/_{0} ¼2pa
Z 1

0

e^{ÿkjyÿgj}J0ðkaÞJ0ðkRÞdk ð1:1Þ

where a is the radius of the ring with centre at ð0;0;gÞ using a cylindrical co-ordinate system ðR;h;yÞ, y-axis being taken as the axis of the ring. However, in a fluid with a boundary at its upper surface, the potential due to a ring source can be decomposed into two parts, the first part representing the potential due to a ring of wave sources present in an unbounded fluid while the second representing its image in the upper boundary and the bottom, if there be any, conditions.

Hulme [2] constructed the velocity potential due to a horizontal ring of wave sources of time- harmonic strength submerged in deep water with a free surfacein terms of multi-valued toroidal harmonics. Rhodes-Robinson [3,4] earlier used a reduction technique to obtain the ring source potential for both deep water and finite depth water in the presence of surface tension at the free surface. Mandal and Kundu [5] obtained the velocity potential due to a ring source of time- dependent strength submerged in deep water with an inertial surface in the presence of surface tension, the inertial surface being composed of uniformly distributed non-interacting floating material. Here we consider the motion due to a submerged horizontal ring of wave sources of time-dependent strength present in water with anice-cover, the ice-cover being modelled as a thin elastic sheet composed of elastic material of uniform area density. The problem is formulated as an initial value problem for the velocity potential describing the motion in the fluid, and the Laplace transform technique is employed to solve it. Three types of source strengths, namely impulsive initially but zero later, the classical case of constant strength and finally the important case of time-harmonic strength are considered. The steady-state development of the potential function for time-harmonic source strength shows the existence of outgoing progressive waves of any frequencyunder the ice-cover. This is in contrast with the case when the ice-cover is modelled as an inertial surface in which case outgoing time-harmonic progressive waves exist under the inertial surface only when the angular frequency is less than a certain constant which depends on the surface density of the inertial surface [6].

2. Mathematical formulation

A cylindrical co-ordinate system ðR;h;yÞ is chosen in which the y-axis is taken vertically downwards into the water which is assumed to be homogeneous with densityqand inviscid. The upper surface of water is covered by a thin layer of ice modelled as an elastic sheet having uniform surface density q, Young’s modulus E and Poisson’s ratio c; being a constant having the dimension of length. A horizontal ring of radiusaof uniformly distributed point sources, each of the same time-dependent strengthmðtÞ, is present at a depthgbelow the mean position of the ice- cover, taken as the y¼0 plane. The axis of the ring coincides with the y-axis. The only external force acting on the system is the gravity g. The motion in water is generated when the point sources on the ring start operating at a given instant simultaneously. Since the motion in water starts from rest, it is irrotational and can be described by a potential function/ðR;y;tÞ. Then / satisfies

1

RðR/_{R}Þ_{R}þ/_{yy}¼0 ð2:1Þ

in the fluid region except at points on the ring. IffðR;tÞ denotes the depression of the ice-cover below its mean position, then the linearised kinematic and dynamic conditions on the ice-cover are given by

/_{y} ¼f_{t} on y¼0 ð2:2Þ

and

ð/ÿ/_{y}Þ_{t} ¼ ðDr^{4}_{R}þ1Þgf on y¼0 ð2:3Þ

where D¼_{12ð1ÿm}^{Eh}^{3}^{0}2Þqg is a constant, h0 being the very small thickness of the ice-cover and
r^{2} ¼_{R}^{1} _{@R}^{@} ðR_{@R}^{@}Þ. Elimination of f between (2.2) and (2.3) produces the linearised ice-cover con-
dition

ð/ÿ/_{y}Þ_{tt} ¼ ðDr^{4}_{R}þ1Þg/_{y} ony ¼0: ð2:4Þ

The initial conditions at the ice-cover are

/ÿ/_{y} ¼0; ð/ÿ/_{y}Þ_{t} ¼0 on y¼0 at t¼0 ð2:5Þ
which are obtained due to continuity offfor all times. Also,/must satisfy the bottom condition

r/!0 as y! 1 ð2:6aÞ

for deep water, or

/_{y} ¼0 on y¼h ð2:6bÞ

for water of uniform finite depthh. Also, at points near the ring

/!mðtÞ/_{0} as fðRÿaÞ^{2}þ ðyÿgÞ^{2}g^{1=2} !0 ð2:7Þ
where/_{0} is given by (1.1).

