Caustic Line Research Articles

The influence of length under blast load on the propagation mechanism of interconnected cracks

In order to study the influence of interconnected defects on the rock breaking effect under blast load, the relevant explosion experiment was carried out by prefabricating interconnected cracks on plexiglass plates (PMMA), and the propagation mechanism of interconnected cracks under blast load was studied from the length level. The dynamic caustic line system and finite element analysis software were combined to analyze the stress intensity factor, propagation velocity of the interconnected crack tip, and full-field stress of the specimen; the results show that with the radial stress wave propagation direction as the axis of symmetry, the two interconnected cracks with symmetrical angles but different lengths are under the action of blast load; there is a certain competition between the two cracks; the longer cracks are more likely to initiate and propagate, and the shorter cracks almost do not propagate. In addition, in terms of energy accumulation at the tips of the two cracks, the longer crack has a certain inhibitory effect on the shorter crack, and when the length of the longer crack remains unchanged, the inhibition effect weakens with the increase of the length of the shorter crack, and when the length of the shorter crack remains unchanged, the inhibitory effect increases with the increase of the length of the longer crack. After the crack initiation of the longer crack, the relationship curve between the stress intensity factor and the crack propagation velocity and time of the longer crack tip decreases first, and then the relationship curve rises to the peak as the energy of the tip of the shorter crack is transferred to the longer crack, and then the oscillation decreases, while the stress intensity factor and crack propagation rate of the shorter side crack tip show an overall downward trend. The numerical simulation is used to supplement the analysis of the stress propagation in the full-field of the specimen, which corroborates with the experiment results. The experimental study provides a theoretical basis for the analysis of the propagation mechanism of interpenetrating cracks under blast load in practical engineering.

Research on the crack formation mechanism of shaped charge blasting and the new dual-shaped charge blasting technology

Shaped charges are extensively utilized in various rock engineering fields for directional fracturing. Numerous scholars have conducted extensive research on this topic, yielding significant results and widespread application in engineering projects. Efforts have focused on improving the directional fracture effects and fully utilizing the energy generated by the explosion to enhance the directional blasting effects of shaped charges and minimize damage to the rock mass in areas not targeted by the shaped energy. This approach aims to preserve the integrity and stability of the rock mass in the areas unaffected by the directional fracture, thereby reducing damage to the surrounding rock. This study builds on the foundations of explosion mechanics. It incorporates a novel digital laser dynamic caustic line experiment system and pressure fields, and numerical analysis techniques. The research systematically analyzes the process of explosive action in shaped charges and proposes a new type of dual polymer charge. This novel charge is compared to traditional and common charges. The findings revealed that, compared to standard-shaped charges, the new dual-shaped charge achieves a more effective directional fracturing, The rock damage is less in the direction of non-gathering energy. Under the conditions of the same amount of explosive and the same decoupling coefficient, The crack propagation length of common charge is 1.34 times of the diameter of the gun hole, the crack propagation length of shaped charge is 10.17 times of the hole diameter, The crack propagation length of the new dual-charge is 19.67 times of the hole diameter. HIGHLIGHTS In this paper, a new type of dual-charge is proposed. The findings revealed that, compared to standard-shaped charges, the new dual-shaped charge achieves a more effective directional fracturing, The rock damage is less in the direction of non-gathering energy. Under the conditions of the same amount of explosive and the same decoupling coefficient, The crack propagation length of common charge is 1.34 times of the diameter of the gun hole, the crack propagation length of shaped charge is 10.17 times of the hole diameter, The crack propagation length of the new dual-charge is 19.67 times of the hole diameter. 1.Following the charge explosion, influenced by the shaped charge tube, the detonation wave and explosive gas alter, expanding outward in an approximate "elliptical" shape. The explosion stress field distribution is approximately "one"-shaped in the direction of energy concentration. The primary explosive crack expands along the direction of energy accumulation, while the secondary cracks predominantly distribute in the perpendicular direction. The seam width and the transmission coefficient (material) influence the condenser’s concentration effect. 2.Compared to ordinary charge blasting, shaped charge blasting exerts a pronounced guiding effect on the explosion stress field. The stress field primarily transfers along the direction of energy concentration, with the stress being markedly greater in this direction than in the direction of non-concentrated energy. In comparison, dual charge blasting undergoes two convergence effects, with longer durations and distances of convergent action. The stress field generated by dual-charge blasting is significantly greater than that of the shaped and common charges, offering enhanced effects. 3.Agglomeration length is a crucial factor in the agglomeration effect. Explosion energy directly transfers to the solid medium via the coalseam, minimizing explosion energy loss in the air medium and controlling the detonation wave shape and explosive gas flow in the energy accumulation direction, resembling "one." The dipolymer charge notably extends crack propagation length in the polymer direction and mitigates damage in the non-polymer direction.

