Perfect Wedge Research Articles

Pseudodifferential adiabatic mode parabolic equations in curvilinear coordinates and their numerical solution

Pseudodifferential mode parabolic equations in curvilinear coordinates are derived from the horizontal refraction equations. An energy-conserving split-step Padé method for their numerical solution is developed. Several examples are considered including the propagation of sound in a perfect wedge, a whispering gallery formation near circular isobath, and the breaking of the whispering gallery waveguide in the vicinity of an inflection point of the isobath.

Analytical methods for perfect wedge diffraction: A review

The subject of diffraction of waves by sharp boundaries has been studied intensively for well over a century, initiated by groundbreaking mathematicians and physicists including Sommerfeld, Macdonald and Poincaré. The significance of such canonical diffraction models, and their analytical solutions, was recognised much more broadly thanks to Keller, who introduced a geometrical theory of diffraction (GTD) in the middle of the last century, and other important mathematicians such as Fock and Babich. This has led to a very wide variety of approaches to be developed in order to tackle such two and three dimensional diffraction problems, with the purpose of obtaining elegant and compact analytic solutions capable of easy numerical evaluation.The purpose of this review article is to showcase the disparate mathematical techniques that have been proposed. For ease of exposition, mathematical brevity, and for the broadest interest to the reader, all approaches are aimed at one canonical model, namely diffraction of a monochromatic scalar plane wave by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The first three approaches offered are those most commonly used today in diffraction theory, although not necessarily in the context of wedge diffraction. These are the Sommerfeld–Malyuzhinets method, the Wiener–Hopf technique, and the Kontorovich–Lebedev transform approach. Then follows three less well-known and somewhat novel methods, which would be of interest even to specialists in the field, namely the embedding method, a random walk approach, and the technique of functionally-invariant solutions.Having offered the exact solution of this problem in a variety of forms, a numerical comparison between the exact solution and several powerful approximations such as GTD is performed and critically assessed.

Benchmarking ocean acoustic models

In this special session honoring the work by Buckingham, we review some of his early research from the 1980s where he developed analytic solutions to various benchmark problems. In particular, he developed elegant solutions for (1) the perfect wedge; (2) a penetrable wedge; and (3) a conical seamount. These were very valuable at the time in resolving errors in early parabolic equation models; they have continued to be important as computational models for fully three-dimensional environments have become more widely used.In this special session honoring the work by Buckingham, we review some of his early research from the 1980s where he developed analytic solutions to various benchmark problems. In particular, he developed elegant solutions for (1) the perfect wedge; (2) a penetrable wedge; and (3) a conical seamount. These were very valuable at the time in resolving errors in early parabolic equation models; they have continued to be important as computational models for fully three-dimensional environments have become more widely used.

Bingham fluid contact line dynamics

Contact line dynamics occurs in wetting and dewetting phenomena. The dynamics of the triple line (line of contact of the gas, liquid, and solid phases) is considered for a Bingham fluid. The previous analysis of de Gennes for Newtonian fluids is extended, considering the moving liquid near the triple line as having a perfect wedge border. The velocity profile near the contact line is obtained along with the size of the plug flow region. The viscous dissipation is found by limiting the liquid thickness to a lower limit of molecular size. The relation between the triple line velocity and the dynamic contact angle is obtained; the result requires trial and error calculations, and is shown to yield de Gennes’ equation as the Bingham fluid yield stress tends to zero. The requirement for the ideal wedge approximation is considered.

Statistical analysis of an experimental compressional sand wedge

The quasi-static deformation of dry sand is widely used as an analogue to the brittle deformation of the upper crust. The quantitative comparison of analogue to natural tectonics, or to mechanical predictions, requires identifying sources of biases and estimating the intrinsic variability of the experimental results. We develop experimental and statistical methods that fulfill these requirements. We consider an initially perfect wedge resting on a flat layer, made of a uniform dry sand in a rectangular glass box. The box is shortened lengthwise by translating one of its end walls towards the other. The lateral walls can remain fixed, or be translated with the moving end wall. Upon shortening, the wedge is thrusted above the flat layer forming classical fore- and backthrusts, as essentially plane-strain, structures. Lifetimes, locations, and dips of all thrusts constitute seven quantifiable output parameters (called observables), in addition to the shortening forces monitored at both end walls during shortening. Up to seventy measurements of each observable were performed in seven final-state cross-sections of ten experiments. A three-step statistical analysis allows us to prove that, first, the observables vary independently, justifying their modeling with independent distributions. Second, the ergodic hypothesis holds, meaning that along strike variations can be used to infer the intrinsic experimental variability. Measurements can thus be repeated on successive cross-sections in each experiment. Third, our data set is free from bias due to friction on the lateral walls, or due to the finite length of the box. We then construct statistical models of each observable using either Gauss or Laplace distributions. For example, forethrusts dip at 38 ° ± 3.2 °, and backthrusts, at 41 ° ± 3.3 °. We finally show how to apply these statistical models to experiments using a different initial geometry. The statistical methods presented here are applicable to experiments with different setups, materials and observables, although the ergodic hypothesis is relevant only to plane-strain experiments.

