Problem of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

Popov, M. M.

doi:10.1007/bf01455058

Cited by 35 publications

(39 citation statements)

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“…Since the proof of Theorem 2.2 can be done in the same way as that of Popov's result (cf. [4]), we omit the proof.…”

Section: Herementioning

confidence: 99%

“…When tc=l, Theorem 2.2 is exactly the same as Popov's result (cf. [4]). Since the proof of Theorem 2.2 can be done in the same way as that of Popov's result (cf.…”

Section: Herementioning

confidence: 99%

“…After some preparations, we state the results of [1], [4] and our results in §2. §3 and §4 are devoted to the proof of our main theorem.…”

Section: § 1 Introductionmentioning

confidence: 99%

“…On the other hand, Popov [4] considered the case /f 0 CP)=0, Ki(P^Q. He constructed an asymptotic solution u to (1.1) and (1.2) in a relatively open neighborhood GdQ of P with the following property.…”

Section: § 1 Introductionmentioning

confidence: 99%

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Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Nakamura¹,

Shibata²,

Tanuma³

1989

Publ. Res. Inst. Math. Sci.

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“…Since the proof of Theorem 2.2 can be done in the same way as that of Popov's result (cf. [4]), we omit the proof.…”

Section: Herementioning

confidence: 99%

“…When tc=l, Theorem 2.2 is exactly the same as Popov's result (cf. [4]). Since the proof of Theorem 2.2 can be done in the same way as that of Popov's result (cf.…”

Section: Herementioning

confidence: 99%

“…After some preparations, we state the results of [1], [4] and our results in §2. §3 and §4 are devoted to the proof of our main theorem.…”

Section: § 1 Introductionmentioning

confidence: 99%

Section: § 1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Nakamura¹,

Shibata²,

Tanuma³

1989

Publ. Res. Inst. Math. Sci.

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“…

The propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point is studied. The function ~o in the present case is given by 0 *We shall not consider here the derivation of the corresponding formulas, since it is altogether analogous to the derivation presented in [3] for the case of a simple zero of the effective curvature of the boundary.

The results of the computations are presented as shadow figures.…”

mentioning

confidence: 99%

Whispering gallery waves in a neighborhood of a flat point of a concave boundary

Popov¹,

Pshenchik²

1983

J Math Sci

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The propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point is studied. As in [4], the problem that arises for an equation of Schrodinger type is solved by the method of grids. The results of the computations are presented as shadow figures.In this work we consider the two-dimensional problem of the propagation of whispering gallery waves near a concave (from the side of the wave field) boundary having a flat point,i.e., a point at which the curvature of the boundary has a zero of multiplicity two. If the medium is inhomogeneous the role of the curvature of the boundary is played by the so-called effective curvature of the boundary (see, e.g., [i, 2]). In this case the problem consists in describing the behavior of whispering gallery waves in neighborhood of the zero (of multiplicity two) of the effective curvature of the boundary.In both these cases we obtain the same problem for the leading term of the asymptotics of the wave field at high frequencies.The initial assumptions consist in the following. It is assumed that the wave field satisfies the Helmholtz equation with a variable speed C (MJ and vanishes on the reflecting (smooth) boundary F , i.e., (~+Lo'~C-~U=0 ,UIF =0 We denote by ~ and ~, respectively, the arc length of the curve ~ and the coordinate along the normal to ~ at the point 6 (~>0 on the side of ~ , where the wave process is considered). Suppose that the effective curvature ~(3) of the boundary ~ in a neighborhood of the point ~=0 has the form ~.~)=~6~ + 0(33) with a constant ~>0. It is assumed that for 8~0 there is one whispering gallery wave, and it is required to describe its behavior for 6~0 9Using the boundary-layer method,* we obtain the following problem for the leading term of the wave field U as oO--oo . In the region ~0 ,~6 ~oo,+oo) it is required to find a solution ~(~,~# of the equation co.1) ~f--~D~ +which vanishes at 3C=0 , foreach te(-oo,+~) is square-integrable on the semiaxis ~=~0 (i.e., W e[~(0,o~) ), and as ~---~o has prescribed asymptotics ~o in the sense of 6~(0,oo) , ao (i.e, Jl~-~ol~d~--0 as ~---oo ). The function ~o in the present case is given by 0 *We shall not consider here the derivation of the corresponding formulas, since it is altogether analogous to the derivation presented in [3] for the case of a simple zero of the effective curvature of the boundary.

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The Boundary Layer Method

Bouche

Molinet

Mittra

1997

Asymptotic Methods in Electromagnetics

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Problem of whispering gallery waves in a neighborhood of a simple zero of the effective curvature of the boundary

Cited by 35 publications

References 0 publications

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Whispering Gallery Waves in a Neighborhood of a Higher Order Zero of the Curvature of the Boundary

Whispering gallery waves in a neighborhood of a flat point of a concave boundary

The Boundary Layer Method

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