Corrosion failure of stainless-steel in caustic soda lines

Abstract The paper deals with the case studies of stainless steel piping corrosion in caustic soda transfer lines at a hydrocarbon processing plant, in the Asian region. The lines were in service of transferring the caustic soda solution from the storage tank farm to different process units, for subsequent use in reaction vessels. To maintain a minimum temperature of the fluid to avoid its precipitation, the lines were accompanied by a steam heat-tracer, throughout the line route. The material of construction (MOC) of the line was stainless steel type 316. After a period of about 3.5 years, 2” caustic line suffered fluid leakage, which was noticed through significant salt accumulation over insulation at one location. A valve was also noticed to leak through a weld at a 1.5” distribution line. The affected components were removed from service and subjected to detailed inspection and laboratory investigation, using advanced tools of metallurgical analysis and characterization. It was found that leakage in 2” line had occurred due to poor workmanship used during the repair of a manufacturing defect at the affected location and leakage of the valve had taken place due to the use of improper welding practice (in the case of subjected weld). In addition, the significant role of metallurgical defects in the parent metal was also disclosed to result in the observed failures. In addition, the contribution of process excursions can’t be completely ruled out. Finally, remedial measures are given to avoid the recurrence of such failure.

Extreme amplification regimes of the Schwarzschild gravitational lens

We investigated a complete set of relativistic images of a small source located at an arbitrary distance from a Schwarzschild black hole gravitational lens. This paper offers a description of a simple and efficient fully relativistic method for calculating the bolometric intensity amplification. We focused our analysis primarily on sources located at small radii and close angular distance from the caustic line, both behind and in front of the compact lens. We term the corresponding large deflection regime ‘extreme lensing’. We approximated the regime of fully-relativistic, extreme amplification of point sources by simple analytical formulae valid over a wide range of source distances. Using such approximations, we also derived formulae for the maximum amplification of extended sources close to or intercepted by the caustic line. Simple analytical approximations of the time delay between the brightest consecutive images in extreme amplification regimes are also presented.

Lower turning point convegence zone in deep water with an incomplete channel

In a deep sea sound channel, rays will bend due to the sound speed profile, and convergence zone will occur when the rays are intensive. Transmission loss in the convergence zone is smaller and it is conducive to acoustic detection and communication. Therefore the study of acoustic characteristics in convergence zone is always the focus of deep-sea acoustics. A long-range sound propagation experiment is conducted in the South China Sea. An equivalent broadband explosive sound source of 1 kg is placed at a depth of 200 m, and the hydrophone receives the data at 3146 m far. The processing and analysis of the experimental data indicate that there is a convergence zone below the sound channel axis in the incomplete deep channel. Compared with the upper turning point convergence zone near the surface, this convergence zone has a high convergence gain at a long distance. The caustic lines of refracted type and refracted surface-refleted type are determined by means of ray-normal mode theory. It is found that the location of the deep convergence zone observed in the experiment is consistent with the position of the refracted caustic line. It is proved that the convergence zone is a lower turning point convergence zone formed by the superposition of a large number of normal modes in the same phase, and it has a convergence effect at a certain depth below the sound channel axis in the deep sea. The formation conditions of the convergence zone and the influence of sound source depth on the caustic structure of the convergence zone are studied. The comparisons of the transmission loss and the width between the upper and lower turning point convergence zone at a long distance aremade. The analysis shows that the convergence gain in the seventh lower turning point convergence zone is still no less than 10 dB. The influence of the vertical structure of sound velocity on the lower turning point convergence zone is studied. The theoretical analysis results are in good agreement with the experimental data.

