Complex Wave Patterns Research Articles

Experiments and molecular-dynamics simulation of elastic waves in a plasma crystal radiated from a small dipole source.

The radiation of elastic waves from a localized source is observed experimentally in a two-dimensional plasma crystal. An initial shear stress applied by a laser forms a small dipole source. The emerging complex wave pattern is shown to consist of outgoing compressional and shear wave pulses. Subsequent structures are identified as inward-going waves due to the finite size of the source region, which reappear on the opposite side. The compressional wave forms a trailing wave train due to strong dispersion, while the nondispersive shear wave evolves into a vortex-antivortex pair on either side. The experiments are compared with a molecular-dynamics simulation.

Analysis of wave patterns in MR elastography of skeletal muscle using coupled harmonic oscillator simulations

The ability to study muscle elasticity in vivo would be of great clinical interest. Magnetic resonance elastography (MRE) has the potential to quantify noninvasively the distribution of the shear modulus in muscle tissue. Elasticity information may be derived by extracting frequencies from the wave patterns of phase-contrast MRE images. In a new approach, MRE wave patterns were reconstructed using 3D coupled harmonic oscillator calculations (CHO). To analyze in vivo MRE measurements of the biceps brachii of healthy volunteers, different anisotropic fibrous structures for the couplings between the muscle elements have to be assumed. V-shaped wave patterns as observed when excitation was applied on the tendon were reproduced by a model, where in a central band of stiff fascicles wave propagation was about twice as fast as that in surrounding tissue. Planar waves were observed for excitation near the muscle surface. They could be reconstructed by assuming a simultaneous wave excitation of all muscle fibers, where fibers along the main muscle axis were coupled more strongly than those perpendicular to the axis. The results show that CHO calculations provide a fast and reliable method for incorporating anatomical information of the investigated tissue in the reconstruction of complex wave patterns.

Experimental and theoretical study of shock wave propagation through double-bend ducts

The complex flow and wave pattern following an initially planar shock wave transmitted through a double-bend duct is studied experimentally and theoretically/numerically. Several different double-bend duct geometries are investigated in order to assess their effects on the accompanying flow and shock wave attenuation while passing through these ducts. The effect of the duct wall roughness on the shock wave attenuation is also studied. The main flow diagnostic used in the experimental part is either an interferometric study or alternating shadow–schlieren diagnostics. The photos obtained provide a detailed description of the flow evolution inside the ducts investigated. Pressure measurements were also taken in some of the experiments. In the theoretical/numerical part the conservation equations for an inviscid, perfect gas were solved numerically. It is shown that the proposed physical model (Euler equations), which is solved by using the second-order-accurate, high-resolution GRP (generalized Riemann problem) scheme, can simulate such a complex, time-dependent process very accurately. Specifically, all wave patterns are numerically simulated throughout the entire interaction process. Excellent agreement is found between the numerical simulation and the experimental results. The efficiency of a double-bend duct in providing a shock wave attenuation is clearly demonstrated.

3-D wave propagation in fluid-filled irregular boreholes in elastic formations

Different seismic testing techniques rely on the propagation of acoustic waves in fluid-filled boreholes from sources placed within the borehole and in the solid media. The interpretation of the signals recorded relies on understanding how waves propagate in the borehole and its immediate vicinity. It is known that very complex wave patterns can arise, depending on the distance between the source and the receiver, and their placement and orientation relative to the axis of a circular borehole. The problem becomes more complex if the cross-section is not circular, conditions for which analytical solutions are not known. In this work, the Boundary Element Method (BEM) is used to evaluate the three-dimensional wave field elicited by monopole sources in the vicinity of a fluid-filled borehole. This model is used to assess the effects of the receiver position on the propagation of both axisymmetric and non-axisymmetric wave modes when different borehole cross-sections are used. Both frequency vs. axial-wave number responses and time-domain responses are calculated.

