Oblique Reflection Research Articles

Limit theorems and the support of SDES with oblique reflections on nonsmooth domains

In this paper we consider the Wong–Zakai type approximation and the support problem of stochastic differential equations with oblique reflections in nonsmooth domains satisfying conditions introduced by Dupuis and Ishii [4]. We first establish the limit theorem for reflected diffusions by proving that the sequence of solutions of approximating equations converge uniformly in Lp to the unique solution of the reflected stochastic differential equation. We also prove the approximate continuity and thus characterize the support of diffusions described by reflected SDEs.

Systems of quasi-variational inequalities related to the switching problem

We prove the existence of weak solution for a system of quasi-variational inequalities related to a switching problem with dynamic driven by operator associated with a semi-Dirichlet form and with measure data. We give a stochastic representation of solutions in terms of solutions of a system of reflected BSDEs with oblique reflection. As a by-product, we prove the existence of an optimal strategy in the switching problem and show regularity of the payoff function.

Stochastic and partial differential equations on non-smooth time-dependent domains

In this article, we consider non-smooth time-dependent domains whose boundary is W1,p in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.

Internal wave attractors in three-dimensional geometries: trapping by oblique reflection

We study experimentally the propagation of internal waves in two different three-dimensional (3D) geometries, with a special emphasis on the refractive focusing due to the 3D reflection of obliquely incident internal waves on a slope. Both studies are initiated by ray tracing calculations to determine the appropriate experimental parameters. First, we consider a 3D geometry, the classical set-up to get simple, two-dimensional (2D) parallelogram-shaped attractors in which waves are forced in a direction perpendicular to a sloping bottom. Here, however, the forcing is of reduced extent in the along-slope, transverse direction. We show how the refractive focusing mechanism explains the formation of attractors over the whole width of the tank, even away from the forcing region. Direct numerical simulations confirm the dynamics, emphasize the role of boundary conditions and reveal the phase shifting in the transverse direction. Second, we consider a long and narrow tank having an inclined bottom, to simply reproduce a canal. In this case, the energy is injected in a direction parallel to the slope. Interestingly, the wave energy ends up forming 2D internal wave attractors in planes that are transverse to the initial propagation direction. This focusing mechanism prevents indefinite transmission of most of the internal wave energy along the canal.

A parametric study of laminar and transitional oblique shock wave reflections

High-resolution particle image velocimetry measurements were performed on laminar and transitional oblique shock wave reflections for a range of Mach numbers ($M=1.6{-}2.3$), Reynolds numbers ($Re_{x_{sh}}=1.4\times 10^{6}{-}3.5\times 10^{6}$) and flow deflection angles ($\unicode[STIX]{x1D703}=1^{\circ }{-}5^{\circ }$ or $p_{3}/p_{1}=1.11{-}1.64$). The laminar interactions revealed a long, flat and triangular shaped separation bubble. For relatively strong interactions ($p_{3}/p_{1}>1.2$), the bubble grows linearly in the upstream direction with increasing shock strength. Under these conditions, the boundary layer keeps an on average laminar velocity profile up to the shock impingement location, followed by a quick transition and subsequent reattachment of the boundary layer. For weaker interactions ($p_{3}/p_{1}<1.2$), the boundary layer is able to remain laminar further downstream of the bubble, which consequently results in a later reattachment of the boundary layer. The pressure distribution at the interaction onset for all laminar cases shows excellent agreement with the free-interaction theory, therefore supporting its validity even for incipiently separated laminar oblique shock wave reflections.

Optimal switching problem and related system of BSDEs with left-Lipschitz coefficients and mixed reflections

This paper studies a system of backward stochastic differential equations with both oblique and normal reflections which arises in the study of the optimal switching problem. The main feature of our system is that its components are interconnected through both the obstacles and the generators which are only assumed to be left-Lipschitz continuous with respect to y. We provide an existence theorem of minimal solutions for RBSDEs in this framework.