Investigation of Corrosion Mechanisms in a Continuous Catalytic Regeneration (CCR) Platformer Unit

— Corrosion is prevalent throughout the world, none less so than in the oil and gas industry. Managing and mitigating corrosion in refining complexes is of paramount importance in order to prevent undesirable consequences such as major fires, explosions and Boiling Liquid Expanding Vapor Explosion (BLEVE) due to Loss of Primary Containment caused by the thinning and ultimate failure of pipelines and vessel walls. Analysis of a platformer unit at a Malaysian refinery has identified the occurrence of thermal fatigue and erosion corrosion on the Vent Gas Tower (VGT) Caustic Circulation line, which in turn has led to the excessive degradation of the walls of a venturi scrubber and the 8’’ caustic circulation line with initial thickness of 12.70mm. The rate of corrosion (0.88 mm/year) of the 8’’ caustic circulation line exceeds the standard allowable carbon steel corrosion rate (0.1 mm/year) as stated in API 571 Damage Mechanisms Affecting Fixed Equipment in the Refining Industry. This indicates that the circulation line has significant potential to fail which would lead to a major HSE incident. Based on these findings it is recommended that the refinery in question modifies the line by increasing the thickness of the line and include a cooling system to reduce temperature swing (ΔT) to below 93°C. Besides that, it is suggested that the implementation of Corrosion-Resistant Alloys (CRA) is conducted on the line. According to ASME B31.3 and supported by ASTM A 193, the line can be replaced with nickel-based alloy, alloy 800H and killed carbon steel which have high resistivity to corrosion than carbon steel. However, more comprehensive studies need to be conducted to identify the viable mitigation methods that are suitable to be implemented on the Vent Gas Tower (VGT) Caustic Circulation line.

Identifying metrics to predict annoyance due to Mach-cutoff flight ground signatures

Theoretically, Mach-cutoff flight under ideal atmospheric conditions could lead to boomless supersonic flight observed under the flight path on the ground. Such ideal atmospheric conditions refract the sonic boom waves upwards at the caustic line, so they do not reach the ground. This presentation describes the perception of the evanescent sound field below the flight path. The work investigates perceptual attributes and metrics related to these unique sounds. Annoyance and three other perceptual factors (“Thunderous,” “Rumbly,” and “Swooshing”) were analyzed through subjective testing using pair-wise comparison. Stimuli used were from recordings made during NASA’s “Farfield Investigation of No-boom Thresholds” (FaINT). Linear regression with principal component analysis indicated which perceptual factors contribute to annoyance, and stepwise regression identified candidate metrics for predicting annoyance. Traditional loudness metrics (i.e., weighted Sound Exposure Level) were analyzed alongside sonic-boom specific metrics (i.e., Perceived Loudness) and sound quality metrics (i.e., Sharpness). [Work supported by the U.S. Federal Aviation Administration Office of Environment and Energy through ASCENT, the FAA Center of Excellence for Alternative Jet Fuels and the Environment, Project 42 through FAA Award No. 13-C-AJFE-PSU under the supervision of Sandy Liu. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the FAA.]

Focusing of ground vibrations generated by high-speed trains travelling at trans-Rayleigh speeds

In the present paper, the effects of focusing of ground vibrations generated by high speed trains travelling at trans-Rayleigh speeds, i.e. under the condition of ground vibration boom, are considered theoretically. These effects are similar to the effects of focusing of sound waves radiated by supersonic aircraft. In particular, if a railway track has a bend, e.g. to provide the possibility of changing direction of train movement, the Rayleigh surface waves generated by high-speed trains under the condition of ground vibration boom may become focused. This results in concentration of their energy along a simple caustic line at one side of the track and in the corresponding increase in ground vibration amplitudes. The effect of focusing of Rayleigh waves may occur also if a train moves along a straight line with acceleration a and its current speed v(t) is higher than Rayleigh wave velocity in the ground. In the present paper, both the above-mentioned focusing mechanisms are investigated in detail using the Green's function formalism and the expressions for space-time distributions of track-train-induced dynamic forces that take into account either track curvature or train acceleration. It is shown that in both these cases the effect of focusing can result in noticeable increase in generated ground vibrations. The obtained results are illustrated by numerical calculations.