It may be noted that for time-harmonic motion of angular frequencyr, the ice-cover condition (2.4) becomes

K/þÿDr^{4}_{R}

þ1ÿK

/_{y} ¼0 on y¼0 ð2:8Þ

whereK¼r^{2}=g. If/has the time-harmonic progressive wave form given by
/¼Refe^{ÿky}H_{0}^{ð1Þ;ð2Þ}ðkRÞe^{ÿirt}g

for deep water, or

/¼RefcoshkðhÿyÞH_{0}^{ð1Þ;ð2Þ}ðkRÞe^{ÿirt}g

for water of uniform finite depthh, thenk satisfies the polynomial equation

DðkÞ kðDk^{4}þ1ÿKÞ ÿK¼0 ð2:9Þ

for deep water, or the transcendental equation

D^{0}ðkÞ kðDk^{4}þ1ÿKÞsinhkhÿKcoshkh¼0 ð2:10Þ
for finite depth water. It can be easily verified that the nature of the zeros of DðkÞ and D0ðkÞ
remains the same whether 1ÿKis positive or negative so long asD6¼0, and that bothDðkÞand
D0ðkÞpossess a unique positive real zero.

ForDðkÞwe denote its positive real zero byk. The other zeros ofDðkÞare two pairs of complex
conjugate numbers denoted by ðk1;k_{1}Þ and ðk2;k_{2}Þ where Rek_{1} >0, Imk_{1}>0 and Rek_{2} <0,
Imk2 >0. Chakrabarti et al. [7] gave an elementary proof for the nature of the zeros ofDðkÞ for
¼0. However, for6¼0, the same elementary proof can be used to find the nature of the zeros of
DðkÞ with obvious modifications.

Again, forD^{0}ðkÞwe denote its positive real zero byl. It can be shown thatD^{0}ðkÞhas a negative
real zero at k¼ ÿl, two pairs of complex conjugate roots l_{1};l_{1} and ÿl_{1};ÿl_{1} with Rel_{1} >0,
Iml_{1} >0 and Rel_{1}<Iml_{1}, and an infinite number of purely imaginary roots ia_{n}ða_{n} >0,

n¼1;2;. . .Þ wherea_{n}h!npasn! 1 (see [8]).

For the caseD¼0, the ice-cover is no longer modelled as an elastic plate, and it becomes an inertial surface, and the ice-cover (inertial surface) condition becomes

K/þ ð1ÿKÞ/_{y} ¼0: ð2:11Þ

This shows that progressive wave is possible only when 1ÿK >0 i.e.r<ðg=Þ^{1=2} (cf. [6]). For
rPðg=Þ^{1=2}, the form (2.11) does not allow any progressive wave.

3. Solution

To solve the initial value problem for/described above, we use Laplace transform defined by

/ðR;y;pÞ ¼ Z 1

0

/ðR;y;tÞe^{ÿpt}dt; p>0; ð3:1Þ

then, /satisfies the boundary value problem described by 1

RðR/_{R}Þ_{R}þ/_{yy} ¼0 ð3:2Þ

in the fluid region except at points on the ring,

/!mðpÞ/_{0} asfðRÿaÞ^{2}þ ðyÿgÞ^{2}g^{1=2} !0; ð3:3Þ

p^{2}/ÿ Dr^{4}_{R}

þ1þp^{2}
g

g/_{y} ¼0 ony ¼0; ð3:4Þ

r/!0 as y! 1 ð3:5aÞ

for deep water, or

/_{y} ¼0 on y¼h ð3:5bÞ

for finite depth water.

We now consider the cases of deep water and finite depth water separately.

