Outgoing Radiation Condition Research Articles

Complex Solutions and Stationary Scattering for the Nonlinear Helmholtz Equation

We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=\varphi+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving the Helmholtz resolvent operator. The key step of this approach is the proof of suitable a priori bounds.

Exceptional points for resonant states on parallel circular dielectric cylinders

Exceptional points (EPs) are special parameter values of a non-Hermitian eigenvalue problem where eigenfunctions corresponding to a multiple eigenvalue coalesce. In optics, EPs are associated with a number of counterintuitive wave phenomena and have potential applications in lasing, sensing, mode conversion, and spontaneous emission processes. For open photonic structures, resonant states are complex-frequency solutions of the Maxwell’s equations with outgoing radiation conditions. For open dielectric structures without material gain or loss, the eigenvalue problem for resonant states can have EPs, since it is non-Hermitian due to radiation losses. For applications in nanophotonics, it is important to understand EPs for resonant states on small finite dielectric structures consisting of conventional dielectric materials. To achieve this objective, we study EPs of resonant states on finite sets of parallel infinitely long circular dielectric cylinders with subwavelength radii. For systems with two, three, and four cylinders, we develop an efficient numerical method for computing EPs, present examples for second- and third-order EPs, and highlight their topological features. Our work provides insight to understanding EPs on more complicated photonic structures, and can be used as a simple platform to explore applications of EPs.

Reconstruction of a crack with the incident waves and measurements inside a penetrable cavity

Abstract The inverse scattering problem of determining a crack outside a known penetrable cavity from the measurements and point sources placed in a closed curve Λ inside the cavity is considered. The factorization method is used to reconstruct the crack on which the mixed-type boundary conditions are posed. To this end, we need to overcome two key difficulties. One of them is to find the relationship between the two data-to-data operators, which map the boundary data on the crack to the values of scattered fields on the closed curve Λ with outgoing radiation condition and incoming radiation condition, respectively. In addition, it is hard to decompose the near field operator in the usual way owing to the mixed-type boundary conditions. By the aid of an auxiliary operator, we can obtain the desired decomposition form in accordance with the theoretical framework of the factorization method.

Perfectly Matched Layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.

Global Seasonal Precipitation Anomalies Robustly Associated with El Niño and La Niña Events—An OLR Perspective*,+

Abstract El Niño–Southern Oscillation (ENSO) events are associated with particular seasonal weather anomalies in many regions around the planet. When the statistical links are sufficiently strong, ENSO state information can provide useful seasonal forecasts with varying lead times. However, using conventional sea surface temperature or sea level pressure indices to characterize ENSO state leads to many instances of limited forecast skill (e.g., years identified as El Niño or La Niña with weather anomalies unlike the average), even in regions where there is considerable ENSO-associated anomaly, on average. Using outgoing longwave radiation (OLR) conditions to characterize ENSO state identifies a subset of the conventional ENSO years, called OLR El Niño and OLR La Niña years herein. Treating the OLR-identified subset of years differently can both usefully strengthen the level of statistical significance in the average (composite) and also greatly reduce the year-to-year deviations in the composite precipitation anomalies. On average, over most of the planet, the non-OLR El Niño and non-OLR La Niña years have much more limited statistical utility for precipitation. The OLR El Niño and OLR La Niña indices typically identify years in time to be of use to boreal wintertime and later seasonal forecasting efforts, meaning that paying attention to tropical Pacific OLR conditions may offer more than just a diagnostic tool. Understanding better how large-scale environmental conditions during ENSO events determine OLR behavior (and deep atmospheric convection) will lead to improved seasonal precipitation forecasts for many areas.

An analysis of Feng’s and other symmetric local absorbing boundary conditions

With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng’s conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng’s conditions depends on the radius of the truncated domain.

