Lim Chee-Seng

Spherical integral representations for finite velocity conduction of an abruptly activated current through a crystal

A source current is abruptly switched on within a crystalline medium with unaligned sets of permittivity and permeability principal axes. The current then propagates with finite velocity along an arbitrarily oriented conductor. The problem incorporates a partial zero condition and is tackled by Fourier analysis. The general solution for a scalar field is derived as a combination of four spherical integrals over the unit sphere. Two of these are time dependent and could represent the switch‐on effect. Their time independent versions relative to the traveling current edge constitute both remaining spherical integrals. These must therefore participate in any eventual steady state and possibly contribute to ?erenkov radiation; their combination is expressible as a single spherical integral directly associated with a basic differential operator. A steady state is found to develop inside an expanding ellipsoid which retreats from the current edge.

Shear crack propagation under cohesive-type tractions

SummaryA model of crack propagation is envisaged wherein space and time dependent cohesive-type tractions act along the crack faces. These tractions terminate away from the arbitrarily moving tips at some time-dependent interior points. Relative to just one crack tip, phenomena differ in three distinct reception regimes. Application of a Barenblatts postulate yields, in these regimes, three nonlinear integral equations consistent, under relevant initial conditions, with continuities of the various boundary paths. Under certain analyticities, both integral equations in two leading adjacent regimes reduce to algebraic ones, while the integral equation in the rear regime becomes transcendental; the corresponding non-singular solutions can be expanded infinitely in terms of hypergeometric functions. Explicit results are deduced for a linear mean stress drop model with uniform tractions, and the recovery of path relations during regime transitions is examined. Finally, the consequence of contracting the traction zones is explored, particularly with reference to (i) a constant traction, (ii) is singularly growing traction.ZusammenfassungEin Modell für die Rißausbreitung wird betrachtet, wobei entlang der Rißflächen orts- und zeitabhängige kohäsive Zugkräfte wirken. Diese Zugkräfte gehen von beliebig bewegten Spitzen zeitabhängiger innerer Punkte aus. Relativ zu so einer Rißspitze unterscheiden sich die Erscheinungen in drei verschiedenen Bereichen. Die Anwendung eines Barenblatt-Postulates führt in diesen Bereichen zu drei nichtlinearen Integralgleichungen, die unter entsprechenden Anfangsbedingungen mit den Stetigkeiten der verschiedenen Randwege verträglich sind. Durch bestimmte analytische Eigenschaften reduzieren sich zwei Integralgleichungen in zwei benachbarten Bereichen zu algebraischen Gleichungen, wogegen die Integralgleichung im letzten Bereich transzendent wird. Die entsprechenden nichtsingulären Lösungen können in Terme hypergeometrischer Funktionen entwickelt werden. Explizite Ergebnisse werden für ein lineares, tropfenförmiges Hauptspannungsmodell mit gleichförmigen Zugkräften angegeben, und weiters wird die Erlangung von Weg-Beziehungen während der Bereichsdurchgänge untersucht. Abschließend wird die Folge der Zugzonenverkleinerung angegeben, besonders in bezug auf (i) eine konstante Zugkraft und (ii) eine singuläre anwachsende Zugkraft.

Water waves generated by an oscillatory surface pressure travelling at critical speed

Abstract A certain perturbation is defined for disturbances caused by a pressure distribution oscillating and travelling on the surface of infinitely deep water without surface tension. Its transverse gradient is then considered along the water surface for a source speed and frequency that combine at a critical level. Boundedness tests become crucial because of this criticality. Quantitative asymptotic results are secured at large distances from a finite source domain. Separate travelling wave solutions hold within two downstream wedges and one upstream-extended wedge. Construction of a typical wavevector is demonstrated. Every wavevector is spatially dependent only on the polar angle. Along that pair of upstream-inclined wedge boundaries, the associted wavefuncion is halved in value and assumes a simplified form. Another wavefunction undergoes the same experience along one pair of downstream inclined wedge boundaries. However, at interior points near the remaining pair, two other wavefunctions combine into another form. This then simplifies along either boundary to a new function which dominates due to its lower attenuation rate. The group velocity vector always points in the outward radial direction. Explicit representations for it indicate a spatial dependence on the radial direction alone. Finally, long range surface approximations of the original perturbation are directly deduced from those of its transverse gradient via substitutions of Fourier transformed source factors. They hold for a class of applied pressures which is somewhat restricted by an integral criterion on the pressure function.