Transmission of sound in a wedge‐shaped layer of fluid with a rigid bottom

AbstractThe theory of generalized rays is applied to the problem of determining the exact shape of the pressure pulse received at very large ranges, in the case of a sound source in a 3° perfect wedge of fluid. The time variation of the pressure at the point source is assumed to be given by a Heaviside unit function H(t). It is found that horizontal refraction and backscattering have a pronounced effect on the overall appearance of the received pulse: at very large down–slope (normal to the apex line) ranges, the shape of the received pulse does not change much, while, at very large cross–slope (parallel to the apex line) ranges, it changes drastically. Thus, in a 3° perfect wedge of fluid, long–range down–slope propagation (exclusively affected by backscattering) seems to be nondispersive, but long–range cross–slope propagation (affected by horizontal refraction and backscattering) seems to be highly dispersive. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Simple Method of Determining the Pitch of a Chiral Nematic Liquid Crystal

The pitch value P of a chiral nematic liquid crystal is a critical parameter for azimuthal anchoring strength measurement. The Grandjean-Cano wedge method is a conventional technique for pitch determination, but the perfect wedge cell with known angle and uniform disclination lines are required. In this work, we have developed an improved Grandjean-Cano method, which is simple and gives accurate pitch value. The sensitivity of this new method is ±λ/2Δn.

An exact representation of the image field in a perfect wedge

A solution is presented for the image component of the three-dimensional (3-D) acoustic wave-field from an impulsive point source in a liquid wedge with impenetrable boundaries (mixed Dirichlet-Neumann) modeling shallow water over sloping bottom (shallow-water wedge). This exact and complete solution is in the form of a sum of partial waves including the pulse emitted from the source plus a finite number of pulses reflected off the wedge boundaries, and it is generally valid for all ranges and depths. The 3-D acoustic propagation effects in an impenetrable wedge are examined for range propagation when the receivers are located directly up-slope and down-slope of the source and at the source-point itself, and for cross-range propagation when the receivers are located directly cross-slope of the source. When the receiver recedes up-slope from the source, the duration of the pressure response is shortened, and the number of pulses arriving along the indirect ray paths (the backscattered pulses) decreases. When the receiver recedes down-slope from the source, the response duration is constant, and the number of backscattered pulses also decreases. As the source and the receiver separate in the cross-range-direction, the response duration is so shortened that the arrival of the last-arriving pulse approaches that of the first-arriving pulse in the limit (i.e., when the separation between the source and the receiver is “large” in the cross-range-direction), and the number of backscattered pulses remains constant.

A new solution for the acoustic pressure‐field in a perfect wedge

AbstractA time domain analytical solution is presented for the acoustic pressure‐field due to an impulsive point source in a liquid wedge with a rigid sloping bottom. The horizontal boundary of the wedge is a pressure‐release surface. The wedge is isovelocity, and it is three‐dimensional (3‐D) in that the source and the receiver are not restricted to the plane perpendicular to the wedge apex. This 3‐D wedge with perfectly reflecting (i.e., impenetrable) boundaries, the 3‐D perfect wedge, is the simplest theoretical model of a shallow water environment in which the ocean bottom has a constant slope, but it is inadequate in that it cannot account for acoustic penetration of the bottom typical for a real ocean floor.In a perfect wedge, the acoustic field produced by a point source consists of two components: the diffraction field due to scattering at the apex, which under certain conditions vanishes [1], and the image field including the wave emitted by the source and a number of waves reflected off the wedge boundaries. These waves with spherical wave fronts propagate along the stationary time paths (the ray paths). The repeated reflections from the sloping bottom introduce the curvature into the projection of the ray path onto the horizontal boundary, provided that the first segment of this path originating at the source has a cross‐slope component. This path curvature in the horizontal is known as “horizontal, or bathymetric refraction” [2]. The ray path that leaves the source propagating up‐slope may be turned around to reach the receiver propagating back down‐slope. This change in propagation direction, known as “backscattering” [2], is due to multiple bottom reflections. The receiver can thus be reached along one of the two ray paths: the direct path or the indirect path which is turned around on approaching the apex. The image field in the wedge is thus three‐dimensional, i.e., it depends on three spatial coordinates: the range (normal to the apex), the cross‐range (parallel to the apex), and the depth coordinate.

