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Key issues and essential features of classical and quantum strings in gravitational plane waves, shock waves and space–time singularities are synthetically understood. This includes the string mass and mode number excitations, energy–momentum tensor, scattering amplitudes, vacuum polarization and wave-string polarization effect. The role of the real pole singularities characteristic of the tree level string spectrum (real mass resonances) and that of the space–time singularities is clearly exhibited. This throws light on the issue of singularities in string theory which can be thus classified and fully physically characterized in two different sets: strong singularities (poles of order ≥ 2, and black holes) where the string motion is collective and nonoscillating in time, outgoing states and scattering sector do not appear, the string does not cross the singularities; and weak singularities (poles of order < 2, (Dirac δ belongs to this class) and conic/orbifold singularities) where the whole string motion is oscillatory in time, outgoing and scattering states exist, and the string crosses the singularities.

Common features of strings in singular wave backgrounds and in inflationary backgrounds are explicitly exhibited.

The string dynamics and the scattering/excitation through the singularities (whatever their kind: strong or weak) is fully physically consistent and meaningful.

The framework of the relativistic quantum mechanics on spatially flat FLRW space–times is considered for deriving the analytical solutions of the Dirac equation in different local charts of these manifolds. Systems of commuting conserved operators are used for determining the fundamental solutions as common eigenspinors giving thus physical meaning to the integration constants related to the eigenvalues of these operators. Since these systems, in general, are incomplete on the FLRW space–times there are integration constants that must be fixed by setting the vacuum either as the traditional adiabatic one or as the rest frame vacuum we proposed recently. All the known solutions of the Dirac equation on these manifolds are discussed in all details and a new type of spherical waves of given energy in the de Sitter expanding universe is reported here for the first time.

We consider monochromatic, plane gravitational waves in a conformally invariant theory of general relativity. We show that the simple, standard ansatz for the metric, usually that which is taken for the linearized theory of these waves, is reducible to the metric of Minkowski spacetime via a sequence of conformal and coordinate transformations. This implies that we have in fact, exact plane wave solutions. However they are simply coordinate/conformal artifacts. As a consequence, they carry no energy.

This article deals with the use of the Ultra Weak Variational Formulation to solve Helmholtz equation and time harmonic Maxwell equations.

The method, issued from domain decomposition techniques, lies in partitioning the domain into subdomains with the use of adapted interface conditions. Going further than in domain decomposition, we make so that the problem degenerates into an interface problem only.

The new formulation is equivalent to the weak formulation. The discretization process is a Galerkin one. A possible advantage of the UWVF applied to wave equations is that we use the physical approach that consists in approximating the solution with plane waves. The formulation allows to use a very large mesh as compared to the frequency, on the contrary to the Finite Element Method when applied to time harmonic equations. Furthermore, the convergence analysis shows the method is a high order one: the order evolves as the square root of the number of degrees of freedom.

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this work, a new type of interpolation for the acoustic field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation has been found in the current work to be highly successful. Results are shown for a variety of classical scattering problems, and also for scattering from nonconvex obstacles. Notable results include a conclusion that, using this new formulation, only approximately 2.5 degrees of freedom per wavelength are required. Compared with the 8 to 10 degrees of freedom normally required for conventional boundary (and finite) elements, this shows the marked improvement in storage requirement. Moreover, the new formulation is shown to be extremely accurate. It is estimated that for 2D Helmholtz problems, and for a given computational resource, the frequency range allowed by this method is extended by a factor of three over conventional direct collocation Boundary Element Method. Recent successful developments of the current method for plane elastodynamics problems are also briefly outlined.

A multidomain pseudospectral time-domain (PSTD) method is developed for acoustic wave equations in lossy media. The method is based on the spectral derivative operator approximated by Chebyshev Lagrange polynomials. In this multidomain scheme, the computational domain is decomposed into a set of subdomains conformal to the problem geometry. Each curved subdomain is then mapped onto a cube in the curvilinear coordinates so that a tensor-product Chebyshev grid can be utilized without the staircasing error. An unsplit-field, well-posed PML is developed as the absorbing boundary condition. The algorithm is validated by analytical solutions. The numerical solutions show that this algorithm is efficient for simulating acoustic wave phenomena in the presence of complex objects in inhomogeneous media. To our knowledge, the multidomain PSTD method for acoustics is a new development in three dimensions, although in two dimensions the method can be made equivalent to the two-dimensional method in electromagnetics.

In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

A discussion is given of the sectional curvature function on a four-dimensional Lorentz manifold and, in particular, on the space–time of Einstein's general relativity theory. Its tight relationship to the metric tensor is demonstrated and some of its geometrical and algebraic properties evaluated. The concept of a sectional curvature preserving symmetry, in the form of a certain smooth vector field, is introduced and discussed.

In this paper we present some new results about $n(\ge 4)$-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for $n>4$ and a pp-wave space-time in $n=4$. In all cases an algebraic classification for the Weyl tensor is provided for $n=4$ and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat ${(PZS)}_n$, $n\ge 4$, space-time is conformal to the Robertson–Walker space-time.

Perforated liners are extensively used in aero-engines and gas turbine combustors to suppress combustion instabilities. These liners, typically subjected to a low Mach number bias flow (a cooling flow through perforated holes), are fitted along the bounding walls of a combustor to convert acoustic energy into flow energy by generating vorticity at the rims of the perforated apertures. To investigate the acoustic damping of such liners with bias flow on plane acoustic waves, a time-domain numerical model is developed to compute acoustic wave propagation in a cylindrical duct with a single-layer liner attached. The damping mechanism of the liner is characterized in real-time by using a 'compliance', developed especially for this work. It is a rational function representation of the frequency-domain homogeneous compliance adapted from the Rayleigh conductivity of a single aperture with mean bias flow in the z-domain. The liner 'compliance' model is then incorporated into partial differential equations of the duct system, which are solved by using the method of lines. The numerical results are then evaluated by comparing with the numerical results of Eldredge and Dowling's frequency-domain model. Good agreement is observed. This confirms that the model and the approach developed are suitable for real-time characterizing the acoustic damping of perforated liners.

In this paper the diffusion model of the linear theory of thermoelasticity of binary mixtures is considered and the following results are obtained: the basic properties of wave numbers of the longitudinal and transverse plane waves are established. The interior boundary value problem (BVP) of steady vibration of binary mixtures of thermoelastic solids is formulated. The existence theorems of eigenoscillation frequencies (eigenfrequencies) of the BVP of steady vibrations are proved. The connection between plane waves and existence of eigenfrequencies is established.