Construction of Bi-Pearcey beams and their mathematical mechanism

We present a theoretical expression in the form of the Pearcey function by deducing the Fresnel diffraction distribution of an elliptic line. Then, we numerically simulate and experimentally generate this kind of new Pearcey beams by using the Fresnel diffraction of optical ellipse line. This kind of beams can be referred to as Bi-Pearcey beams because their appearance of the topological structure is very similar to the combination of two face-to-face classical Pearcey beams. It is no doubt that so-called Bi-Pearcey beams are the new member of a family of form-invariant Pearcey beams. Subsequently, we also provide the theoretical mechanism of generating Bi-Pearcey beams based on the Zeeman catastrophe machine of catastrophic theory. By solving the critical equation of potential function of Bi-Pearcey beams generated by an ellipse line, we find that the optical morphogenesis of Bi-Pearcey beams is determined by the number of roots of the critical equation. The critical equation of potential function of Bi-Pearcey beams is a classical Cartan equation, which has at most three real roots. For the Fresnel diffraction of ellipse line, three real roots of the critical equation are corresponding to three stable points and represent three diffraction lines, hence they can be used to examine the optical topological structure of Bi-Pearcey beams. By choosing the appropriate control variable of Bi-Pearcey beams, two diffraction lines of an ellipse line overlap, and the strong caustic line of Bi-Pearcey beams is correspondingly generated when the two of the three real roots of the critical equation are equal. If the three real roots of the critical equation are all equal, the strongest cusps of Bi-Pearcey beams are generated, accordingly. Moreover, the equation of the caustic line and their positions of four cusps of Bi-Pearcey beams are given by solving the control variable equation of Bi-Pearcey beams. In conclusion, we elucidate the mathematical mechanism of topical morphogenesis of Bi-Pearcey beams based on catastrophic theory.

Singularities in a teacup

Standard presentations of optics concentrate on ideal systems made for imaging which bring all rays from a point source to one focus. But, in Nature, or in realistic optical systems with defects, rays do not behave precisely in this way. Rather than the focus simply being blurred, the rays, after reflection or refraction, form beautiful and rather universal patterns of bright lines known as caustics. Mathematically speaking, a family of rays is best viewed as a surface in a higher-dimensional space where we keep track of both the position and direction of rays. The intensity enhancement on approaching the caustic line is a singularity, arising from projection of a smooth surface from higher dimensions to lower dimensions. The universal features of such singularities, which arise in many contexts beyond optics, formed a major theme of Vladimir Arnold’s work after 1965, when he was exposed to Rene Thom’s vision of ‘catastrophe theory’. Arnold and his school made seminal contributions to singularity theory.

Studying wave optics in the light curves of exoplanet microlensing

We study the wave optics features of gravitational microlensing by a binary lens composed of a planet and a parent star. In this system, the source star near the caustic line produces a pair of images in which they can play the role of secondary sources for the observer. This optical system is similar to the Young double-slit experiment. The coherent wave fronts from a source on the lens plane can form diffraction pattern on the observer plane. This diffraction pattern has two modes from the close- and wide-pair images. From the observational point of view, we study the possibility of detecting this effect through the Square Kilometer Array (SKA) project in the resonance and high magnification channels of binary lensing. While the red giant sources do not seem satisfy the spatial coherency condition, during the caustic crossing, a small part of source traversing the caustic line can produce coherent pair images. Observations of wave optics effect in the longer wavelengths accompanied by optical observations of a microlensing event provide extra information from the parameter space of the planet. These observations can provide a new basis for study of exoplanets.

Caustic echoes from a Schwarzschild black hole

We present the first numerical approximation of the scalar Schwarzschild Green function in the time domain, which reveals several universal features of wave propagation in black hole spacetimes. We demonstrate the trapping of energy near the photon sphere and confirm its exponential decay. The trapped wavefront passes through caustics resulting in echoes that propagate to infinity. The arrival times and the decay rate of these caustic echoes are consistent with propagation along null geodesics and the large $\ensuremath{\ell}$ limit of quasinormal modes. We show that the fourfold singularity structure of the retarded Green function is due to the well-known action of a Hilbert transform on the trapped wavefront at caustics. A twofold cycle is obtained for degenerate source-observer configurations along the caustic line, where the energy amplification increases with an inverse power of the scale of the source. Finally, we discuss the tail piece of the solution due to propagation within the light cone, up to and including null infinity, and argue that, even with ideal instruments, only a finite number of echoes can be observed. Putting these pieces together, we provide a heuristic expression that approximates the Green function with a few free parameters. Accurate calculations and approximations of the Green function are the most general way of solving for wave propagation in curved spacetimes and should be useful in a variety of studies such as the computation of the self-force on a particle.