Case (a): deep water

A solution for / satisfying (3.2), (3.3) and (3.5a) is constructed as

/ðR;y;pÞ ¼mðpÞ /_{0}

þ Z 1

0

AðkÞe^{ÿky}J0ðkRÞdk

ð3:6Þ
whereAðkÞ is an unknown function of k to be determined such that the integral in (3.6) is con-
vergent. Using the form of /_{0} given in (1.1), it is seen that the condition (3.4) is satisfied if we
choose

AðkÞ ¼2paJ_{0}ðkaÞfgkð1þDk^{4}Þ ÿ ð1ÿkÞp^{2}ge^{ÿkg}

ð1þkÞp^{2}þgkð1þDk^{4}Þ : ð3:7Þ

Thus /ðR;y;pÞ in this case is obtained as

/ðR;y;pÞ ¼mðpÞXðR;yÞ þmðpÞ Z 1

0

X^{2}

X^{2}þp^{2}YðR;y;kÞdk ð3:8Þ

where

XðR;yÞ ¼2pa Z 1

0

e^{ÿkjyÿgj}

ÿ1ÿk

1þke^{ÿkðyþgÞ}

J0ðkaÞJ0ðkRÞdk;

YðR;y;kÞ ¼4paJ0ðkaÞ

1þkJ0ðkRÞe^{ÿkðyþgÞ} ð3:9Þ

and

X^{2}ðkÞ ¼gkð1þDk^{4}Þ

1þk : ð3:10Þ

Laplace inversion of (3.8) produces

/ðR;y;tÞ ¼mðtÞXðR;yÞ þ Z 1

0

XðkÞYðR;y;kÞ

Z t

0

mðsÞsinXðt

ÿsÞds

dk:

ð3:11Þ

For a ring source of impulsivestrength we take mðtÞ ¼dðtÞ, and in this case (3.11) produces

/^{imp}ðR;y;tÞ ¼dðtÞXðR;yÞ þ
Z 1

0

XðkÞYðR;y;kÞsinXtdk ð3:12Þ For large t, the expression in (3.12) vanishes. This has the interpretation that since the sources around the ring act instantaneously at t¼0, they have no effect on the fluid motion after a long lapse of time.

For a ring source ofconstant strength mðtÞ ¼1, (3.11) gives

/^{const}ðR;y;tÞ ¼XðR;yÞ þ
Z 1

0

YðR;y;kÞð1ÿcosXtÞdk: ð3:13Þ For a ring of wave sources oftime-harmonicstrength, we takemðtÞ ¼sinrtwhereris the circular frequency. In this case, (3.11) gives

/ðR;y;tÞ ¼sinrtXðR;yÞ þ Z 1

0

XðkÞYðR;y;kÞXsinrtÿrsinXt

X^{2}ÿr^{2} dk ð3:14Þ

To determine the form of (3.14) ast! 1, we introduce a Cauchy principal value atk ¼kwhich
is the real positive zero of X^{2}ÿr^{2} i.e. DðkÞ, in the integral in (3.14), and following Rhodes-
Robinson [6], we obtain, ast! 1

/!2pasinrt Z

--

1 0

e^{ÿkjyÿgj}

þkðDk^{4}þ1ÿKÞ þK

DðkÞ e^{ÿkðyþgÞ}

J_{0}ðkaÞJ0ðkRÞdk

ÿ4p^{2}acosrtkðDk^{4}þ1ÿKÞ

1ÿKþ5Dk^{4} e^{ÿkðyþgÞ}J0ðkaÞJ0ðkRÞ ð3:15Þ

where the integral is in the sense of Cauchy principal value. This integral can be simplified
by using the relation 2J_{0}ðkRÞ ¼H_{0}^{ð1Þ}ðkRÞ þH_{0}^{ð2Þ}ðkRÞ, and rotating the contour in the complex
k-plane for the integral involving H_{0}^{ð1Þ}ðkRÞ in the first quadrant and for the integral involving
H_{0}^{ð2Þ}ðkRÞin the fourth quadrant. Thus an alternative representation for the expression in (3.15) is
given by