Solutions to range-dependent benchmark problems by the finite-difference method

An explicit second-order finite-difference scheme has been used to solve the elastic-wave equation in the time domain. Solutions are presented for the perfect wedge, the lossless penetrable wedge, and the plane parallel waveguide that have been proposed as benchmarks by the Acoustical Society of America. Good agreement with reference solutions is obtained if the media is discretized at 20 gridpoints per wavelength. There is a major discrepancy (up to 20 dB) in reference-source level because the reference solutions are normalized to the source strength at 1 m in the model, but the finite-difference solutions are normalized to the source strength at 1 m in a homogeneous medium. The finite-difference method requires computational times between 10 and 20 h on a super minicomputer without an array processor. The method has the advantage of providing phase information and, when run for a pulse source, of providing insight into the evolution of the wave field and energy partitioning. More complex models, including velocity gradients and strong lateral heterogeneities, can be solved with no additional computational effort. The method has also been formulated to include shear wave effects.

The finite difference method for time domain solutions to range‐dependent bottom‐interaction problems

An explicit second‐order finite difference scheme has been used to solve the elastic wave equation in the time domain. Solutions are presented for the perfect wedge, the lossless penetrable wedge, and the plane parallel waveguide, which have been proposed as benchmarks by the Acoustical Society of America. Good agreement with reference solutions is obtained if the media is discretized at 20 grid points per wavelength. The principle disadvantage of the technique is long computational times that are between 10 and 20 h on a minicomputer without an array processor. The method has the advantage of providing phase information and, when run for a pulse source, of providing insight into the evolution of the wave field and energy partitioning. Arbitrarily more complex models including velocity gradients, strong lateral heterogeneities, and random media can be solved with no additional computational effort. The method has also been formulated to include shear wave effects. [Work supported by ONR.]

Diffraction of sound by hard strips and truncated wedges

Starting from rigorous diffraction solutions for single, hard, semi‐infinite wedges, it has long been possible to construct approximate solutions for more complex angular surfaces [A.D. Pierce, J. Acoust. Soc. Am. 55, 941–955 (1974)]. Demonstrated here is an exact procedure that formulates the multiple scatter between vertices of an angular body via the standard self‐consistent algorithm. This allows one to obtain the correct diffracted field of each vertex. Summing these together with the relevant incident and image fields leads to an exact solution for the full sound field. This is done here for the case of a plane harmonic incident sound field for (1) a truncated wedge and (2) a hard strip of width l. Numerical results for the latter model are shown to compare satisfactorily for all kl with experimental data [H. Medwin et al., J. Acoust. Soc. Am. 72, 1005–1013 (1982)]. This theoretical procedure yields, in the plane wave case, a formal estimate of the error incurred in the “double diffraction” approximation (i.e., the procedure that neglects all but the first two terms of the multiple scatter series). This error decreases like elkl/kl for increasing kl. It is then possible to confirm theoretically the observation of Medwin et al. who showed that, for the geometry of their strip experiment, this approximation is actually very good for kl≳ 3 and quite adequate in the region 1 < kl < 3, using the Biot‐Tolstoy rigorous impulsive point source solution for the perfect wedge [M. A. Biot and I. Tolstoy, J. Acoust. Soc. Am. 29, 381–391 (1957)] for their double‐diffraction solution. [Work supported by ONR.]

Theory of three-dimensional acoustic propagation in a wedgelike ocean with a penetrable bottom

A new analytical solution is presented for the three-dimensional acoustic field due to a monotonic point source in an ocean wedge with a penetrable bottom. The field consists of a sum of normal modes which depend weakly on range (out from the apex) and frequency. The mode coefficients depend strongly on range, on z (cross range parallel to the apex), and on frequency. A dispersion relation is derived which gives the range-dependent mode shapes. Then a notional ‘‘effective’’ wedge is constructed with perfectly reflecting boundaries whose apex is displaced from that of the penetrable wedge. The offset is such that the range-independent modes in the effective wedge match the range-dependent modes in the penetrable wedge. A known solution for the perfect wedge is then modified, by excluding rays which intersect the bottom at greater than the critical grazing angle, to give the spatial and frequency dependence of the mode coefficients in the penetrable wedge. The new solution, which satisfies the Helmholtz equation and the boundary conditions, exhibits prominent 3-D features uniquely associated with the penetrable bottom. These features conform with ray-theoretic predictions. Toward the end of the article the transient field in the penetrable wedge due to an impulsive point source is derived by Fourier transforming the cw solution. The Green’s function for the velocity potential is a transient of extended duration which contains evidence of image smearing, a phenomenon due directly to the phase shift experienced by rays undergoing total internal reflection from the penetrable interface.

Accurate Determination of a Low-Order Fourier Coefficient of the Potential of Aluminium Single Crystals by a Modified Thickness Fringe Method

A modified thickness fringe method is proposed for the determination of a low-order Fourier coefficient of the crystal potential. This method can be applied not only to single crystals with perfect wedge shape but also to those with some irregularities by introducing a `surface function', which can automatically correct the irregularity of the fringe periods due to the wedge surfaces. Electro-thinned aluminium single crystals are used as a reference substance in experiments with and without a filter technique. Comparison of the experimental data with n-slice calculations can give the coefficient with an error of less than 1 percent.