Structure of phonon focusing patterns in tetragonal crystals

A set of critical conditions for the characteristic caustic points in the phonon focusing patterns in tetragonal crystals is formulated. A caustic line segment in the focusing pattern is generally associated with either a fold or a cusp on the wave surface. Most of the caustic lines are symmetrical with respect to the principal symmetry plane and can be characterized by the caustic points at the centers of the caustic lines. These characteristic caustic points originated from inflection/parabolic points with zero in-plane/ex-plane curvature, respectively. By employing the Stroh formalism, the inflection/parabolic points on the slowness surface are studied in terms of the so-called zero-curvature transonic states. Since these transonic states are related to extraordinary degeneracies in the Stroh eigenvalue equation, the conditions for the degeneracies can be regarded as critical conditions for the characteristic caustic points. These conditions provide an overview of global structure of the phonon focusing patterns in tetragonal crystals. A set of caustic lines in vicinity of (001) plane is also investigated and exemplified.

Corrosion failure analysis of spent caustic drainage line pipe

A mid country oil refinery requested for the analysis of spent caustic 4 in. damaged line pipe leading to spent caustic storage tank which failed after installation by localize corrosion attack within 8 months. The refinery also reported that the failures were frequent and were occurring at same region between continuous flow line (CFL) and intermittent flow line (IFL) at 6 O’ Clock position. The line duty cycle was half filled used as and when required basis. The chemistry of spent caustic solution, operating conditions, metallurgy and electrochemical behavior of line pipe in spent caustic was carefully reviewed. The design flow was also examined and certain flaws in design were identified. It was concluded that by modifying the diameter of 4 in. diameter existing line pipe in the design, the problem was solved. The original suggestions have been validated after more than 3 years of satisfactory service of said line pipe.

Corrosion of carbon steel caustic header in the presence of chloride ions

Caustic soda produced in caustic cells at a petrochemical plant was loaded into a crude tank by means of a caustic header line made of plain carbon steel. After a fraction of expected service life, the header was severely corroded. Scanning electron microscopy (SEM) coupled with energy dispersive X-ray spectroscopy (EDS) and X-ray diffraction (XRD) was used to identify the material and the corrosion product. Metallurgical evaluation suggested that the mode of failure was localized attack by chlorine and/or hydrochloric acid at pipe connections (flanges) where concentrated solutions could accumulate. Most evidence pointed out that the failure resulted from corrosion attack due to condensation of hydrochloric acid. Misalignment of the pipe at the flange led to cold work which is also thought to accelerate the corrosion rate. It was recommended to consider a better pipe alignment practice as well as the addition of an inhibitor as a short-term solution. Since the piping system was used close to the critical operating conditions, it was recommended as a long-term solution to replace the caustic header material by one of the higher grades Ni–Cr–Mo alloys.

Multiwavelength focusing with the Sun as gravitational lens

The light deviation caused by the gravitational potential in the vicinity of the sun could be used as a means of focussing radiation that cannot be focussed easily otherwise. The gravitational lens formed by the sun is not stigmatic, but does have the advantage of being achromatic and acts identically on all types of mass-less radiations. For a source at infinity, its geometrical characteristics present a “caustic” line starting at 550 astronomical units (UA) downstream from the sun. In a plane perpendicular to that caustic line, images of distant objects are formed.

The relation between the spherical aberration of a lens and the spun cusp diffraction catastrophe

A lens with spherical aberration, illuminated with an axial plane wave, produces a rotationally-symmetric cusped caustic together with an axial caustic line. Both caustics are truncated by the finite aperture of the lens, and they are decorated by diffraction. One may pass continuously from the limit of small aperture, where the diffraction pattern consists simply of Airy rings around the focus, to the limit of infinite aperture, where the diffraction pattern is that of the three-dimensional spun cusp. This contains ring zeros both inside and outside the cusped caustic. The rings are structurally stable phase singularities (wave dislocations), whose progress out of the focal plane can be traced as the aperture is enlarged. In any axial plane the dislocations are points. Before reaching their final destinations these dislocation points invariably trace out spirals, whose detailed form may be deduced by a perturbation theory. Apart from this, their trajectories, births and deaths are different from those encountered in the analogous case of the two-dimensional Pearcey pattern.