/!8asinrt Z 1

0

Lðk;yÞLðk;gÞI_{0}ðkaÞ

k^{2}ð1ÿKþDk^{4}Þ^{2}þK^{2}K_{0}ðkRÞdk
þ2p^{2}iasinrt fn ðy;g;k_{1}ÞH_{0}^{ð1Þ}ðk_{1};RÞ

ÿfðy;g;k_{1}ÞH_{0}^{ð2Þ}ðk_{1}RÞo

ÿ2p^{2}afðy;g;kÞfsinrtY0ðkRÞ þcosrtJ0ðkRÞg ð3:16Þ
where

Lðk;yÞ ¼kð1ÿKþDk^{4}ÞcoskyÿKsinky; ð3:17Þ

fðy;g;kÞ ¼2kð1ÿKþDk^{4}Þ

1ÿKþ5Dk^{4} e^{ÿkðyþgÞ}J0ðkaÞ: ð3:18Þ

It may be noted that the second term in the expression in (3.16) is real. For largeR, we find from (3.16) that ast! 1.

/! ÿ4p^{2}akð1ÿKþDk^{4}Þ

1ÿKþ5Dk^{4} e^{ÿkðyþgÞ}J0ðkaÞ 2
pkR
1=2

coskR

ÿrtÿp 4

: ð3:19Þ

This shows that/represents outgoing progressive waves as R! 1.

Case (b): finite depth water

In this case a solution for /satisfying (3.2) and (3.3) is constructed as

/¼mðpÞ /_{0}

ÿ2pa Z 1

0

e^{ÿkðyþgÞ}J0ðkaÞJ0ðkRÞdk
þ

Z 1

0

fBðkÞcoshkðhÿyÞ þCðkÞsinhkyg J0ðkaÞ

coshkhJ0ðkRÞdk

ð3:20Þ where the functions BðkÞ and CðkÞ, for the satisfaction of the conditions (3.4) and (3.5b), are chosen as

BðkÞ ¼4pa

gk1þDk^{4}þ^{p}_{g}^{2}

MðkÞðX^{2}_{0}þp^{2}Þ coshkðhÿgÞ;

CðkÞ ¼4pae^{ÿkh}sinhkg ð3:21Þ

with

MðkÞ ¼coshkhþksinhkh;

X^{2}_{0}ðkÞ ¼gkð1þDk^{4}Þsinhkh

MðkÞ : ð3:22Þ

Thus /is obtained as

/¼mðpÞPðR;yÞ þmðpÞ Z 1

0

X^{2}_{0}

X^{2}_{0}þp^{2}QðR;y;kÞdk ð3:23Þ

where

PðR;yÞ ¼2pa Z 1

0

e^{ÿkjyÿgj}

ÿe^{ÿkðyþgÞ}þ 2
coshkh

k

MðkÞ coshkðh

ÿyÞcoshkðhÿgÞ
þe^{ÿkh}sinhkysinhkg

J0ðkaÞJ0ðkRÞdk ð3:24Þ

and

QðR;y;kÞ ¼4pacoshkðhÿyÞcoshkðhÿgÞ

MðkÞsinhkh J0ðkaÞJ0ðkRÞ: ð3:25Þ

Laplace inversion of (3.23) gives

/ðR;y;tÞ ¼mðtÞPðR;yÞ þ Z 1

0

X_{0}ðkÞQðR;y;kÞ
Z t

0

sinX_{0}ðt

ÿsÞmðsÞds

dk: ð3:26Þ

For impulsive source strengthmðtÞ ¼dðtÞ, and (3.26) gives

/^{imp}ðR;y;tÞ ¼dðtÞPðR;yÞ þ
Z 1

0

X_{0}ðkÞQðR;y;kÞsinX_{0}tdt ð3:27Þ
which tends to zero as t! 1, as in the case of deep water.

