Rajdeep Maiti, Uma Basuand B. N. Mandal

1. Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India

2. Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700035, India

Oblique Wave-free Potentials for Water Waves in Constant Finite Depth

Rajdeep Maiti1, Uma Basu1and B. N. Mandal2*

1. Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India

2. Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700035, India

In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform finite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.

wave-free potentials; modified Helmholtz equation; free surface; surface tension; ice-cover; water wave

1 Introduction1

Problems involving generation or scattering of surface water waves by a body of any geometrical configuration present in water are of immense importance in ocean related industry and are generally investigated mathematically assuming linear theory. Lamb (1932), Stoker (1957), Weahausen and Laitone (1960) and others investigated various water wave problems by employing a general expansion of the wave potential describing the motion in water. For example, when an infinitely long horizontal cylinder submerged or half-immersed in water undergoes some sort of oscillatory motion or is under the action of a train of surface water waves incident on it normally or obliquely, waves are generated or reflected and transmitted, and the resulting motion can be described by a velocity potentialfunction which can be expanded in terms of a a regular wave, a wave source, a wave dipole and a wave-free part. The wave-free part can be expressed as a linear combination of what are termed as wave-free potentials. These are singular solutions of the governing partial differential equation satisfying the usual boundary conditions of the water wave potential and decaying rapidly away from the point of singularity. Ursell (1949; 1950) first investigated water waves generated due to a half-immersed or completely submerged long horizontal circular cylinder employing the method of multipole expansion of the time harmonic stream function as well as the velocity potential. Thorne (1953) constructed velocity potentials due to line and point singularities in infinitely deep water and also in water of uniform finite depth. Athanassonlis (1984) considered an infinitely long horizontal cylinder of arbitrary cross section floating on the free surface and derived a general multipole expansion for the wave potential which is convergent throughout the fluid domain. Bolton and Ursell (1973) investigated the problem of wave radiation by an infinitely long horizontal cylinder and by utilising the radiation condition, they obtained an expression of the series expansion of the heaving potential which is exponentially small at infinity. Mandal and Goswami (1984) studied oblique scattering by a half-immersed circular cylinder by using two methods, one based on integral equation formulation and other based on expansion of the scattered velocity potential by the method of multipoles. The reflection and transmission coefficients were computed by both the methods and depicted graphically against the wave number to compare the results.

All the above studies are mostly focused on water with a free surface. Rhodes-Robinson (1970) considered the effect of surface tension at the free surface and obtained expressions for harmonic potential functions describing both two and three-dimensional fundamental wave source and multipole singularities for infinitely deep water as well as finite depth water. Gayen and Mandal (2006) considered the motion due to fundamental singularities with time-dependent source strengths present in finite depth water with an ice-cover modelled as a floating thin elastic plate (cf. Fox and Squire, 1994).

Construction of wave-free potentials is of some interest in theory of water waves due to their usefulness in the expansionof velocity potentials for water wave problems involving a circular cylinder or a sphere submerged or floating (half-immersed) in water of both infinite and finite depths. Ursell (1961, 1968) showed that for an infinitely long horizontal circular cylinder floating on water, the general expansion of the potential function includes a wave-free part which can further be expanded in terms of wave-free multipoles. As mentioned earlier, these are termed as wave-free potentials and decay rapidly from the point of singularity. Weahausen and Laitone (1960) obtained expressions for wave source potentials for infinitely deep water and mentioned for three dimensions a particular linear combination of these potentials which decay rapidly far away from the source, so that the combination produces wave-free potentials. This also gives a clue in the construction of wave-free potentials in various situations. Exploiting this clue, Mandal and Das (Das and Mandal, 2010; Mandal and Das, 2010) described a systematic method for construction of wave-free potentials in two dimensions in deep as well as finite depth water with a free surface taking into account the effect of surface tension, and also in water with a floating thin ice-cover modelled as a thin elastic plate. Also, recently Dhilon and Mandal (2013) extended these to three dimensions for both finite and infinite depth water and depicted graphically the profile of the upper surface related to each of the wave-free potentials. It may be noted that a brief description regarding the derivation of wave-free potentials in water with a free surface is given in the monograph of Linton and McIver (2001).

In the present paper, wave-free potentials are constructed for the case when the governing partial differential equation is the modified Helmholtz equation which arises in the linearised theory of water waves if the dependence of the velocity potential on one horizontal coordinate is harmonic throughout, and water is of uniform finite depth, has either a free surface or an ice-cover. At the free surface, the effect of surface tension is first neglected and then taken into account. In the case of ice-cover, it is modelled as a thin elastic plate floating on water. For different boundary conditions at the upper surface, the problem is solved by using multipole expansion and by taking appropriate linear combition of these multipoles, the oblique wave-free potentials are constructed. The main purpose of the present study is to construct these wave-free potentials for various types of boundary conditions at the upper surface. Finally, the forms of the upper surface in connection with these wave-free potentials are depicted graphically in a number of figures for each case and appropriate conclusions are made.

2 Mathematical formulation of the problem

The present problem is formulated mathematically by using a Cartesian coordinate system in which the origin is taken at the mean position of the upper surface of water which corresponds toy=0and the direction of theyaxis is taken vertically downwards into the water region. The motion in water is assumed to be irrotational and time harmonic with angular frequencyσand sinusoidal with respect to the horizontal coordinatezso that it can be described by a potential function：

where φ（x, y）having singularity at （0,f ）,( f＞ 0,fis the distance of the point of singularity measured from the origin) situated on theyaxis, satisfies the two dimensional Helmholtz’s equation

in the fluid region except at（0, f）,

where h is the depth of water, and φ（ x, y）behaves like an outgoing wave as→∞.We introduce the angle θ and θ′ by

andr, r′are the radial distances from the point （x, y） to the point （0,f） and the image （0,-f）, respectively.

