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Journal ArticleDOI

On the radiation of short surface waves by a heaving circular cylinder

G. Alker1
01 Jul 1975-Journal of Engineering Mathematics (Springer Science and Business Media LLC)-Vol. 9, Iss: 3, pp 197-205
TL;DR: In this paper, it was proved that there are no eigensolutions of the infinite vertical barrier problem containing waves which are purely outgoing, and it was shown how this can be used to predict the wave amplitude to a higher order than that of the matching solution.
Abstract: A long circular cylinder half immersed in the free surface of an ideal fluid undergoes small time periodic motions. The method of matched asymptotic expansions is used to give a solution in the high frequency limit. Of particular interest are the surface waves generated by this motion, and a three term asymptotic series for their amplitude is found. It is proved that there are no eigensolutions of the infinite vertical barrier problem containing waves which are purely outgoing, and it is shown how this can be used to predict the wave amplitude to a higher order than that of the matching solution.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential.
Abstract: The problem of the scattering of a surface wave in a nonviscous, incompressible fluid of infinite depth by a fully submerged, rigid, stationary sphere has been reduced to the solution of an infinite set of linear algebraic equations for the expansion coefficients in spherical harmonics of the velocity potential. These equations are easily solved numerically, so long as the sphere is not too close to the surface. The approach has been to formulate the problem as an integral equation, expand the Green's function, the velocity potential of the incident wave, and the total velocity potential in spherical harmonics, impose the boundary condition at the surface of the sphere, and carry out the integrations. The scattering cross section has been evaluated numerically and is shown to peak for values of the product of radius and wave number somewhat less than unity. Also, the Born approximation to the cross section is obtained in closed form.

17 citations

Journal ArticleDOI
G. Alker1

5 citations


Cites methods from "On the radiation of short surface w..."

  • ...The method of solution is the systematic method of matched asymptotic expansions developed by Van Dyke (1964) as applied t o problems involving short surface waves by Leppington (1972, 1973a, b ) , Ayad & Leppington (1977) and Alker (1974, 1975)....

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Journal ArticleDOI
TL;DR: In this paper, the authors extend the known asymptotic forms, in heave and sway, of the wave amplitude radiated by bodies at high frequency to other geometries by way of the solution of certain potential problems.
Abstract: The aim of this work is to extend the known asymptotic forms, in heave and sway, of the wave amplitude radiated by bodies at high frequency. Both two- and three-dimensional geometries will be considered, the prototype problems being the circular cylinder and sphere respectively, each with its centre in the mean free surface. The method is, in principle, applicable to other geometries by way of the solution of certain potential problems much simpler than the finite-frequency surface-wave problem.

4 citations


Cites methods from "On the radiation of short surface w..."

  • ...T he h e a v i n g s e m i c i r c u l a r c y l i n d e r As an example of the method, consider the geometry studied by Ursell (1953), Alker (1975) and Rhodes-Robinson (1982); later the generalization to other bodies will be described....

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  • ...For the heaving semicircular cylinder this was first derived by Ursell (1953) using rigorous arguments, and later by Rhodes-Robinson (1970a, b), Hermans (1972) and Alker (1975) using simpler, but non-rigorous, methods....

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  • ...(11.4) The result (11.1), given here for the first time, was derived by integrating the matched asymptotic solution of Alker (1975); this is therefore not a possible technique for other two-dimensional geometries without an amount of work at least as great as his....

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Journal ArticleDOI
TL;DR: In this article, matched asymptotic expansions were used to extend the short-wave transmission coefficient T by the addition of the terms of order 1/N 5, (log N ) 2 /N 6 and log N / N 6 as N → ∞ (where N = the length of the cylinder radius).
Abstract: The method of matched asymptotic expansions is used to extend the short-wave asymptotics of the transmission coefficient T by the addition of the terms of order 1/ N 5 , (log N ) 2 / N 6 and log N / N 6 as N → ∞ (where N = wavenumber times cylinder radius). The result is the formula \begin{eqnarray*} T &=& \frac{2{\rm i}}{\pi N^4}\exp (-2{\rm i}N)\left[1+\frac{4\log N}{\pi N}-\frac{4}{\pi N} \bigg(2-\gamma-\log 2+\frac{{\rm i}\pi}{8}\bigg)+\frac{8(\log N)^2}{\pi^2N^2}\right.\\ && \left.-\frac{8\log N}{\pi^2N^2}\bigg(5-2\gamma - \log 4+\frac{{\rm i}\pi}{4}\bigg)\right] + O\bigg(\frac{1}{N^6}\bigg)\quad {\rm as}\;N\rightarrow \infty \end{eqnarray*} (where γ = Euler's constant). The first term above is that derived rigorously by Ursell (1961) using an integral-equation method; the second term is that added by Leppington (1973) using matched asymptotic expansions; and the next three terms are those derived in this paper. Significant agreement between numerical values of T obtained from the completed fifth-order asymptotics and those obtained using Ursell's multipole expansions is demonstrated for 8 [les ] N [les ] 20 (table 2). The extensions of the perturbation expansions for the potential in the various fluid sub-domains (used in the method of matched expansions) provide some interesting cross-checks, between the solutions for potentials occurring later in the series and determined at advanced matching stages, with those for potentials occurring earlier on and determined independently at an earlier stage in the matching process. Some examples are given.

