Existence theorems for trapped modes

Evans, D. V.; Levitan, M. A.; Vassiliev, Dmitri

doi:10.1017/s0022112094000236

Cited by 326 publications

(288 citation statements)

References 6 publications

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“…It might therefore be informative to consider the effect of a cut perpendicular to the guide but which only partially extends across the guide, a relatively simple problem to investigate mathematically. Finally, it should be possible to provide rigourous existence and non-existence proofs for trapped waves, possibly using variational methods similar to those used by [25] in proving the existence of trapped waves in acoustic waveguides. …”

Section: Discussionmentioning

confidence: 99%

Trapped waves in thin elastic plates

Porter

2007

Wave Motion

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In this paper we present new results which illustrate the existence of trapped modes on a thin elastic plate, which is infinitely long, of finite uniform width and simply-supported along the two parallel edges. The plates contain a circular cut-out on the centreline of the strip whose edges are either free or clamped. The trapped waves describe time-harmonic vibrations of a frequency below a cut-off frequency which are localised to the circular cut-out and decay along the strip. Results show that whilst no trapped waves occur for a circular hole with a clamped edge, for a hole with a free edge trapped waves occur in all four possible modes of symmetry.

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Section: Discussionmentioning

confidence: 99%

Trapped waves in thin elastic plates

Porter

2007

Wave Motion

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“…Next, in §3, we use the variational principle to prove the existence of eigenvalues located off the continuous spectrum. This way of argumentation is traditional; see, e.g., [5][6][7], where it was used for the study of eigenfunctions in waveguide-like domains.The author is deeply grateful to V. M. Babich for setting the problem and for his permanent attention, and to M. Sh. Birman for valuable remarks.…”

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confidence: 99%

Surface wave running along the edge of an elastic wedge

Камоцкий¹

2008

St. Petersburg Math. J.

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Abstract. The existence of the waves mentioned in the title is proved for the case of an acute wedge. §1. IntroductionFor the first time, the existence of surface waves in elastic wedges was conjectured in the papers [1,2] on the basis of results of numerical calculations. It was discovered that such waves oscillate along the edge of the wedge, decaying exponentially far from the edge, and that the velocity of their propagation depends on the wedge's angle and can be considerably less than that of the Rayleigh waves. This latter fact opens a way to use such waves in lag lines.After the surface waves had been discovered, the corresponding studies developed in several directions; see, e.g., [3,4]. However, the proof that surface waves do exist remained an open question. In the present paper we prove the existence of such waves in acute wedges. Our method is based on variational considerations. Namely, we show that the frequencies for which surface waves may arise correspond to the eigenvalues of a certain selfadjoint operator. Next, in §3, we use the variational principle to prove the existence of eigenvalues located off the continuous spectrum. This way of argumentation is traditional; see, e.g., [5][6][7], where it was used for the study of eigenfunctions in waveguide-like domains.The author is deeply grateful to V. M. Babich for setting the problem and for his permanent attention, and to M. Sh. Birman for valuable remarks. §2. Setting of the problemLet Ω = {x 1 , x 2 ) : r > 0, φ ∈ (0, Φ)} be an angle on the plane R 2 ; here (r, φ) are polar coordinates. We consider the problem of steady-state oscillations of an isotropic elastic wedge K = Ω × R with free boundary:where n, m = 1, 2, 3,are the components of the stress tensor, λ and µ are the Lamé coefficients, δ nm is the Kronecker symbol, U = (U 1 , U 2 , U 3 ) is the vector of displacements, and (ν 1 , ν 2 , ν 3 ) is the outward normal vector. The operators on the left-hand sides in (2.1) and (2.2) will be denoted 2000 Mathematics Subject Classification. Primary 74J15.

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“…This is because, even if the existence of a trapped mode in a given subregion of the fluid is established, in general this trapped mode does not correspond to a trapped mode in the full fluid domain. This is in contrast to the situation in a wave guide, where Evans, Levitin & Vassiliev (1994) were able to exploit the symmetry of the guide and body configuration to show that a trapped mode in the half-guide which satisfies φ = 0 on the centreline is also a trapped mode in the full guide. Further work is in progress to determine whether or not the existence of nodal lines can be used to prove the existence of trapped modes in the two-dimensional water-wave problem.…”

Section: Discussionmentioning

confidence: 74%

“…Ursell (1951) also showed that waves may be trapped above a submerged, horizontal cylinder and propagate along the cylinder. More recently Evans, Levitin & Vassiliev (1994) proved that trapped modes exist in the presence of bodies which are symmetrically placed in waterwave channels or guides. Unlike the modes found by McIver (1996), these other types of trapped modes occur at frequencies which are less than a 'cut-off' value, below which waves cannot propagate to infinity.…”

Section: Introductionmentioning

confidence: 99%

“…There is not, in fact, a cut-off frequency below which no waves can propagate to infinity in a waveguide but Evans, Levitin & Vassiliev (1994) were able to introduce an artificial cut-off frequency by restricting attention to motion which is antisymmetric about the centreline of the channel. The restriction to antisymmetric motion is equivalent to the requirement that the centreline of the channel should be a nodal line on which φ = 0.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness below a cut-off frequency for the two-dimensional linear water-wave problem

McIver

1999

Proc. R. Soc. Lond. A

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The two-dimensional velocity potential associated with the scattering of linear waves in water of finite but non-uniform depth is shown to be unique for Kh max ≤ 1. Here h max is the maximum depth of the fluid and K = ω 2 /g, where ω is the angular frequency and g is the acceleration due to gravity. Uniqueness is established by proving that the homogeneous boundary value problem for the velocity potential φ has only the trivial solution φ = 0 when Kh max ≤ 1.It is first shown that there is a nodal line in the fluid on which φ = 0 and that this line has one end on the mean free surface and asymptotes to a horizontal line as x → ∞. The line forms the lower boundary of a fluid region contained between itself and the free surface. The potential in this region is shown to be identically equal to zero for Kd max < 1, where d max is the maximum depth of the layer, by bounding the potential energy in the region by a fraction of the kinetic energy. By analytic continuation the potential is zero in the whole fluid for this range of frequencies and as d max < h max , uniqueness is established for Kh max ≤ 1. The critical value of Kd max = 1 corresponds to a cut-off frequency below which waves cannot propagate in a uniform layer of depth d max with φ = 0 on the lower boundary.The analysis is extended to prove uniqueness for the same range of frequencies when any number of non-bulbous, surface-piercing bodies are in the fluid or a single surface-piercing or submerged body with arbitrary shape.

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Existence theorems for trapped modes

Cited by 326 publications

References 6 publications

Trapped waves in thin elastic plates

Trapped waves in thin elastic plates

Surface wave running along the edge of an elastic wedge

Uniqueness below a cut-off frequency for the two-dimensional linear water-wave problem

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