# A note on the singularities in the theory of water waves with an inertial surface

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167 citations

### "A note on the singularities in the ..." refers background in this paper

...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....

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73 citations

### "A note on the singularities in the ..." refers methods in this paper

...8) This coincides with the expression obtained by Finkelstein [1] for e = 0....

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...References [1] A. B. Finkelstein, "The initial-value problem for transient water waves", Comm. Pure Appl....

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...X) 1 — cosfit D(k) sinh kh k (3.8) This coincides with the expression obtained by Finkelstein [1] for e = 0....

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70 citations

### "A note on the singularities in the ..." refers background in this paper

...[6] Y. S. Yu and F. Ursell, "Surface waves generated by an oscillating circular cylinder on water of finite depth: theory and experiment", J. Fluid Mech., 11 (1961), 529-551....

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...Following Yu and Ursell [6], we note that the integral /•«> coshk(h - Y)coshk(h - y)exp(ik\x - X\) (k sinh kh — K* cosh kh ) cosh kh along the real k axis (indented at k = k$, where the integrand has a simple pole) is equal to the integral along arg/c = w/4, and thus clearly tends to zero as \x - X\ -» oo....

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...Following Yu and Ursell [6], we note that the integral /•«> coshk(h -...

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32 citations

### "A note on the singularities in the ..." refers background in this paper

...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....

[...]

...Rhodes-Robinson [3] considered a two-dimensional wave source submerged in a liquid of infinite depth with an inertial surface....

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...[4] P. F. Rhodes-Robinson, "Note on the effect of surface tension on water waves at an inertial surface", J. Fluid Mech....

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...[3] P. F. Rhodes-Robinson, "On the generation of water waves at an inertial surface", J. Austral....

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...[2] P. F. Rhodes-Robinson, "Fundamental singularities in the theory of water waves with surface tension", Bull Austral....

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26 citations

### "A note on the singularities in the ..." refers background or result in this paper

...As discussed in [3] 9 can be shown to satisfy the initial conditions 9 - edq>/dy = 0, 3/3f(<p - edtp/dy) = 0 on y = 0 at t = 0, (2....

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...Hence as t -» oo, following [3], we obtain after simplification, coshkjh - Y)coshk{h - y) G - sinaf lln-^ - f , k sinh kh - K* cosh kh , exp(/rft) ....

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...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....

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...It is believed that the potential functions derived here and in [3] due to different types of singularities will be of use when problems associated with obstacles of different geometrical shapes present in a liquid with an inertial surface are considered....

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...Rhodes-Robinson [3] considered a two-dimensional wave source submerged in a liquid of infinite depth with an inertial surface....

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