1 citations

References
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Book
01 Jan 1975

2,966 citations


"On the radiation of short surface w..." refers methods in this paper

  • ...This method was intuitively similar to Van Dyke's [ 3 ] method of matched asymptotic expansions in that the solution was approximated by different asymptotic expansions in different regions....

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Journal ArticleDOI
TL;DR: In this paper, a modified version of Van Dyke's matching principle was used to match the inner and outer asymptotic expansions of a semi-infinite rigid plate under irradiation by a plane acoustic wave.
Abstract: Singular perturbation methods are used to determine the field diffracted by a semi-infinite rigid plate of thickness $2a$ under irradiation by a plane acoustic wave at wavenumber $k$. Six terms of both an outer and an inner expansion in the small parameter $\epsilon $ = $ka$ are calculated in closed form, yielding simple results for the far-field directivity pattern. The outer series is determined by certain eigensolutions, and by a sequence of straightforward Wiener-Hopf problems, while the inner terms are all obtained from a simple conformal mapping. Previous discussions of this problem (e.g. Jones 1953) involve the formulation of a modified Wiener-Hopf equation, and reduce the problem to that of inverting an infinite matrix with elements dependent upon $\epsilon $. Jones has given a numerical inversion of the truncated $4\times 4$ matrix in the limit $\epsilon $ $\rightarrow $ $0$. Here we obtain exact expressions A$\_{2n+1}$ = $\frac{1}{2(2n+1)}$ {J$\_{n}$($n+\frac{1}{2}$) - J$_{n+1}$($n+$$\frac{1}{2}$)} for Jones's variables, and prove that they satisfy his infinite system of linear equations. It is also shown that to O($\epsilon $$^{2}$ ln$^{2}$ $\epsilon $) the plate may be replaced by a duct longer than the plate by an amount $L$ = ($a$/$\pi $) $ln2$, in agreement with Jones's numerical value of $L=0.22a$, together with the monopole field necessary to annul the effect of plane wave propagation down the duct. The principle used here to match the inner and outer asymptotic expansions is a slightly, though significantly, modified version of the one used extensively by Van Dyke (1964). In the present problem Van Dyke's matching principle appears to hold, in that matching can be formally accomplished, but leads to erroneous results violating the reciprocal theorem. Accordingly, an appendix here gives a discussion and justification, in elementary terms, of the proposed modified asymptotic matching principle.

83 citations


"On the radiation of short surface w..." refers methods in this paper

  • ...The matching principle to be used is a modified version of that proposed by Van Dyke [-3] : the modification due to Crighton and Leppington [ 9 ] stipulates that all terms of the form e~ log e, e~ log log e must be grouped with e~ for matching purposes....

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Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of finding the resulting fluid motion when the parameter N = Ω(n σ 2 n/g ) is large; the method of an earlier paper (Ursell 1949) is then unworkable; the present solution is made to depend on an integral equation (3$\cdot $15) which can be chosen to have a kernel tending to zero with N$^{-1}$, and which is solved by iteration.
Abstract: A long circular cylinder of radius a, with its axis horizontal, is half-immersed in a fluid under gravity and is making periodic vertical oscillations of small constant amplitude and of period 2$\pi $/$\sigma $ about this position. It is required to find the resulting fluid motion when the parameter N = $\sigma ^{2}$a/g is large; the method of an earlier paper (Ursell 1949) is then unworkable. The present solution is made to depend on an integral equation (3$\cdot $15) which can be chosen to have a kernel tending to zero with N$^{-1}$, and which is solved by iteration. Successive terms in the iteration are of decreasing order, and the convergence of the method for sufficiently large N is proved. Expressions are given for the virtual-mass coefficient (5$\cdot $1) and for the wave amplitude at infinity (5$\cdot $7). The present work appears to be the first practical and rigorous solution of a short-wave problem when a solution in closed form is not available. It is suggested that a similar technique may be applicable to the diffraction problems of acoustics and optics, which have hitherto been treated by the approximate Kirchhoff-Huygens principle.

69 citations


"On the radiation of short surface w..." refers methods in this paper

  • ...A rigorous treatment of this problem has been given by Ursell [ 1 ], who obtained a first order estimate of the amplitude of the generated surface waves and the virtual mass of the cylinder....

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Journal ArticleDOI

30 citations


"On the radiation of short surface w..." refers methods in this paper

  • ...by Van Dyke [-3] as applied to short surface wave problems by Leppington [ 4 , 5, 6], Ayad [7] and Alker [8]....

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Journal ArticleDOI

17 citations


"On the radiation of short surface w..." refers background or methods in this paper

  • ...As Leppington [ 5 ] has already given a formulation of this problem to first order, details will be kept to a minimum....

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  • ...by Van Dyke [-3] as applied to short surface wave problems by Leppington [4, 5 , 6], Ayad [7] and Alker [8]....

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