XVI.—Creeping Waves in an Inhomogeneous Medium

SynopsisThis paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.

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7.—High-frequency Scattering in a Certain Stratified Medium

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0080454100009444 ◽

1974 ◽

Vol 72 (2) ◽

pp. 93-107

Author(s):

W. G. C. Boyd

Keyword(s):

Real Axis ◽

High Frequency ◽

Fourier Integral ◽

Stratified Medium ◽

Plane Boundary ◽

Airy Functions ◽

Asymptotic Field ◽

The Real ◽

Asymptotic Nature ◽

Residue Series

SynopsisThe propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.

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Femtosecond laser interaction with photo-refractive crystals: High frequency field and internal field contributions to the refractive index changes

CLEO/Europe - EQEC 2009 - European Conference on Lasers and Electro-Optics and the European Quantum Electronics Conference ◽

10.1109/cleoe-eqec.2009.5196280 ◽

2009 ◽

Author(s):

S. Juodkazis ◽

E.G. Gamaly ◽

V. Mizeikis ◽

H. Misawa ◽

A.V. Rode ◽

...

Keyword(s):

Refractive Index ◽

Femtosecond Laser ◽

High Frequency ◽

Internal Field ◽

High Frequency Field ◽

Laser Interaction ◽

Frequency Field ◽

Index Changes

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Integrals with a large parameter: water waves on finite depth due to an impulse

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1995.0072 ◽

1995 ◽

Vol 450 (1938) ◽

pp. 67-87 ◽

Cited By ~ 12

Keyword(s):

Asymptotic Expansion ◽

Stationary Phase ◽

Water Waves ◽

Three Dimensional ◽

Finite Depth ◽

Three Dimensions ◽

Dimensional Problem ◽

Airy Functions ◽

Asymptotic Results ◽

Conventional Solution

The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.

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Uniform asymptotic representation of a field of reflected and head waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1969.0206 ◽

1969 ◽

Vol 313 (1515) ◽

pp. 477-490 ◽

Cited By ~ 5

Keyword(s):

Asymptotic Expansion ◽

Wave Equation ◽

Asymptotic Representation ◽

Asymptotic Series ◽

Spherical Wave ◽

Plane Boundary ◽

Airy Functions ◽

Uniform Asymptotic Expansion ◽

Head Waves ◽

Ray Series

A uniform asymptotic expansion for integrals of the type ʃ +∞ -∞ u ½ F ( u ) exp {i kψ ( u )}d u has been obtained in terms of generalized Airy functions, which are the solutions of the equation V"( z )+ z 2 V ( z ) = 0. This result is applied to the construction of a uniform asymptotic representation of a solution of the wave equation in the case of reflexion of a spherical wave from a plane boundary in a region including a critical ray. This asymptotic series may be divided into two series corresponding to reflected and head waves respectively, which are transformed into the ray series for these waves far from the critical ray and reduced to the expressions given by Brekhovskikh (1960) in the vicinity of the critical ray.

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