2D finite‐difference viscoelastic wave modelling including surface topography

We have pursued two‐dimensional (2D) finite‐difference (FD) modelling of seismic scattering from free‐surface topography. Exact free‐surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free‐surface topography. The wave equations are numerically discretized by an eighth‐order FD method on a staggered grid in space, and a leap‐frog technique and the Crank–Nicholson method in time.In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back‐scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter‐crust heterogeneities.

3-D versus 2-D finite-difference seismic synthetics including real surface topography

We have pursued and compared two-dimensional (2-D) and three-dimensional (3-D) finite-difference (F-D) modeling of scattering from free surface topography. A velocity–stress formulation of the full elastic wave equations are combined with exact boundary conditions for the surface topography and numerically discretized by an eighth-order F-D method on a staggered grid. We have simulated scattering in 2-D and 3-D from teleseismic P-waves using a plane, vertically incident P-wave and real topography from a 60×60 km 2 area including the NORESS array in southeastern Norway. Many field observations that are not easily explained by simpler 2-D cases are shown to better match qualitative effects from 3-D surface topography modeling. These include strong amplifications at hills, complex wave pattern caused by scattering, and directivity of scattered waves. Snapshots and seismograms show clear conversion from P- to Rg- (short period fundamental mode Rayleigh) waves in an area of rough topography in the vicinity of the array site. All results are consistent with numerous observations. By parallellization of the original software, possibilities have been opened for modeling with higher resolution and/or larger areas than before.

Shock phenomena in high speed aerodynamics: still a source of major concern

AbstractShockwaves are present in a flow as soon as the Mach number becomes supersonic. Being viscous phenomena, Shockwaves are a source of drag which can be predominant when the Mach number is significantly higher than one. In supersonic air intakes, the production of entropy by shocks is felt as a loss in efficiency. At high Mach numbers, Shockwaves produce a considerable temperature rise leading to severe heating problems, complicated by real gas effects. The intersection - or interference - of two shocks gives rise to complex wave patterns containing slip-lines and associated shear layers whose impingement on a nearby surface can cause detrimental pressure and heat transfer loads. The impact of a Shockwave on a boundary layer is the origin of strong viscous interactions which remain a limiting factor in the design of transonic wings, supersonic air intakes, propulsive nozzles and compressor cascades. More effort is needed to improve prediction of these interactions and to devise new techniques to control such phenomena.

Shock wave propagation in a branched duct

The propagation of a planar shock wave in a 90° branched duct is studied experimentally and numerically. It is shown that the interaction of the transmitted shock wave with the branching segment results in a complex, two-dimensional unsteady flow. Multiple shock wave reflections from the duct's walls cause weakening of transmitted waves and, at late times, an approach to an equilibrium, one-dimensional flow. While at most places along the branched duct walls calculated pressures are lower than that existing behind the original incident shock wave, at the branching segment's right corner, where a head on-collision between the transmitted wave and the corner is experienced, pressures that are significantly higher than those existing behind the original incident shock wave are encountered. The numerically evaluated pressures can be accepted with confidence, due to the very good agreement found between experimental and numerical results with respect to the geometry of the complex wave pattern observed inside the branched duct.

Charge-density oscillations onBe(101¯0): Screening in a non-free two-dimensional electron gas

The surface state on Be(10{bar 1}0) has been investigated using a low-temperature scanning tunneling microscope (STM). The Fermi contour of this surface state is located at one boundary of the surface Brillouin zone, and surface-state electrons provide the main part of the charge density near the Fermi energy. Be(10{bar 1}0), therefore, corresponds closely to a non-free two-dimensional electron gas. We have observed standing waves of the surface charge density on Be(10{bar 1}0) near step edges and point defects. Such wave patterns derive from the interference of incoming and scattered electrons; they demonstrate the screening characteristics of the surface state. On Be(10{bar 1}0) these waves were found to be highly anisotropic. It is shown that calculating the Fourier transforms of topographic STM images is a powerful method for determining the Fermi contour of the surface state. This method could even be applied to images that display a complex wave pattern arising from a random distribution of point scatterers. Fourier analysis also revealed that the charge density oscillations on Be(10{bar 1}0) contain multiple periods that differ by reciprocal lattice vectors. These multiperiodic oscillations relate to the non-free character of the surface-state electrons and constitute an interference pattern of Bloch states. Fourier filteringmore » was used to separate the charge-density oscillations from the topographic corrugation and to visualize their shape and spatial range. The experimental data are qualitatively discussed using a model calculation based on the scattering of Bloch electrons from planar obstacles in a two-dimensional conductor. Experimental results and model calculations highlight how the screening characteristics on Be(10{bar 1}0) significantly deviate from the behavior expected for a free two-dimensional electron gas. {copyright} {ital 1998} {ital The American Physical Society}« less