Effects of Different Equations of State on the Oblique Shock Wave Reflection in Solids

The equations of state of solids under high pressure are more complicated than that of gases in a variety of forms. While the existing studies on the oblique shock wave reflection usually take just one of the equations of state, lacking of the comparisons among them. This paper aims at the investigation of oblique shock wave reflection in solids through shock polar methodology. Four different forms of equations of state are considered, principal shock taking with vN particle velocity relationship and second shock taking with Grüneisen equation of state(1Du2Grüneisen), principal and second shock both taking with vN particle velocity relationship (1Du2Du), principal shock taking with vN particle velocity relationship and second shock taking with stiffened gas equation of state (1Du2Gamma), and principal and second shock both taking with stiffened gas equation of state (1Gamma2Gamma). The effects of different equations of state on the pressure behind the reflected shock show: different combination of EOS may lead to diverse results. In our opinion, the 1Du2Grüneisen and 1Du2Du shows almost the same trends in most conditions and is suitable for the theoretical analysis of oblique reflection, and 1Du2Gamma and 1Gamma2Gamma should be carefully employed in studies to avoid the incorrect result.

Existence and uniqueness of reflecting diffusions in cusps

We consider stochastic differential equations with (oblique) reflection in a 2-dimensional domain that has a cusp at the origin, i.e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leq \delta _0,\psi _1(x_1)<x_2<\psi _ 2(x_1)\}$, with $\psi _1(0)=\psi _2(0)=0$, $\psi _1'(0)=\psi _2'(0)=0$. Given a vector field $g$ of directions of reflection at the boundary points other than the origin, defining directions of reflection at the origin $g^i(0):=\lim _{x_1\rightarrow 0^{+}}g (x_1,\psi _i(x_1))$, $ i=1,2$, and assuming there exists a vector $e^{*}$ such that $\langle e^{*},g^i(0)\rangle >0$, $i=1,2$, and $e^{*}_1>0$, we prove weak existence and uniqueness of the solution starting at the origin and strong existence and uniqueness starting away from the origin. Our proof uses a new scaling result and a coupling argument.

Formation, evolution and present-day activity of offshore sand ridges on a narrow, tideless continental shelf with limited sediment supply

An integrated approach combining swath bathymetry, high-resolution seismic reflection and seabed sediment sampling was performed to characterize a field of sand ridges on the Murcia continental shelf in the western Mediterranean Sea. The aim of this work was to improve knowledge about the formation, evolution and present-day dynamics of these bedforms on a narrow, tideless continental shelf receiving little sediment input. Sand ridges are observed on the middle shelf at water depths ranging between 58 and 78m. The sand ridges are 1.5 to 3m high and show a predominant E-W orientation oblique to the present-day shoreline. Smaller-scale subaqueous dunes (0.3–1.3m high) appear superimposed on the sand ridges and beyond the ridge field. The subaqueous dunes show a NW-SE orientation and asymmetric profiles, with the lee side facing southwest. The comparison of two bathymetric surveys 10years apart revealed that the subaqueous dunes are migrating towards the southwest at very low rates (~3myr−1). Internally, both sand ridges and dunes display southwestward dipping oblique reflections, indicating long-term migration in that direction.The morphology, architecture and distribution of the Murcia sand ridges suggest that they were formed in a shallow-water environment during the Holocene transgression and were later detached from the coast due to the subsequent sea level rise. The shallow architecture of the sand ridges reveals the presence of small, mound-like features within the sand ridges, most probably associated with coastal deposits, which could have served as a precursor for the development of the sand ridges. The preservation of these deposits within the sand ridges evidences limited offshore ridge migration. At present, active subaqueous dunes superimposed on the sand ridges suggest that they may play a significant role in the ridge evolution. At a regional scale, the comparison of the Murcia sand ridges with those described in the western Mediterranean allows us to propose a gradation from partially evolved to fully evolved sand ridges with increasing water depth.

Non-invasive spectroscopic techniques in the diagnosis of non-melanoma skin cancer

The number of non-melanoma skin cancers is increasing worldwide and has become an important health and economic issue. Early detection and treatment of skin cancer can significantly improve patient outcome. Therefore there is an increase in the demand for proper management and effective non-invasive diagnostic modalities in order to avoid relapses or unnecessary treatments. Although the gold standard of diagnosis for non-melanoma skin cancers is biopsy followed by histopathology evaluation, optical non-invasive diagnostic tools have obtained increased attention. Emerging non-invasive or minimal invasive techniques with possible application in the diagnosis of non-melanoma skin cancers include high-definition optical coherence tomography, fluorescence spectroscopy, oblique incidence diffuse reflectance spectrometry among others spectroscopic techniques. Our findings establish how those spectrometric techniques can be used to more rapidly and easily diagnose skin cancer in an accurate and automated manner in the clinic.