Forconstant source strengthmðtÞ ¼1, and (3.26) gives

/^{c}ðR;y;tÞ ¼PðR;yÞ þ
Z 1

0

QðR;y;kÞð1ÿcosX_{0}tÞdk: ð3:28Þ
For time-harmonicsource strengthmðtÞ ¼sinrt and in this case (3.26) produces

/ðR;y;tÞ ¼sinrtPðR;yÞ þ Z 1

0

X0ðkÞQðR;y;kÞX0sinrtÿrsinX0t

X^{2}_{0}ÿr^{2} dk: ð3:29Þ

As in the case of deep water, the steady-state development of/, given by (3.29), can be obtained
by introducing a Cauchy principal value at k ¼l which is the real positive zero of X^{2}_{0}ÿr^{2} i.e.

D^{0}ðkÞ, in the integral in (3.29). Then as t! 1, we find
/!sinrt PðR;yÞ

þ2 Z

--

1 0

kð1þDk^{4}Þ

D^{0}ðkÞMðkÞ coshkðhÿyÞcoshkðhÿgÞJ0ðkaÞJ0ðkRÞdk

ÿ8p^{2}alð1ÿKþDl^{4}ÞcoshlðhÿyÞcoshlðhÿgÞJ0ðlaÞJ0ðlRÞ

2lhð1ÿKþDl^{4}Þ þ ð1ÿKþ5Dl^{4}Þsinh 2lh ð3:30Þ
where the integral is in the sense of CPV.

In the right-hand side of (3.30), combining the integral representation ofPðR;yÞgiven in (3.24) and the CPV integral and changing the contour along the real axis with indentations above the pole at k¼ ÿland below the pole at k¼l, the following alternative representation is obtained:

/!8pasinrtX^{1}

n¼1

gðy;g;ia_{n}ÞK0ðanRÞ þ4p^{2}iasinrtfgðy;g;l_{1}ÞH_{0}^{ð1Þ}ðl_{1}RÞ

ÿgðy;g;l_{1}ÞH_{0}^{ð2Þ}ðl_{1}RÞg ÿ4p^{2}agðy;g;lÞfsinrtY0ðlRÞ þcosrtJ0ðlRÞg ð3:31Þ

where

gðy;g;kÞ ¼2kð1ÿKþDk^{4}ÞcoshkðhÿyÞcoshkðhÿgÞJ0ðkaÞ

ð1ÿKþ5Dk^{4}Þsinh 2khþ ð1ÿKþDk^{4}Þ2kh ð3:32Þ
and the second term in (3.31) is real. For large R, we find that, ast! 1

/! ÿ8p^{2}alð1ÿKþDl^{4}ÞcoshlðhÿyÞcoshlðhÿgÞJ0ðlaÞ
ð1ÿKþ5Dl^{4}Þsinh 2lhþ ð1ÿKþDl^{4}Þ2lh

2 plR 1=2

coslR

ÿrtÿp 4

:

ð3:33Þ This shows that/represents progressive outgoing waves as R! 1.

4. Conclusion

The velocity potential due to a horizontal circular ring of wave sources of time-dependent strength submerged in water with an ice-cover has been obtained for both infinite and finite depth of water. For the case of time-harmonic sources, the steady-state development of the potential function shows the existence of outgoing progressive waves at large distances from the ring source. If the elastic parameterDis put equal to zero, then the results for deep water coincide with the results obtained in [5] for deep water with an inertial surface in the absence of surface tension.

If bothDand are put equal to zero, then the results obtained above can be identified with the results obtained earlier in [3]. The effect of surface tension at the ice-cover can be incorporated in the above results.

Acknowledgement

This work is supported by CSIR, New Delhi.

References

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[2] A. Hulme, The potential of a horizontal ring of wave sources in a fluid with a free surface, Proc. R. Soc. London A375 (1981) 295–306.

[3] P.F. Rhodes-Robinson, On surface waves in the presence of immersed vertical boundaries. I, Q. J. Mech. Appl.

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[4] P.F. Rhodes-Robinson, On surface waves in the presence of immersed vertical boundaries. II, Q. J. Mech. Appl.

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