Scattering of sound by a cylindrically symmetric seamount

A three-dimensional waveguide is used to model scattering of sound by a cylindrically symmetric seamount. The waveguide is “closed” in the sense that it has a boundary below the ocean bottom. This makes the spectrum of local wavenumbers discrete. Partial differential equations govern the modal expansion coefficients of the acoustic pressure and of the radial pressure gradient. Their solutions must be finite-valued at the axis of symmetry and satisfy outgoing radiation conditions at the perimeter of the base of the seamount. Direct numerical integration of this problem is unstable, but with variation of parameters and a Riccati transformation an equivalent problem is obtained for which numerical integration is stable. This approach is compared with an alternative [Pannatoni, POMA 14, 070003 (2011)] that uses leaky modes of an “open” waveguide having no boundary below the ocean bottom. It is shown how coupling between nearest-neighbor modes of the “closed” waveguide relates to the leaky modes of the “open” waveguide. This coupling restricts the step size that numerical integration can use.

Cartesian PML approximation to resonances in open systems in [formula omitted

In this paper, we consider a Cartesian PML approximation to resonance values of time-harmonic problems posed on unbounded domains in R2. A PML is a fictitious layer designed to find solutions arising from wave propagation and scattering problems supplemented with an outgoing radiation condition at infinity. Solutions obtained by a PML coincide with original solutions near wave sources or scatterers while they decay exponentially as they propagate into the layer. Due to rapid decay of solutions, it is natural to truncate unbounded domains to finite regions of computational interest. In this analysis, we introduce a PML in Cartesian geometry to transform a resonance problem (characterized as an eigenvalue problem with improper eigenfunctions) on an unbounded domain to a standard eigenvalue problem on a finite computational region. Truncating unbounded domains gives rise to perturbation of resonance values, however we show that eigenvalues obtained by the truncated problem converge to resonance values as the size of computational domain increases. In addition, our analysis shows that this technique is free of spurious resonance values provided truncated domains are sufficiently large. Finally, we present the results of numerical experiments with simple model problems.

Diffraction on the Two-Dimensional Square Lattice

We solve the thin-slit diffraction problem for two-dimensional lattice waves. More precisely, for the discrete Helmholtz equation on the semi-infinite square lattice with data prescribed on the left boundary (the aperture), we use lattice Green's functions and a discrete Sommerfeld outgoing radiation condition to derive the exact solution everywhere in the lattice. The solution is a discrete convolution that can be evaluated in closed form for the wave number $k=2$. For other wave numbers, we give a recursive algorithm for computing the convolution kernel.

Wave absorbing system using inclined perforated plates

The interaction of oblique monochromatic incident waves with horizontal/inclined/dual porous plates is investigated in the context of two-dimensional linear potential theory and Darcy's law (the normal velocity of fluid passing through a thin porous plate is linearly proportional to the pressure difference across it). The developed theory is verified by both small-scale and full-scale experiments. First, matched eigenfunction expansion (MEE) solutions for a horizontal porous plate are obtained. The relationship between the plate porosity and the porous parameter is obtained from systematic model tests by using six porous plates with different sizes and spacing of circular holes. Secondly, a multi-domain boundary-element method (BEM) using simple-sources (second-kind modified Bessel function) is developed to confirm the MEE solutions and to apply to more general cases including inclined or multiple porous plates. The BEM-based inner solutions are matched to the eigenfunction-based outer solutions to satisfy the outgoing radiation condition in the far field. Both analytical and BEM solutions with the developed empirical porous parameter agree with each other and correlate well with both small-scale data from a two-dimensional wave-tank test and full-scale measurement in a large wave basin. Using the developed predictive tools, wave-absorption efficiency is assessed for various combinations of porosity, water depth, submergence depth, wave heading, and plate/wave characteristics. In particular, it is found that the performance can be improved by imposing the proper inclination angle near the free surface. The optimal porosity is near porosity P=0.1 and the optimal inclination angle is around 10° as long as the plate length is greater than 20% of the wavelength. Based on the selected optimal parameters (porosity=0.1 and inclination angle=11.3°), the effective wave-absorption system for MOERI's square basin is designed.

The Helmholtz equation in a locally perturbed half-plane with passive boundary

In this article, we study the existence and uniqueness of outgoing solutions for the Helmholtz equation in locally perturbed half-planes with passive boundary. We establish an explicit outgoing radiation condition which is somewhat different from the usual Sommerfeld's one due to the appearance of surface waves. We work with the help of Fourier analysis and a half-plane Green's function framework. This is an extended and detailed version of the previous article Duran et al. (2005, The Helmholtz equation with impedance in a half-plane. C. R. Acad. Sci. Paris, Ser. 1, 3f0, 483-488).