Event-Related Potentials Elicited by Passive Movements in Humans: Characterization, Source Analysis, and Comparison to fMRI

Cortical areas responsive to proprioceptive stimulation were assessed by ERP technique in normals and in selected patients with stroke and were compared to fMRI data. Repetitive extension of right and left forefinger elicited a P1/N1/P2 complex wave pattern. This pattern was absent in patient with complete sensory loss and present but spatially modified in patient with recovered sensory deficit. Source localization with a simple model showed three sources starting in the contralateral rolandic area (SI), then involving the inferior parietal lobe unilaterally and the supplementary motor area (10 to 134 ms). It was followed by a bilaterally distributed pattern of two sources located in the ipsilateral parietal region and in the contralateral insula. Right and left stimulation led to very symmetrical patterns. Comparison to fMRI obtained from passive extension of the wrist in normals showed very compatible data. We described in this paper, a sequential processing of proprioceptive inputs after passive movements involving primary and secondary sensory motor areas.

Amplitude Control of Chemical Waves in Catalytic Membranes. Asymmetric Wave Propagation between Zones Loaded with Different Catalyst Concentrations

Chemical waves of the Belousov−Zhabotinsky type are studied applying bathoferroin catalyst fixed on a polysulfone membrane. A new method is developed to create contacting high- (H) and low-amplitude (L) regions for chemical waves. The amplitude is high in zones (H) loaded with high catalyst concentrations, and it is low in zones (L) loaded with low catalyst concentrations. An asymmetric wave propagation is found: waves coming from region H can initiate waves in region L across the HL boundary with a higher frequency than vice versa. The ratio of the cross-recovery times R(L → H) and R(H → L) is 1.7 in the experiments reported here. To measure this ratio, rotating chemical waves were applied. The waves propagate in two concentric annular zonesthe inner zone with low and the outer with high catalyst concentrationand the HL boundary forms a circle. It was found that in such a reactor complex wave patterns (so-called chemical pinwheels) can rotate nearly independently in the H and L zones, interacting only w...

Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock–spherical fast/slow bubble interactions

Collapsing shock-bounded cavities in fast/slow (F/S) spherical and near-spherical configurations give rise to expelled jets and vortex rings. In this paper, we simulate with the Euler equations planar shocks interacting with an R12 axisymmetric spherical bubble. We visualize and quantify results that show evolving upstream and downstream complex wave patterns and emphasize the appearance of vortex rings. We examine how the magnitude of these structures scales with Mach number. The collapsing shock cavity within the bubble causes secondary shock refractions on the interface and an expelled weak jet at low Mach number. At higher Mach numbers (e.g. M=2.5) ‘vortical projectiles’ (VP) appear on the downstream side of the bubble. The primary VP arises from the delayed conical vortex layer generated at the Mach disk which forms as a result of the interaction of the curved incoming shock waves that collide on the downstream side of the bubble. These rings grow in a self-similar manner and their circulation is a function of the incoming shock Mach number. At M=5.0, it is of the same order of magnitude as the primary negative circulation deposited on the bubble interface. Also at M=2.5 and 5.0 a double vortex layer arises near the apex of the bubble and moves off the interface. It evolves into a VP, an asymmetric diffuse double ring, and moves radially beyond the apex of the bubble. Our simulations of the Euler equations were done with a second-order-accurate Harten–Yee-type upwind TVD scheme with an approximate Riemann Solver on mesh resolution of 803×123 with a bubble of radius 55 zones.