Terahertz beam switching by electrical control of graphene-enabled tunable metasurface

Controlling the terahertz wave, especially the dynamical and full control of terahertz wavefront, is highly demanded due to the increasing development of practical devices and application systems. Recently considerable efforts have been made to fill the ‘terahertz gap’ with the help of artificial metamaterial or metasurface incorporated with graphene material. Here, we propose a scheme to design tunable metasurface consisting of metallic patch array on a grounded polymer substrate embedded with graphene layers to electrically control the electromagnetic beam reflection at terahertz frequency. By adjusting geometric dimension of the patch elements, 360 degree reflection phase range may be achieved, thus abrupt phase shifts can be introduced along the metasurface for tailoring the reflected wavefront. Moreover, the reflective phase gradient over the metasurface can be switched between 90 and 360 degree by controlling the Fermi energy of the embedded graphene through voltage biasing, hence dynamically switching the reflective beam directions. Numerical simulations demonstrate that either single beam or dual beam dynamically switching between normal and oblique reflection angles can be well attained at working frequency. The proposed approach will bring much freedom in the design of beam manipulation devices and may be applied to terahertz radiation control.

Obliquely reflected Brownian motion in nonsmooth planar domains

We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including nonsmooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.

Numerical and experimental investigation of oblique shock wave reflection off a water wedge

Shock wave interaction with solid wedges has been an area of much research in past decades, but so far very few results have been obtained for shock wave reflection off liquid wedges. In this study, numerical simulations are performed using the inviscid Euler equations and the stiffened gas equation of state to study the transition angles, reflection patterns and triple point trajectory angles of shock reflection off solid and water wedges. Experiments using an inclined shock tube are also performed and schlieren photography results are compared to simulations. Results show that the transition angles for the water wedge cases are within 5.3 % and 9.2 %, for simulations and experiments respectively, compared to results obtained with the theoretical detachment criterion for solid surfaces. Triple point trajectory angles are measured and compared with analytic solutions, agreement within $1.3^{\circ }$ is shown for the water wedge cases. The transmitted wave in the water observed in the simulation is quantitatively studied, and two different scenarios are found. For low incident shock Mach numbers, $M_{s}=1.2$ and 2, no shock wave is formed in the water but a precursor wave is induced ahead of the incident shock wave and passes the information from the water back into the air. For high incident shock Mach numbers, $M_{s}=3$ and 4, precursor waves no longer appear but instead a shock wave is formed in the water and attached to the Mach stem at every instant. The temperature field in the water is measured in the simulation. For strong incident shock waves, e.g. $M_{s}=4$, the temperature increment in the water is up to 7.3 K.

Optical device for thermal diffusivity determination in liquids by reflection of a thermal wave.

In this work, we present a device for determination of the thermal diffusivity using the oblique reflection of a thermal wave within a solid slab that is in contact with the medium to be characterized. By using the reflection near a critical angle under the assumption that thermal waves obey Snell's law of refraction with the square root of the thermal diffusivities, the unknown thermal diffusivity is obtained by simple formulae. Experimentally, the sensor response is measured using the photothermal beam deflection technique within a slab that results in a compact device with no contact of the laser probing beam with the sample. We describe the theoretical basis and provide experimental results to validate the proposed method. We determine the thermal diffusivity of tridistilled water and glycerin solutions with an error of less than 0.5%.

Electrical tunability due to coalescence of exceptional points in parity-time symmetric waveguides.

We demonstrate theoretically the electric tunability due to the coalescence of exceptional points in PT-symmetric waveguides bounded by imperfect conductive layers. Owing to the competition effect of multimode interaction, multiple exceptional points and PT phase transitions could be attained in such a simple system, and their occurrences are strongly dependent on the boundary conductive layers. When the conductive layers become very thin, it is found that the oblique transmittance and reflectance of the same system can be tuned between zero and one by a small change in the carrier density. The results may provide an effective method for fast tuning and modulation of optical signals through electrical gating.