3 Method of solution and wave-free potentials

Solutions of （1） which are singular atr=0are given by

for symmetric and anti-symmetric multipoles respectively, where Kn（lr） is the modified Bessel function of second kind. It is known that (cf. Gradstein and Ryzhik, 1980)

and

where, ν= lcoshμ.

3.1 Water with a free surface

If the upper surface of the ocean is a free surface, then the linearised boundary condition is given by

The symmetric and anti-symmetric multipoles are constructed as

where, A（μ） ,A1（μ） ,B （μ）and B1（μ） are functions of μto be determined. Now, by utilising the boundary condition (2) and (8), we obtain

Thus, for the case of symmetric multipoles, using (9), we obtain

where,

and

when n is even, and

when n is odd. By a similar procedure the oblique anti-symmetric wave-free potentials can be obtained, by utilising (10), as

when n is even,

whennis odd and

If l→0the limiting forms of the wave-free potentials are given by

when n is even, and

when n is odd and the anti-symmetric wave-free potentials are

whenn is even,

when n is odd. These forms were obtained by Linton and McIver (2001) in case of two dimensional motion in finite depth water.

If the singularity is situated at the upper surface of water, then the wave-free potentials can be obtained from (14)-(17) by makingf→0and they are given by

when n is even, and

when n is odd. For the anti-symmetric wave-free potentials,

when n is even,

3.2 Effect of surface tension at the upper surface

If the effect of surface tension is included at the free surface, then the linearised free surface condition is given by

Proceeding as before, the symmetric multipoles can be constructed as

when n is even and

when n is odd. Similarly, the oblique anti-symmetric wave-free potentials are obtained as

whenn is even,

whenn is odd. Here

As before, if l→0the limiting forms of the wave-free potentials are given by

when n is even and

when n is odd. Similarly the anti-symmetric wave-free potentials become as l→0

whennis even and

whennis odd.

Making f→0the free surface oblique wave-free potentials are obtained from (28)-(31) as

when n is even and

whenn is odd. The oblique anti-symmetric wave-free potentials are

whenn is even and

when n is odd. Here,

and

3.3 Water with an ice cover

If the upper surface of the ocean is covered by an ice-cover modelled as a thin elastic plate, then the boundary condition at the upper surface is given by

In this case, the symmetric multipoles are constructed as

where

and

do not contribute anything as→∞so that they are wave-free. The oblique symmetric wave-free potentials are constructed as

whenn is even,

whenn is even,

when n is odd and

As before, if l→0the limiting forms of the wave-free potentials are given by

when n is even and

when n is odd. Similarly the anti-symmetric wave-free potentials become as l→0

when n is even and

whenn is odd. Makingf→0the free surface oblique wave-free potentials are obtained from (41)-(45) as

when n is even and

whenn is odd. Similarly, the oblique anti-symmetric wave-free potentials are

whenn is even,

whenn is odd. Here

and

4 Numerical results

The velocity potential Φ（ x, y, z, t）describing the wave motion corresponding to the wave-free potential Ψn（x, y, z, t ）can be written as

Now, we write

and

so that the non-dimensional form of the depression of the upper surface of water is

Writing we find

Fig. 1 ξsfor different Kf

Fig. 2 ξafor different Kf

Fig. 3 ξsfor different lf

Fig. 4 ξafor different lf

Fig. 5 ξsfor n=2, 5

Fig. 6 ξafor n=2, 5

Here, the amplitudes of are seen to decrease due to the effect of the surface tension at the upper surface. For a sufficiently small value of M′= 0.001, the profiles ofare similar to those in Figs. 1 and 2.

Fig. 7 ξsfor differentM′

Fig. 8 ξafor differentM′

Fig. 9 ξsfor different lf andM'=0.1

Fig. 10 ξsfor different Kf andM'=0.1

Fig. 11 ξsfor differentD′

Fig. 12 ξafor differentD′

Fig. 13 ξsfor different Kf andD′=0.1

Fig. 14 ξsfor different lf andD′=0.1

5 Conclusions

In the linearised theory of water waves wave-free potentials have been derived in the literature by using multipole expansion of the potential function for finite or infinitely deep water with a free surface. Taking the effect of surface tension into account or for water with an ice-cover, the expressions of the oblique wave-free potentials having singularity at a point submerged in water of uniform finite depth are constructed here in a systematic manner by taking appropriate linear combinations of the multipoles. These wave-free potentials decay as one moves away from the point of singularity. The depression of the upper surface related to these wave-free potentials is depicted graphically for each case in a number of figures. In each figure, it is observed that the wave profile of the depression of the upper surface rapidly diminishes with the distance from the origin which is plausible since the wave-free potentials decay rapidly from the point of singularity. These figures also show that the coefficient of surface tension and the stiffness parameter of the elastic ice-cover affect the resulting wave motion. The results for non-oblique wave-free potentials are derived by making l→0 i. e α=0and the upper surface wave-free potentials are obtained by making f→0.The limiting case l→0for non-oblique wave-free potentials in finite depth water with a free surface coincide with those given by Linton ans McIver (2001).

The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely submerged are used in the construction of offshore structures for extraction of crude oil deposited below the ocean floor. Further extension of the above results can be made by considering generalised boundary condition involving higher order derivatives at the upper surface of water.

Acknowledgement

The authors thank to the Reviewer for his valuable comments and suggestions to revise the paper.

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1671-9433(2015)02-0126-12

10.1007/s11804-010-1019-0

DOI： 10.1007/s11804-015-1308-8

Received date： 2014-10-31.

Accepted date： 2015-01-22.

*Corresponding author Email： birenisical@gmail.com

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015