Resonant excitations of the 't Hooft-Polyakov monopole.

The spherically symmetric magnetic monopole in an SU(2) gauge theory coupled to a massless Higgs field is shown to possess an infinite number of resonances or quasinormal modes. These modes are eigenfunctions of the isospin 1 perturbation equations with complex eigenvalues, E(n)=omega(n)-igamma(n), satisfying the outgoing radiation condition. For n--> infinity, their frequencies omega(n) approach the mass of the vector boson, M(W), while their lifetimes 1/gamma(n) tend to infinity. The response of the monopole to an arbitrary initial perturbation is largely determined by these resonant modes, whose collective effect leads to the formation of a long living breatherlike excitation with an amplitude decaying at late times as t(-5/6).

A UNIQUENESS THEOREM OF THE 3-DIMENSIONAL ACOUSTIC SCATTERING PROBLEM IN A SHALLOW OCEAN WITH A FLUID-LIKE SEABED

This paper shows that under the assumption of the out-going radiation conditions at infinity, the time-harmonic acoustic scattered field off a sound-soft solid in a shallow ocean with a fluid-like seabed is unique in [Formula: see text]. Here M1 is the water part, M2 the seabed, [Formula: see text] the waveguide and Ω the solid object. The associated modal problem is studied and a representation formula for the solution in terms of the Green's function is derived.

Fluid-loaded elastic plate with mean flow: point forcing and three-dimensional effects

The unsteady behaviour of an infinitely long fluid-loaded elastic plate subject to a single-frequency line forcing in the presence of a uniform mean flow is known to exhibit a number of interesting phenomena. These include the onset of absolute instability for non-dimensional flow speeds U in excess of some critical speed Uc, and various interesting propagation effects when U < Uc. In the latter respect Crighton & Oswell (1991) have shown that over a particular frequency range there exists an anomalous neutral mode with group velocity directed towards the driver, in violation of the usual Lighthill outgoing radiation condition. Similar results have been found by Peake (1997) when transverse curvature effects are included. In this paper we seek to extend these results and consider the substantially harder problem of a fluid-loaded elastic plate with uniform mean flow which is subject to a point forcing, thereby resulting in a two-dimensional structural problem. A systematic method for determining the absolute instability threshold is developed, and it is shown that the flow is absolutely unstable for flow speeds U > Uc, where Uc is the one-dimensional value found by Crighton & Oswell. At flow speeds U < Uc the flow is marginally stable and convective growth is found to occur downstream of the driver, over a particular frequency range depending on the transverse Fourier wavenumber ky, within a wedge-shaped region. Outside this wedge-shaped region there is only neutral mode behaviour. Asymptotic forms are found for the dominant large-distance causal flexion response downstream of the driver inside and outside the wedge region, and the appropriate critical angle for the wedge region is identified. Within the convective instability wedge the flexion and critical angle take two different forms depending on whether the frequency ω is greater or less than U2/√5. In addition to this interesting behaviour, the flow also exhibits the usual anomalous neutral mode behaviour and, as with Peake's problem, we also find an extra stability (hoop) resulting in neutral mode behaviour over a small frequency range. Asymptotic forms are also found for the threshold frequencies which divide up the various regions of stability of the system (neutral, neutral anomalous, convectively unstable), as a function of ky, and are compared with the results of both Crighton & Oswell and Peake.

A Comment on the Outgoing Radiation Condition for the Gravitational Field and the Peeling Theorem

The connection between the Bondi-Sachs (BS) andthe Newman-Penrose (NP) framework for the study of theasymptotics of the gravitational field is done. Inparticular the coordinate transformation relating the BS luminosity parameter and the NP affineparameter is obtained. Using this coordinatetransformation it is possible to express BS quantitiesin terms of NP quantities, and to show that if theOutgoing Radiation Condition is not satisfied then thespacetime will not decay in the way prescribed by thePeeling theorem.

