Water waves trapped by thin submerged cylinders: Exact solutions

Garibay, F.; Zhevandrov, P.

doi:10.1134/s1061920815020041

Cited by 4 publications

(14 citation statements)

References 18 publications

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“…In all our considerations, α , β , k , a , b are fixed numbers; we do not investigate, for example, the limit α →0 (see Nazarov et al) since this limit is singular. We note only that does not pass into the corresponding formula for the one‐layer case since for α = 0 the part of the continuous spectrum corresponding to the interval ( λ 1 , λ 2 ) simply disappears.…”

Section: Formulation and Main Resultsmentioning

confidence: 96%

“…The procedure of obtaining the system for ϕ , ψ and θ is quite standard (one has to calculate the limit for as ( x , y ) tends to a point of the boundary, cf Romero Rodríguez and Zhevandrov and Garibay and Zhevandrov). Using the formulas for the Fourier transform of the function K 0 and its derivative, one finally comes to the following system for

\overset{ϕ}{true}

\overset{ψ}{true}

and θ (we omit these tedious calculations):

\overset{ψ}{true} + \frac{α}{β λ} e^{- b τ} \overset{ψ}{true} = ε \int_{- π}^{π} M_{4} false(p, t false) θ false(t false) d t,

- e^{- b τ} \overset{ϕ}{true} + \frac{1}{λ} false(1 - \frac{1}{β} \frac{τ - λ}{τ + λ} false) \overset{ψ}{true} = ε \int_{- π}^{π} M_{5} false(p, t false) θ false(t false) d t,

θ + {\overset{M}{true}}_{1} θ = \int M_{2} false(t, p false) \overset{ϕ}{true} false(p false) d p + \int M_{3} false(t, p false) \overset{ψ}{true} false(p false) d p,

where

M_{2} (t, p) = \frac{1}{2 π} \frac{λ + τ}{τ} e^{- (a - ε Y (t)) τ + i ε p X (t)},

M_{3}

…”

Section: Exact Solution and The Proof Of Theorem 21mentioning

confidence: 99%

“…Our final goal is to find the leading term of the solution of with respect to ε . This is done exactly as in Garibay and Zhevandrov by means of the Taylor expansions in ε , p , σ for the functions M 2,3,4,5 , the fact that

{(1 + {\hat{M}}_{1})}^{- 1} = {(1 + {\hat{M}}_{1}^{(0)})}^{- 1} + O (ε^{2} \ln ε),

the use of formulas and the use of lemma 3.4 from Garibay and Zhevandrov . We omit the corresponding rather lengthy calculations which lead to formula .…”

Section: Exact Solution and The Proof Of Theorem 21mentioning

confidence: 99%

“…The procedure of obtaining the system for , and is quite standard (one has to calculate the limit for (16) as (x, y) tends to a point of the boundary, cf Romero Rodríguez and Zhevandrov 13 and Garibay and Zhevandrov 15 ). Using the formulas for the Fourier transform of the function K 0 and its derivative, one finally comes to the following system for̃, and (we omit these tedious calculations):…”

Section: Reduction To Integral Equationsmentioning

confidence: 99%

“…In Garibay and Zhevandrov, the problem of waves trapped by a submerged thin cylinder with fairly arbitrary cross‐section in a one‐layer fluid without any symmetry conditions was considered. This resulted in finding exact solutions in the form of series in powers of ε and

ε \ln ε

, where ε characterises the “thinness” of the cylinder, by means of a technique similar to previous studies .…”

Section: Introductionmentioning

confidence: 99%

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Water waves trapped by thin submerged cylinders in a two‐layer fluid: Discrete eigenvalue

Rodríguez

Zhevandrov

2018

Math Methods in App Sciences

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Exact solutions of the linear water‐wave problem describing oblique water waves trapped by a submerged horizontal cylinder of small (but otherwise fairly arbitrary) cross‐section in a two‐layer fluid are constructed in the form of convergent series in powers of the small parameter characterising the “thinness” of the cylinder. The terms of this series are expressed through the solutions of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder.

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Section: Formulation and Main Resultsmentioning

confidence: 96%

\overset{ϕ}{true}

\overset{ψ}{true}

and θ (we omit these tedious calculations):

\overset{ψ}{true} + \frac{α}{β λ} e^{- b τ} \overset{ψ}{true} = ε \int_{- π}^{π} M_{4} false(p, t false) θ false(t false) d t,

- e^{- b τ} \overset{ϕ}{true} + \frac{1}{λ} false(1 - \frac{1}{β} \frac{τ - λ}{τ + λ} false) \overset{ψ}{true} = ε \int_{- π}^{π} M_{5} false(p, t false) θ false(t false) d t,

θ + {\overset{M}{true}}_{1} θ = \int M_{2} false(t, p false) \overset{ϕ}{true} false(p false) d p + \int M_{3} false(t, p false) \overset{ψ}{true} false(p false) d p,

where

M_{2} (t, p) = \frac{1}{2 π} \frac{λ + τ}{τ} e^{- (a - ε Y (t)) τ + i ε p X (t)},

M_{3}

…”

Section: Exact Solution and The Proof Of Theorem 21mentioning

confidence: 99%

{(1 + {\hat{M}}_{1})}^{- 1} = {(1 + {\hat{M}}_{1}^{(0)})}^{- 1} + O (ε^{2} \ln ε),

the use of formulas and the use of lemma 3.4 from Garibay and Zhevandrov . We omit the corresponding rather lengthy calculations which lead to formula .…”

Section: Exact Solution and The Proof Of Theorem 21mentioning

confidence: 99%