Compressor cascade flow with strong shock-wave/boundary-layer interaction

A two-dimensional multiblock Navier-Stokes solver has been successfully applied to a supersonic compressor cascade flow with strong shock-wave/boundary-layer interaction. The cascade model investigated is typical of a fan blade section with precompression design, low flow turning, and a static pressure ratio around 2.0. Numerical calculations have been performed on a very fine grid to guarantee sufficient resolution of the supersonic/transonic flowfield with a complex wave pattern and shock-induced boundary-layer separation. High accuracy and robustness of the numerical scheme is achieved by the second-order Roe upwind-total variation diminishing scheme and a one-equation turbulence model. Investigations were performed for inlet Mach numbers from 1.28 to 1.53 and a Reynolds number of 2.6 x 106. The results are compared with those of previous experiments in a supersonic cascade wind tunnel that provide profile Mach number distributions, wake traverse data, and measured suction surface boundary-layer profiles throughout the strong interaction region. Furthermore, calculated discontinuities of Mach number and flow direction across the shock system at blade passage entrance are compared with laser anemometer data

Array tracking of the volcanic tremor source at Deception Island, Antarctica

We have found experimental evidence which shows that the volcanic tremor recorded at Deception Island (South Shetland Islands, Antarctica) is a superposition in time of overlapping hybrid events. We studied data from a small aperture seismic array. Data analysis for tremor and hybrids included: (1) spectral analysis; (2) apparent slowness and back‐azimuth determination by using the zero‐lag cross‐correlation method; and (3) polarization analysis. Both types of events share these common features: (a) two dominant spectral bands at frequencies 1–3 Hz (the most energetic) and 4–8 Hz; (b) several coherent phases with the same back‐azimuth to the source and apparent slowness along the whole signal; (c) in the high frequency band, the apparent slowness is very low (around 0.17 s/km), indicating the propagation of body waves; (d) in the low frequency band, the apparent slowness is high (around 1.6 s/km), consistent with the presence of surface waves; and (e) clear P‐wave onset followed by a complex pattern of Rayleigh waves. Therefore, both types of events are strongly related because they share the same source region, the same wave‐propagation properties, and the same wave composition. Moreover, several arrivals, that resemble a single hybrid event, have been found along the tremor signals. Due to these reasons, we hypothesize that volcanic tremor of Deception Island is a superposition of hybrid type events. The source of both types could be the interaction between thaw water and hot materials in a shallow aquifer.

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.

Computation of flow in open-channel transitions

The supercritical flow in open-channel transitions often produces discontinuities in the flow variables. Depending on the size, shape and kind of channels, the flow can generate normal/oblique shock waves, expansion waves, hydraulic jumps and sometimes complex wave patterns due to multiple reflection of the waves at the boundary and their subsequent interactions with one another. In the numerical simulation of these flows, uniform grid distribution may introduce detrimental effects in the prediction and resolution of the How details. The use of the presently available numerical schemes to solve these problems on a uniformly spaced grid systems fail to resolve the characteristic flow features and hence do a poor job in simulating the flows. In this paper, MacCormack second-order accurate explicit predictor-corrector scheme is used to solve the two-dimensional depth averaged shallow water equations to numerically simulate the supercritical free-surface flows in open-channel transitions. However, instead of a fixed grid, an adaptive grid system which adjusts itself as the solution evolves is used for a better resolution of the flow properties. In the present approach, Rai and Anderson's method based on grid speed is used for grid adaptation. In a flow with shock, best solutions are obtained when the coordinate axes are aligned along the shock. Rai and Anderson's method to cluster the grids are found to fulfill this criteria and thus produces better quality solutions, as compared to those obtained with uniformly distributed grid with a specified number of grid points.