Transient Response of the 'Multiple Water-Bag' Plasma

A charge activates impulsively and then decays temporally within a MWB (multiple water-bag)-modelled warm plasma. The transient problem is formulated and asymptotically resolved for large time. The response potential comprises two characteristically distinct quantities W and WN: W is a superposition of spherically expanding, moderately attenuated Kelvin waves contributed by certain points on a subset of dispersion curves; WN is a superposition, associated with two other dispersion curves, of three spherical wavefunctions, one of which incorporates the Fresnel integrals. A transient state feature of the MWB discretization is the partitioning of the response field by growing (fast) fronts, (trailing) slow caustics and aj¯, the fastest among these being an aN¯ surface (thermal front) which pushes back a quasi-static exterior. Contrary to expectations, there is no response jump across any of those growing partitions. Wavefunctions near the slow caustics possess Airy factors. A rest state ultimately develops behind the slowest slow caustic. An application is made to the fluid plasma.

Coupled Quasiisotropic Systems

A coupled system involving isotropic operators and anisotropic source functions is posed for several vectorial and scalar unknowns. The first objective is to resolve the system, basically, into two scalar-like matrix equations, one governing a column of the vectorial unknowns, the other governing a column of the scalar unknowns. The resolution is first performed on a fully nondegenerate system and, subsequently, modified and applied to partially degenerate systems. The occurrence of common operator factors is treated. Examples in coupled elastodynamical field theory are indicated.The second objective is to investigate the system in the time-harmonic state induced by purely oscillatory anisotropic source functions. General solutions are established. They incorporate a radiation condition and appear as superpositions over coexisting real and complex eigenvalues of arbitrary orders. A far-field approximated solution, however, ranges over either a subset of maximal order real eigenvalues or, should there be n...

Switched-on evolution due to a temporal electron charge drifting subthermally through a warm collisional plasma

An electron charge is suddenly switched on while drifting through a warm collisional plasma. It acts thereafter with an arbitrary time-dependence. The evolution of the plasma, initially at rest, is considered in two and three dimensions. A general solution is first established for drift-modified slow plasma modes and then employed in subthermal drift analysis. Its abrupt switch-on causes the electron charge to release, during subthermal drift, a fully symmetric thermal front Г which subsequently expands ahead of it into an undisturbed receding expanse. A transversely symmetric response develops inside Г. On switch-off, the drifting charge releases another thermal front Г 0 which also precedes it but trails non-concentrically behind Г. Plasma response continues after switch-off. Response properties between Г and Г 0 differ strikingly from those inside Г 0 . Applications are next considered, first in three dimensions, for a subthermal pulsating electron with a generally complex frequency ω . The prepermanent state response inside Г comprises an axisymmetric dominant component o ∞ plus a spherically symmetric transient component o tr . o ∞ acquires the frequency ω . o tr has amplitudes dependent on and wave crests independent of both frequency ω and drift; it suffers a fast (slow) ω –independent temporal attenuation in a collisional (collisionless) plasma. The geometrical drift wave structure of o ∞ is closely examined for real ω and a collisionless plasma. Every energy surface associated with o ∞ nucleates at the electron; from there, it evolves slower than a phase surface, which eventually ‘disappears’ past Г. The leading energy surface, which nucleated at switch-on, develops permanently inside Г; it serves as an energy front that seals off the energy sustaining o ∞ since switch-on. Finally, the two-dimensional evolving drift field of an activated antenna line is computed.

Certain relativistic phenomena in crystal optics

Relativistic unsteady phenomena are established for a crystalline medium with unaligned sets of permittivity and permeability principal axes, but incorporating a compounded uniaxiality about some nonprincipal direction. All effects originate from a suddenly activated, arbitrarily oriented, maintained line current conducted with a finite velocity v. Integral representations studied in another paper (Chee‐Seng) are applied. The original coordinate system is subjected to a series of rotational and translational, scaled and unscaled transformations. No specific coordinate frame is strictly adhered to. Instead, it is often expedient and advantageous to exploit several reference frames simultaneously in the course of the analysis and interpretations. The electric field is directly related to a net scalar field Δ involving another scalar Ψ and its complement Ψ which can be deduced from Ψ; Ψ and Ψ are associated with two expanding, inclined ellipsoidal wavefronts ξ and ξ; these are cocentered at the current or...

Extension of a semiinfinite crack against an antisymmetric quasiuniform body force

SummaryA semiinfinite crack extends nonuniformly against a body force that is antisymmetric about the fault plane but independent of the distance from the latter. Conversion of the inhomogeneous wave equation for SH motion into a radiation equation incorporates fault plane distributed mechanisms. An integral equation is then formulated for fault plane observations preceding the crack. Both the body force and the stress drop are assumed to be bounded over finite histories. Multiple inversion yields a formal exterior solution which behaves singularly near the crack edge. An analogy with a Barenblatt postulate cancels the singularity through a nonlinear integral equation governing the edge coordinate. The resultant non-singular displacement gradient turns out to be continuous at the crack edge; here, the force-induced contribution vanishes. The interior problem is also solved for the displacement along either crack face. An application is illustrated for an impulsive body force; this leads to an interesting corollary with totally uniform results. Finally, if the body force exists within a half-space away from which extension occurs, it contributes nothing to the exterior solution; furthermore, the interior solution can be evaluated in terms of only the stress drop and the edge locus, i.e. without explicit prescription of such a body force.ZusammenfassungEin halbunendlicher Riß weitet sich ungleichförmig zufolge einer Volumskraft, die antisymmetrisch zur Rißebene wirkt, aber unabhängig vom Abstand dieser Ebene ist, aus. Die Umwandlung der inhomogenen Gleichung für die SH-Bewegung in eine Strahlengleichung beinhaltet in der Rißebene verteilte Mechanismen. Eine Integralgleichung für das Verhalten der Rißebene vor dem Riß wird formuliert. Sowohl die Volumskraft, als auch die Spannungsspitze werden über einen endlichen Zeitbereich als beschränkt vorausgesetzt. Mehrfache Inversion führt zu einer formalen äußeren Lösung, die sich nahe der Rißspitze singulär verhält. Eine Analogie mit einem Barenblatt-Postulat ersetzt die Singularität durch eine nichtlineare Integralgleichung für die Rißspitze. Der resultierende nichtsinguläre Verschiebungsgradient ergibt sich als kontinuierlich an der Rißspitze; der durch die Kraft hervorgerufene Beitrag verschwindet. Das innere Problem wird ebenso für die Verschiebung entlang einer Rißfläche gelöst. Eine Anwendung wird für eine impulsartig wirkende Volumskraft angegeben; dies führt zu einem interessanten Satz mit gänzlich gleichartigen Ergebnissen. Tritt die Volumskraft innerhalb eines Halbraumes, entfernt von der Stelle wo eine Erweiterung auftritt, auf, so trägt sie nicht zur äußeren Lösung bei. Die innere Lösung kann in Abhängigkeit von den Spannungsspitzen und den Rißspitzen ermittelt werden, d. h. ohne explizite Beschreibung der Volumskraft.

A class of problems of pulsatory radiation from a travelling source distribution

A vectorial source distribution is pulsating and travelling through an unbounded medium. The class of radiation problems posed is subjected to an augmented Fourier transform. A hyperbolicity-into-ellipticity condition is imposed on the propagator (which supports two “flow” vectors) for a related evolution into an ultimate state, so as to indirectly influence the existence and proper behaviour of the pulsatory medium response. A general radiation-conditioned solution is established under a double-slowness criterion. It is interpreted in detail through its temporal geometric development. It is next expanded, reduced and asymptotically approximated. Applications are made to (i) plasma physics, (ii) crystal optics, (iii) elastodynamics, (iv) quantum fluid dynamics.

Certain analytic source mechanisms in crack propagation theory

SummaryMultiple series solutions are extracted from certain recently established integral convolutions (Chee-Seng [5]) pertaining to crack extension against an antisymmetric quasiuniform body force. Analyticity of the source mechanism plays the crucial role. The nonlinear integral equation consistent with a nonsingular crack-preceding reception is likewise converted to a series equation. Results inside and outside the crack domain are represented by, principally, infinite series of hypergeometric functions. The source mechanism contributes to their coefficients which generally depend on reception coordinates. An application is illustrated for a linear source distribution with analytic coefficients. If these are uniform, the infinite series solutions degenerate to finite combinations involving inverse trigonometric functions; moreover, under compatible conditions, the crack edge describes a hyperbolic locus. Another application concerns a body force with an exponentially diminishing intensity; it leads to a logarithmic locus for the crack edge. Finally, general criteria are determined for a zero initial edge-velocity.ZusammenfassungMannigfaltige Reihenlösungen werden von kürzlich eingeführten Integralfaltungen (Chee-Seng [5]), die sich auf die Rißausdehnung durch eine antisymmetrische, quasigleichförmige Volumskraft beziehen, abgeleitet. Die analytische Eigenschaft dieser Ursprungsmechanismen spielt eine entscheidende Rolle. Die nichtlineare Integralgleichung, die mit einem nichtsingulären, dem Riß vorausgehenden, Eingang verträglich ist, wird ebenfalls in eine Reihengleichung umgewandelt. Die Ergebnisse innerhalb und außerhalb des Rißbereiches werden durch unendliche Reihen hypergeometrischer Funktionen dargestellt. Die Koeffizienten hängen von den Entstehungsmechanismen und im allgemeinen von den Eingangskoordinaten ab. Eine Anwendung wird für eine lineare Entstehungsverteilung mit analytischen Koeffizienten angegeben. Sind diese gleichförmig, dann entarten die Lösungen mit unendlichen Reihen in endliche Kombinationen inverser trigonometrischer Funktionen; und weiter beschreiben die Rißspitzen, unter passenden Bedingungen einen hyperpolischen geometrischen Ort. Eine weitere Anwendung betrifft eine volumskraft mit exponentiell abnehmender Intensität, die führt zu einem logarithmischen geometrischen Ort für die Rißspitze. Schließlich werden noch allgemeine Kriterien für eine anfänglich in Ruhe befindliche Rißspitze angegeben.