An explanation of a difficulty with Huygens' secondary wavelets

1996 ◽  
Vol 74 (5-6) ◽  
pp. 236-239 ◽  
Author(s):  
J. M. Daniels
Keyword(s):  
Wave Front ◽  
Travelling Wave ◽  
Diffraction Integral

Huygens' construction for the propagation of a wave front using secondary wavelets gives rise to a paradox, that a backward travelling wave is predicted, but this wave does not exist. This paradox is resolved by postulating two wavelets that cancel in the backward direction. Kirchhoff's diffraction integral is examined in the light of this postulate.

scholarly journals Thermally controlled glacier surging

2001 ◽  
Vol 47 (159) ◽  
pp. 527-538 ◽  
Author(s):  
A. C. Fowler ◽  
Tavi Murray ◽  
F. S. L. Ng
Keyword(s):  
Simple Model ◽  
Wave Front ◽  
Thermal Regime ◽  
Thick Layer ◽  
Travelling Wave ◽  
Yukon Territory ◽  
Wave Fronts ◽  
Ice Flow ◽  
Trapridge Glacier ◽  
Travelling Wave Front

AbstractBakaninbreen in Svalbard and Trapridge Glacier in Yukon Territory, Canada, are two prominent examples of surging glaciers which are thought to be controlled by their thermal regime. Both glaciers have developed large bulges which have propagated forward as travelling wave fronts, and which are thought to divide relatively stagnant downstream cold-based ice from faster-moving warm-based upstream ice. Additionally, both glaciers are underlain by a wet, metres thick layer of deforming till. We develop a simple model for the cyclic surging behaviour of these glaciers, which interrelates the motion of the ice and till through a description of the subglacial hydrology. We find that oscillations (surges) can occur if the subglacial hydrological transmissivity is sufficiently low and the till layer is sufficiently thin, and we suggest that these oscillations are associated with the development and propagation of a travelling wave front down the glacier. We therefore interpret the travelling wave fronts on both Trapridge Glacier and Bakaninbreen as manifestations of surges. In addition, we find that the violence of the surge in the model is associated with the resistance to ice flow offered by undulations in the bed, and the efficiency with which occasional hydrological events can release water accumulated at the glacier sole.

Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion

2004 ◽  
Vol 134 (3) ◽  
pp. 579-594 ◽  
Author(s):  
Ming Mei ◽  
Joseph W.-H. So ◽  
Michael Y. Li ◽  
Samuel S.P. Shen
Keyword(s):  
Asymptotic Stability ◽  
Wave Front ◽  
Nonlinear Stability ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Nicholson’S Blowflies Equation ◽  
Travelling Wave Front

This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.

Existence and uniqueness of travelling waves for a neural network

1993 ◽  
Vol 123 (3) ◽  
pp. 461-478 ◽  
Author(s):  
G. Bard Ermentrout ◽  
J. Bryce McLeod
Keyword(s):  
Neural Network ◽  
Wave Front ◽  
Existence And Uniqueness ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Steady States ◽  
Stable States ◽  
One Dimensional ◽  
Travelling Wave Front

SynopsisA one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.

TRAVELLING WAVES IN INVASION PROCESSES WITH PATHOGENS

2008 ◽  
Vol 18 (03) ◽  
pp. 325-349 ◽  
Author(s):  
ARNAUD DUCROT ◽  
MICHEL LANGLAIS
Keyword(s):  
Wave Front ◽  
Allee Effect ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Host Population ◽  
Travelling Wave Solutions ◽  
Front Speed ◽  
Bistable Dynamics ◽  
Si Model

This work is devoted to the study of a singular reaction–diffusion system arising in modelling the introduction of a lethal pathogen within an invading host population. In the absence of the pathogen, the host population exhibits a bistable dynamics (or Allee effect). Earlier numerical simulations of the singular SI model under consideration have exhibited stable travelling waves and also, under some circumstances, a reversal of the wave front speed due to the introduction of the pathogen. Here we prove the existence of such travelling wave solutions, study their linear stability and give analytical conditions yielding a reversal of the wave front speed, i.e. the invading host population may eventually retreat following the introduction of the lethal pathogen.

Diffraction of Plane Electromagnetic Waves by a Cylinder

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Wave Front ◽  
Diffraction Pattern ◽  
Electromagnetic Waves ◽  
Plane Waves ◽  
Spherical Wave ◽  
Electric Polarization ◽  
Secondary Wave ◽  
Diffraction Integral ◽  
Incident Plane ◽  
Radio Signals

This chapter discusses the mathematics underlying the diffraction of plane electromagnetic waves by a cylinder. Radio signals may undergo diffraction. A signal that encounters an obstacle tends to travel around it, suggesting that a signal may be received from a transmitter even though it may be “shaded” by a large object between them. This phenomenon is best explained by the Huygen's principle, according to which each point on a spherical wave front can be interpreted as a source of a secondary wave front. The chapter first examines electric polarization by considering normally incident plane waves incident on the cylinder (parallel to the z-axis of the cylinder) before turning to classical diffraction, Huygen's principle, and the Kirchhoff-Huygens diffraction integral. It also shows the derivation of the generalized Airy diffraction pattern.

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