a result, two comparisons were made: the first one, between the new model and the Mindlin–Reissner model without constitutive curvature with the shear correction factor αn=5/6 as a constant, and the other, between the new model and the tridimensional numerical models (solids and shells) obtained by finite element method for different slenderness ratios (h/R). The limitation of the proposed model is that it is to be formulated for a general linear-elastic and axial-symmetrical state with continuous distribution of the mass.

Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin-Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an expression of the shear correction factor variable (αn) in terms of the thickness (h) and the radius of curvature (R). The main advantage of the proposed model is that it has the possibility to analyze thin, medium and thick tunnel ventilation shafts. As a result, two comparisons were made: the first one, between the new model and the Mindlin-Reissner model without constitutive curvature with the shear correction factor (α_n=6/5) as a constant, and the other, between the new model and the tridimensional numerical models (solids and shells) obtained by finite element method for different slenderness ratios (h/R). The limitation of the proposed model is that it is to be formulated for a general linear-elastic and axial-symmetrical state with continuous distribution of the mass.

COMPUTER ENGINEERING ANALYSIS OF DOUBLE-SIDED MULTI-ARC WELDING OF VERTICAL JOINTS OF TANKS FOR STORING OIL AND OIL PRODUCTS

The possibilities of computer engineering analysis of special features of double-sided multi-arc welding of vertical joints of tanks with thick shells for storing oil and oil products are shown. The analysis was carried out on the basis of a numerical implementation of the unconventional physical and mathematical model of the formation of a melt pool and a weld seam with two pairs of arcs on each side of the joint with a double-sided bevel. It was established that during weld formation with the movement of the electrode down (vertical down), the possibility of supplying the first pair of arcs with pulsed current should be taken into account, and for the second pairs, the possibility of lateral oscillation of the arcs in the groove with delays at the edges. During groove filling with the movement of the electrodes up (vertical up), their lateral oscillations and delays at the edges should be carried out according to an aperiodic law in order to minimize the effects of “magnetic blow”. For root formation, it is recommended to use pulsed welding, and during groove filling - high current welding with low welding speed. The results of the study can be used in the formation of scientifically-based requirements for the welding process and equipment for double-sided multi-arc welding of the vertical joints of tanks for storing oil and oil products.

Study of Vibration Behaviour of Stiffened Polymer Composite Shells for Underwater Structural Applications

This paper presents vibration behavior of ring stiffened polymer composite thick shells used for underwater structures. Filament wound shells stiffened with internal and external rings and with hemispherical ends were tested for vibration in air and water in free-free boundary condition using roving hammer and fixed response method. Modal testing of the shells was performed under hydrostatic loading in a custom designed buckling tester for determining natural frequency at higher sea depths. Accelerometer was mounted on the inner surface of the shell. It was excited using a plumbob, rope and pulley arrangement. Experimental results were validated by modal analysis using Hyperworks and ANSYS. Vibration behavior in water was simulated by Fluid structure interaction approach. Experimental first natural frequency in water was lesser than that in air. With increase in hydrostatic pressure, the shell showed moderate variation in natural frequency. The experimental and numerical results of natural frequency and mode shapes were in good agreement with each other. Natural frequencies were lower in long and thick shells.