The Characteristic Treatment of Black Holes

The characteristic initial value problem has been successfully implemented as a robust computational algorithm (the PITT NULL CODE) to evolve 4-dimensional vacuum spacetimes. It has been applied to the calculation of gravitational waveforms emitted by black holes and to the event horizon structure in the merger of black holes. The characteristic code also has potential application to the binary black hole problem via Cauchy-characteristic matching. Because the event horizon is itself a characteristic hypersurface, it can be analyzed by characteristic techniques as a stand-alone object. We have developed an analytic conformal model of null hypersurfaces which gives new insight into the intrinsic geometry of the pairof-pants horizon found in the numerical simulation of the head-on collision of black holes and into the initially toroidal horizon found in the simulation of a collapsing, rotating cluster. Most studies of black hole formation and merger have been restricted to axisymmetry. However, axisymmetric horizons, like the Schwarzschild horizon, are non-generic. When applied to a non-axisymmetric horizon, the characteristic approach reveals substantially new features. In particular, coalescing black holes generically go through a toroidal phase before they become spherical. The conformal structure of the event horizon supplies part of the data for a simulation of the exterior space-time. This provides a new way to calculate the post-merger waveforms from a binary black hole inspiral. §1. Null cone evolution A longterm project 1) to develop the pioneering work of Bondi 2) and Penrose 3) into a computational algorithm for the characteristic initial value problem has recently culminated in a highly accurate,efficient and robust code — the PITT NULL CODE. 4) Because the evolution proceeds on a space-time foliation by null cones which are generated by the characteristic rays of the theory,this approach offers several advantages for numerical work. I will describe here the fruits of these investigations relevant to black hole physics. In null cone coordinates,Einstein’s equations reduce to propagation equations along the radial light rays,which can be integrated in hierarchical order for one variable at a time. This leads to a highly efficient characteristic marching algorithm. There is one complex evolution variable which describes the free degrees of freedom of the gravitational field and four auxiliary variables. A compactified grid,based upon Penrose’s conformal description of null infinity,removes the necessity of an artificial outgoing radiation condition and makes possible a rigorous description of geometrical quantities such as the Bondi mass and news function. The news function supplies both the true waveform and polarization of the gravitational radiation incident on a distant antenna. Furthermore the grid domain is exactly the region in which waves propagate,which is ideally efficient for the purpose of radiation studies. Since each

The surface projection approach to vector field calculations

A finite element method for solving the vector Helmholtz equation can be derived by means of a variational approach in which the trial functions within each element are required to satisfy the Helmholtz equation, subject to the boundary condition that they match the surface projections of the vector potential on the element boundaries as determined by the nodal values of the surface projections and the surface basis functions. The contribution of an element to the global function is then a quadratic form in the nodal surface projections, with coefficients corresponding to the elements of the admittance matrix for the element. The admittance matrix can be determined by a generalized form of infinitesimal scaling which involves shell matrices of somewhat greater complexity than in scalar problems. It has nevertheless been possible to successfully apply the methodology both to closed regions and to open regions with the outgoing radiation condition at infinity.

On accuracy conditions for the numerical computation of waves

The Helmholtz equation ( Δ+ K 2 n 2) u= f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain, imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by ( Kh) −1, is not sufficient to determine the accuracy of a given discretization. For example, the quantity K 3 h 2 is shown to determine the accuracy in the L 2 norm for a second-order discretization method applied to several propagation models.

Null infinity from a quasi-Newtonian view

A scheme is elaborated which generates unique characteristic initial value data for a general relativistic fluid in terms of initial value data for a Newtonian fluid. The chief ingredient is the demand that the evolution of the general relativistic fluid yield the evolution of the Newtonian fluid as a parameter λ, corresponding to 1/c, goes to zero. The resulting gravitational null data is examined in detail to the first three nontrivial orders in λ. To this order, the data is asymptotically flat at future null infinity and appears to properly incorporate an outgoing radiation condition. The implications of these results for the calculation of gravitational radiation from quasi-Newtonian systems is discussed.

Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation

The reduced scalar Helmholtz equation for a transversely inhomogeneous half-space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial-value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first-order Weyl pseudodifferential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories which extend the narrow-angle, weak-inhomogeneity, and weak-gradient ordinary parabolic (Schrödinger) approximation. The analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies. The wave theories foreshadow computational algorithms, the inclusion of range-dependent effects, and the extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence.