Oblique Interaction Research Articles

On the oblique interaction of a large and a small solitary wave

The steady oblique interaction of two solitary waves on the surface of water of constant depth is considered. One wave is taken to be of arbitrary (large) amplitude and the other is small with a (non-dimensional) amplitude measured by the parameter ε. A solution is sought as an asymptotic expansion, based on ε, that assumes that in some region the solution is the sum of the two waves plus interaction terms. It is shown that this expansion is not uniformly valid close to a critical angle. This angle varies from zero (parallel waves) up to about 63°, as the amplitude of the larger wave increases from infinitesimally small to the largest-possible solitary wave. I n the limit of two small waves, the details agree precisely with the results obtained by Miles (1977a).When the angle between the two waves is dose to the critical angle, for a given large wave, an alternative asymptotic expansion is required. In this strong-interaction case, the dominant term is just the large wave but with a phase shift that is an arbitrary function of the characteristic variable associated with the small wave. This function is determined by matching to an appropriate far field, and it turns out to be proportional to the logarithm of a hypergeometric function (which itself can be expressed in terms of the associated Legendre function Pv−μ). The phase shift is then well-defined (finite, real) provided the angle is not very close to the critical value. When this occurs the phase shift can be infinite at a specific angle (which corresponds to the case |μ| = v > 0), and even closer to the critical angle (|μ| < v) the phase shift is undefined (no longer real). A real solution for the wave profile is still possible if negative amplitudes are allowed, but the resulting solution is unacceptable since the surface is then not undis- turbed at infinity. It is shown that the criterion |μ| < v matches exactly (for two small waves) with Miles’ criterion for the non-existence of a regular reflection.For the strong interaction (and |μ| ≥ v i t is argued that the small wave cannot penetrate the large wave unchanged. The large wave suffers significant distortion (bending) in the interaction, but the small wave, if it penetrates at all, must have an amplitude O(ε). This is the main aspect of the problem which cannot be completely determined using the present methods. The difficulty can be traced to the large solitary wave which is not known in closed form: only the exponential behaviour in the ‘tail’ is used explicitly in this work.

Oblique interaction of waves with shocks

This note investigates the oblique interaction between plane waves and shocks in materials described by a system of conservation equations. The interaction turns out to be accounted for by a relation allowing the explicit determination of the amplitudes of the emergent modes together with the acceleration and the angular velocity of the shock.

Computer simulation of a Cerenkov interaction between obliquely propagating whistler mode waves and an electron beam

A Cerenkov type wave-particle interaction between an obliquely propagating whistler mode wave and an electron beam has been investigated by a computer simulation experiment. A one-dimensional electromagnetic simulation code which is especially designed to deal with the obliquely propagating wave is used. Two cases of such a wave-particle interaction have been investigated. In the first case no initial wave is assumed to exist in the simulation model space. In the second case, an obliquely propagating whistler mode wave with a finite amplitude is initially assumed to exist. In both cases after linear wave growth, nonlinear saturation and amplitude oscillation are observed. This nonlinear behavior of the interaction can be interpreted by particle trapping in the wave potential well just as in the case of longitudinal electrostatic waves. Waves excited by Cerenkov radiation from the beam electrons develop as a coherent wave due to the instability and propagate in the magnetized plasma. By the oblique beam-plasma interaction, electromagnetic waves can be amplified. Amplification of the waves observed by this computer simulation may be closely related to VLF hiss noise occurring in the earth’s magnetosphere.

Application of the wavenumber jump condition to the normal and oblique interaction of a plane acoustic wave and a plane shock

A kinematic approach is considered whereby the wavenumber jump conditions in conjunction with the appropriate dispersion relations is applied to the investigation of the normal and oblique interaction of a plane acoustic wave with a plane shock wave. For the normal interaction of an acoustic wave with a stationary plane shock a logarithmic shift in the wave spectra is obtained. For the normal interaction with a moving shock front it is shown that for shock Mach numbers above a critical value, the frequency of the transmitted wave becomes negative. This results in the fact that the crests of the transmitted signal arrive at a fixed observer in a reverse order to their generation. Finally, the oblique interaction of an acoustic wave with a stationary shock is considered. The “Snell's Law” for the transmitted wave is derived and two special angles of incidence are identified. The first is a no-refraction angle: i.e., the transmitted wave angle is the same as the incident wave angle. The second is a critical angle such that for incident angles greater than this critical angle there is no transmitted wave. A necessary and sufficient condition for the existence of a transmitted wave is derived in terms of the speed of sound and Mach number of the fluid and the frequency and tangential wavenumber component of the incident wave. The dynamics aspects of the interaction concerning the determination of the frequency independent transmission coefficients and shock displacements are determined for the simple case of the normal interaction with a moving shock as an illustration.

Obliquely interacting solitary waves

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α1 and α2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin2ϕ α1,2 or strong if ϕ2 = O(α1,2). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O(α1α2), then is determined from these basic solutions.Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α2-α1 > (2ϕ)2 but not if |α2-α1| (2ϕ)2 (e.g. the reflexion problem, for which the interacting waves are images and α2 = α1). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech2 profiles yield singular outgoing waves with - csch2 profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.

Nonlinear interaction of discontinuities in the solar wind and the origin of slow shocks

Nonlinear oblique interactions between tangential discontinuities (T), rotational discontinuities (A) and fast shocks (S/subf/) are investigated with the aim of explaining the origin of slow shocks in the solar wind. First, it is shown that the interaction processes S/subf/T and possibly S/subf/A are attractive candidates for explaining the origin of slow shocks consistent with observations, i.e., at the right place in the rising portion of solar wind streams and with large deviations of their normals from the radial direction. Second, heating of the solar wind by damping of fast waves is strongly enhanced by their partial conversion into slow waves. Third, it is expected theoretically that the downstream region of flare-produced shock will be considerably richer in fast and slow waves than the upstream region. Fourth, corotating shocks may produce increased numbers of slow shocks just outside 1 AU. Finally, the A moving through the solar wind ensemble of T will break up into isolated surface segments the edges of which radiate all types of MHD waves. (AIP)

Nonlinear oblique interaction of interplanetary tangential discontinuities with magnetogasdynamic shocks

The nonlinear oblique interaction between a plane tangential discontinuity (TD) and a plane fast magnetogasdynamic shock is treated generally and then applied to the interplanetary TD-bow shock interaction. A number of similarity relations are found for the general problem. For the frequent directional TD's the interaction is strongest if the directional change is 90° and one of the magnetic field vectors is parallel to the shock. Bow shock motions of up to several tens of kilometers per second are induced by these TD's. Tangential discontinuities with density jumps held responsible for geomagnetic sudden impulses lead to a much stronger interaction with the bow shock with nonnegligible strength of the newly generated wave. The oblique interaction requires weaker TD's to produce observed bow shock velocities. It is also shown that complex nonlinear wave patterns very often develop, which have not been predicted in the past in this context.

Microplate tectonics, oblique collisions, and evolution of the Hercynian orogenic systems

Plate boundaries are rarely straight lines, and interplate movements are rarely at right angles. The consequences of oblique interactions at both continent-ocean and continent-continent irregular margins are considered theoretically, and the results are compared with various active plate boundaries. We use an oblique-collision model to explain the evolution of the late Paleozoic orogens of the “Atlantic-Mediterranean‘ region. In particular, the model reconciles paleontologic requirements for a “Central European Ocean” with the lack of geotectonic evidence for closure of such an ocean. In addition, the model explains the inhomogeneous nature of these orogens, from the southern Appalachians and the Mauritanides to the Urals. The model is highly speculative, but it does rationalize the irregular distributions (in time and in space) of ophiolite, island-arc sequences, major strike-slip faults, and metallogenic belts.

Oblique shock-sound interaction at a freestream Mach number of about20 in helium.

Detailed experimental measurements of the characteristics and level of pre- and postshock disturbances obtained for typical disturbance modes and shock angles are presented to aid in the evaluation of transition studies conducted in 'noisy' hypersonic wind tunnels. The presented data suggest that the shock is probably generating total temperature fluctuations in a fashion directly related to shock strength. Since in high supersonic and hypersonic flows the total temperature fluctuations are mainly contained in velocity fluctuations, it may be concluded that the predominant disturbances generated are vorticity as long as the postshock flow is at a sufficiently high Mach number.-

Laminar boundary layer oblique shock wave interaction on flat and curved plates

Ce memoire decrit quelques recherches experimentales et theoriques sur l'interaction des ondes de choc obliques avec les couches limites laminaires. Une serie de mesures detaillees de pressions et du frottement a ete faite dans la region d'interaction dans un ecoulement le long d'une plaque plane. Un modele du mecanisme en a ete deduit, dont les predictions s'accordent raisonnablement bien avec les resultats des experiences. Les memes idees ont ete appliquees a l'interaction sur une surface courbe, et les resultats s'accordent encore avec les experiences. Une etude simplifiee de la region de stabilite de l'interaction sous l'action d'oscillations de faible amplitude a ete faite.