Experimental Study of Bilinear Initiating System Based on Hard Rock Pile Blasting

It is difficult to use industrial explosives to excavate hard rock and achieve suitable blasting effect due to the low energy utilization rate resulting in large rocks and short blasting footage. Thus, improving the utilization ratio of the explosive energy is important. In this study, a novel bilinear initiation system based on hard rock blasting was proposed to improve the blasting effects. Furthermore, on the basis of the detonation wave collision theory, frontal collision, oblique reflection, and Mach reflection during detonation wave propagation were studied. The results show that the maximum detonation pressure at the Mach reflection point where the incident angle is 46.9° is three times larger than the value of the explosive complete detonation. Then, in order to analyze the crack propagation in different initiation forms, a rock fracture test slot was designed, and the results show that bilinear initiating system can change the energy distribution of explosives. Finally, field experiment was implemented at the hard rock pile blasting engineering, and experimental results show that the present system possesses high explosive energy utilization ratio and low rock fragments size. The results of this study can be used to improve the efficiency in hard rock blasting.

Plasma-based polarization modulator for high-intensity lasers

Manipulation of laser pulses at high intensities is an important yet challenging issue. New types of plasma-based optical devices are promising alternatives to achieve this goal. Here we propose to modulate the polarization state of intense lasers based on oblique reflection from solid-plasma surfaces. A new analytical description is presented considering the plasma as an uniaxial medium that causes birefringence effect. Particle-in-cell simulation results numerically demonstrate that such a scheme can provide a tunable polarization control of the laser pulses even in the relativistic regime. The results are thus relevant for the design of compact, easy to use, and versatile polarization modulators for high-intensity laser pulses.

Optical Microscopy for Detecting Binding on Small Molecule Microarrays.

Small molecule microarrays (SMMs) have revolutionized parallel screening of multiple biological interactions. SMMs are a high-throughput technique for simultaneous detection of up to thousands of biological interactions. Conventional detection technique employed in SMMs is fluorescence spectroscopy. However, the possibility of fluorescent labels affecting the binding properties of small molecules drives the development of label-free screening techniques. Here, we describe two label-free detection techniques using optical microscopy for the detection of binding on SMMs. The first optical detection technique-ellipsometry, and more specifically, oblique incidence reflectivity difference to study the kinetics of the association between small molecule targets and protein probes is discussed. Then, we describe another optical detection technique-surface plasmon resonance microscopy. The sample preparation steps and protocols of the two techniques to study the interactions between SMMs and their protein targets are elaborated in great detail.

Pure shear backward waves in the X-cut and Y-cut piezoelectric plates of potassium niobate

The problem of the existence of pure shear backward waves and waves with zero group velocity is investigated for X-cut and Y-cut plates of orthorhombic crystal of potassium niobate, which distinguishes from the other crystals by exceptionally strong piezoelectric effect. The secular equations of the problem are derived in an explicit analytical form and they are studied numerically in the case of crystal-vacuum interface. Two factors contributing to the appearance of backward waves are identified: the negative displacement of the rays of bulk shear waves at oblique reflection from the surface in piezoelectric crystals and the concavity in cross section of the slowness surface for bulk shear waves near the X axis of the potassium niobate plate. The wide frequency ranges of the existence of symmetric and antisymmetric backward waves of the first and second orders are numerically found in the X-cut plate, for which both of these factors affect. To study the dispersion spreading of backward shear wave pulses, it is suggested to apply the parabolic equation corresponding to the second approximation in the dispersion theory. It is demonstrated that the “diffusion” coefficient in this equation vanishes at certain frequencies, leading to significant suppression of the dispersion distortions in the pulses of the waves under study.

An existence theorem for multidimensional BSDEs with mixed reflections

In this note, we consider the pricing problem for a type of real option, which gives the right to switch investment modes and abandon the investment project before its maturity. The value of this option can be characterized by solutions to multidimensional backward stochastic differential equations (BSDEs) with both normal and oblique reflections, whose coefficients are of linear growth and are left-Lipschitz with respect to (w.r.t) y and Lipschitz w.r.t. z. We provide an existence theorem of minimal solutions for BSDEs in this framework.