Audition Laboratory Activities for Teaching Sensation and Perception

This article describes exercises for a sensation and perception laboratory using a computer application called SoundWave. The exercises were conducted in two 1-hr sessions. Goals of the lab included giving students hands-on experience with how sound wave shape changes as frequency and amplitude vary, the continuousness of the speech stream, and determining whether a complex wave pattern represents noise or music. These exercises could also be adapted for use in introductory psychology classes.

Measurements of the triple shock wave/turbulent boundary-layer interaction

A joint experimental and computational study has investigated the flowfield structure created by three crossing oblique shock waves interacting with a turbulent boundary layer. Such an interaction is of practical importance in the design of high-speed sidewall-compression inlets. The interaction is created by a test model consisting of two vertical sharp fins mounted at 15-deg angle of attack to a horizontal flat plate. A third compression surface of 10-deg angle is mounted to the plate between the two vertical fins. The Mach 3.85 flowfield is examined experimentally through kerosene lampblack and planar laser scattering visualizations. The results are used to develop a flowfield model of the interaction structure. This structure is found to consist of a complex wave pattern overlying a viscous separated region. The presence of the 10-deg compression ramp reduces the severity of the boundary-layer separation compared to interactions created by the two fins alone. Additionally, the experimental flowfield data are compared with a computational solution using a modified κ-e (Rodi) turbulence model.

On the origin of wave patterns in fluidized beds

In this report, a model designed for the description of the flow of two miscible phases in a fluidized bed is discussed. Apart from basic problems of modelling accurately such multi-phase flows, little analytical progress had been achieved in the investigation of a certain standard model based on the theory of interacting continua. It turns out, however, that the model under consideration can be investigated with the help of bifurcation theory. In particular, the methods of the theory of bifurcation with symmetry can be applied owing to the symmetries of the system. In general, a stationary homogeneous state exists in fluidized beds which can become unstable when the physical parameters of the system are varied. Then pattern formation takes place, e.g. in the form of one- and/or two-dimensional waves, bubbles, or convection patterns; also turbulent behaviour has been observed. In order to understand the occurrence of wave patterns and other phenomena as an inherent feature of the system, a finite, but periodically continued two-dimensional bed is investigated. While this suppresses certain boundary effects, it gives us thorough insight into the principal behaviour of this complicated system. In particular, it allows us not only to perform easily a linear stability analysis of the basic state of uniform fluidization, but also to conclude that bifurcation of travelling waves occurs when this state becomes unstable. Well-known patterns like vertical and oblique travelling waves (OTW) of the form $u(x,y,t) = \tilde{u}(x-\omega t\pm ky)$ , k [ges ] 0, are discovered. Owing to symmetry, the existence of standing travelling waves (STW) of the form $u(x,y,t) = \tilde{u}(x-\omega t, y)$ is also expected, but regrettably no mathematically rigorous proof of this last conjecture is presently available. Bubble formation can also be approached via the instability of a vertical travelling plane wave train to transverse perturbations. Then a secondary stationary bifurcation to another kind of standing travelling waves takes place. This scenario is also in agreement with experimental observations. In addition, the occurrence of bifurcations of higher order, which lead to more and more complex wave patterns and are to be found on the route to turbulence, can be deduced.

The stresses produced in an elastic half-space by a normal step loading over a circular area analytical and numerical results

The sudden application of a pressure over a circular area on the surface of an elastic half-space produces a complex wave pattern. An exact solution for the stresses at interior points of the solid is obtained by the use of a Laplace-Hankel integral transform. The expression for the axial stress is simplified for points on the axis of symmetry, and is evaluated numerically. Respective contributions of the static solution, the Rayleigh wave and other elastic waves are compared. Also, the additional complexity as compared to the acoustic case is illustrated. Analytical expressions are given for the discontinuities at the wave fronts, and the singularities due to the head waves are pointed out. The space variation of the stress along the axis, and the time variation at a given point are discussed.