Determination of the properties of the central engine in microlensed QSOs

We study a recently observed gravitational microlensing peak in the V-band light curve of Q2237+0305A using a relatively simple physical model, but one that is highly consistent with the data (the best-fit reduced is very close to 1). The source quasar is assumed to be a Newtonian geometrically-thin and optically-thick accretion disk. The disk has an arbitrary orientation, and both blackbody and greybody emission spectra are considered. When the electron-photon scattering plays a role, the greybody spectrum will be a simplified version of the exact one. In our model, the microlensing variability results from the source crossing a caustic straight line. The main goal of our work is to estimate the black hole mass and the mass accretion rate in QSO 2237+0305 as well as to discuss the power and the weakness of the technique, some possible improvements, and future prospects from multifrequency monitoring of new microlensing peaks. We also put into perspective the new methodology and the results on the central engine in QSO 2237+0305. From the fitted microlensing parameters and reasonable dynamical/cosmological constraints, it is concluded that QSO 2237+0305 harbours a central massive black hole: 107 6 108 . While the information about the central dark mass is very interesting, the mass accretion rate is not so well constrained. The typical values of the disk luminosity/Eddington luminosity ratio are in the range, where 1 is the emissivity relative to a blackbody and the highest ratio corresponds to the largest deflector motion. Therefore, in order to verify 1, a relatively small projected peculiar motion of the lens galaxy and a greybody emission seem to be favored.

Semiclassical study on multidimensional effects in tunneling chemical reactions: Tunneling paths and tunneling tubes

The effects of multidimensionality in the quantum mechanical tunneling of chemical reactions are investigated. The aim of the present report is twofold. In the first place, we construct a new semiclassical theory to describe the tunneling by incorporating nonclassical solutions of the time-dependent Hamilton–Jacobi equation into the Feynman kernel. A systematic class of complex-valued (nonclassical) solutions for the time-independent Hamilton–Jacobi equation has been found that are generated along non-Newtonian paths in real-valued configuration space [K. Takatsuka and H. Ushiyama, Phys. Rev. A 51, 4353 (1995)]. In the present paper, the straightforward extension is applied to the time-dependent Hamilton–Jacobi equation, the solutions of which describe the tunneling in chemical reactions. It is shown that no damping factor due to the tunneling arises from the preexponential factor in the thus obtained nonclassical kernel, since it is still real valued, aside from the complex phase due to the Maslov index, and moreover its functional form is essentially the same as in the nontunneling case. Thus only the imaginary part of the action integral is responsible for the damping. A quasiclassical treatment of the semiclassical mechanics is developed to characterize the real-valued tunneling paths. In the second-half of this paper, some typical tunneling reactions in collinear three atomic systems on the LEPS (London–Eyring–Polanyi–Sato) potential surface are investigated in terms of our semiclassical theory. The effect of the initial energy distribution among the vibrational and translational modes is investigated asking which is preferable for tunneling and what is the resultant distribution of the energy in the product molecules. The following two factors to control the tunneling reactions are mainly examined as our first case study: (a) the mass effects featuring heavy–light–heavy and light–heavy–light patterns and (b) the anisotropy of the potential surface, namely, the early or late barrier. Tunneling paths of the types of Marcus–Coltrin and Miller–George are both generated spontaneously. A path of Marcus–Coltrin type takes a major role when the translational energy dominates in tunneling, while that of Miller–George type is dominant in a case where the vibrational excitation is important. As a distinguished feature of the multidimensionality in tunneling, we have identified what we call a tunneling tube, in which a bunch of the tunneling paths are involved emanating from the so-called caustic line. It turns out that the width of the tunneling tube determines in part the final energy distribution among the product vibrational modes.

The two-component representation of time-harmonic elastic body waves in the high- and intermediate-frequency regimes

Propagation and diffraction of time-harmonic elastic body waves through homogeneous and isotropic materials is revisited. The two-component representation of body waves is given which is applicable in the high- and intermediate-frequency regimes when the family of rays is (i) regular or (ii) possesses an irregularity due to the presence of a wedge, an acoustic point source located on the boundary, simple caustic, or focal line. Both physical and mathematical description of the phenomena under consideration is offered. The resulting expressions are asymptotic in character and easy to compute. They should prove useful in producing algorithms for describing acoustic fields in homogeneous and isotropic materials with isolated defects and in providing benchmark cases for testing numerical codes designed to solve elastic wave equations in more complicated situations.