/0 ¼2pa Z 1
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  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  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 .
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
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 ¼ft on y¼0 ð2:2Þ
ð/ÿ/yÞt ¼ ðDr4Rþ1Þgf on y¼0 ð2:3Þ
where D¼12ð1ÿmEh302Þqg is a constant, h0 being the very small thickness of the ice-cover and r2 ¼R1 @R@ ðR@R@Þ. Elimination of f between (2.2) and (2.3) produces the linearised ice-cover con- dition
ð/ÿ/yÞtt ¼ ðDr4Rþ1Þg/y ony ¼0: ð2:4Þ
The initial conditions at the ice-cover are
/ÿ/y ¼0; ð/ÿ/yÞt 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 fðRÿaÞ2þ ðyÿgÞ2g1=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
/y ¼0 on y¼0 ð2:8Þ
whereK¼r2=g. If/has the time-harmonic progressive wave form given by /¼RefeÿkyH0ð1Þ;ð2ÞðkRÞeÿirtg
for deep water, or
for water of uniform finite depthh, thenk satisfies the polynomial equation
DðkÞ kðDk4þ1ÿKÞ ÿK¼0 ð2:9Þ
for deep water, or the transcendental equation
D0ðkÞ kðDk4þ1ÿKÞsinhkhÿKcoshkh¼0 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;k1Þ and ðk2;k2Þ where Rek1 >0, Imk1>0 and Rek2 <0, Imk2 >0. Chakrabarti et al.  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, forD0ðkÞwe denote its positive real zero byl. It can be shown thatD0ðkÞhas a negative real zero at k¼ ÿl, two pairs of complex conjugate roots l1;l1 and ÿl1;ÿl1 with Rel1 >0, Iml1 >0 and Rel1<Iml1, and an infinite number of purely imaginary roots ianðan >0,
n¼1;2;. . .Þ whereanh!npasn! 1 (see ).
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=Þ1=2 (cf. ). For rPðg=Þ1=2, the form (2.11) does not allow any progressive wave.
To solve the initial value problem for/described above, we use Laplace transform defined by
/ðR;y;pÞ ¼ Z 1
/ðR;y;tÞeÿptdt; p>0; ð3:1Þ
then, /satisfies the boundary value problem described by 1
RðR/RÞRþ/yy ¼0 ð3:2Þ
in the fluid region except at points on the ring,
/!mðpÞ/0 asfðRÿaÞ2þ ðyÿgÞ2g1=2 !0; ð3:3Þ
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
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 ¼2paJ0ðkaÞfgkð1þDk4Þ ÿ ð1ÿkÞp2geÿkg
ð1þkÞp2þgkð1þDk4Þ : ð3:7Þ
Thus /ðR;y;pÞ in this case is obtained as
/ðR;y;pÞ ¼mðpÞXðR;yÞ þmðpÞ Z 1
XðR;yÞ ¼2pa Z 1
1þk : ð3:10Þ
Laplace inversion of (3.8) produces
/ðR;y;tÞ ¼mðtÞXðR;yÞ þ Z 1
For a ring source of impulsivestrength we take mðtÞ ¼dðtÞ, and in this case (3.11) produces
/impðR;y;tÞ ¼dðtÞXðR;yÞ þ Z 1
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
/constðR;y;tÞ ¼XðR;yÞ þ Z 1
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
X2ÿr2 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 X2ÿr2 i.e. DðkÞ, in the integral in (3.14), and following Rhodes- Robinson , we obtain, ast! 1
1ÿKþ5Dk4 eÿkðyþgÞJ0ðkaÞJ0ðkRÞ ð3:15Þ
where the integral is in the sense of Cauchy principal value. This integral can be simplified by using the relation 2J0ðkRÞ ¼H0ð1ÞðkRÞ þH0ð2ÞðkRÞ, and rotating the contour in the complex k-plane for the integral involving H0ð1ÞðkRÞ in the first quadrant and for the integral involving H0ð2ÞðkRÞin the fourth quadrant. Thus an alternative representation for the expression in (3.15) is given by
/!8asinrt Z 1
k2ð1ÿKþDk4Þ2þK2K0ðkRÞdk þ2p2iasinrt fn ðy;g;k1ÞH0ð1Þðk1;RÞ
ÿ2p2afðy;g;kÞfsinrtY0ðkRÞ þcosrtJ0ðkRÞg ð3:16Þ where
Lðk;yÞ ¼kð1ÿKþDk4ÞcoskyÿKsinky; ð3:17Þ
1ÿKþ5Dk4 eÿkðyþgÞJ0ðkaÞ: 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.
1ÿKþ5Dk4 eÿkðyþgÞJ0ðkaÞ 2 pkR 1=2
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
ÿ2pa Z 1
fBðkÞcoshkðhÿyÞ þCðkÞsinhkyg J0ðkaÞ
ð3:20Þ where the functions BðkÞ and CðkÞ, for the satisfaction of the conditions (3.4) and (3.5b), are chosen as
CðkÞ ¼4paeÿkhsinhkg ð3:21Þ
MðkÞ : ð3:22Þ
Thus /is obtained as
/¼mðpÞPðR;yÞ þmðpÞ Z 1
PðR;yÞ ¼2pa Z 1
ÿeÿkðyþgÞþ 2 coshkh
MðkÞsinhkh J0ðkaÞJ0ðkRÞ: ð3:25Þ
Laplace inversion of (3.23) gives
/ðR;y;tÞ ¼mðtÞPðR;yÞ þ Z 1
X0ðkÞQðR;y;kÞ Z t
For impulsive source strengthmðtÞ ¼dðtÞ, and (3.26) gives
/impðR;y;tÞ ¼dðtÞPðR;yÞ þ Z 1
X0ðkÞQðR;y;kÞsinX0tdt ð3:27Þ which tends to zero as t! 1, as in the case of deep water.
Forconstant source strengthmðtÞ ¼1, and (3.26) gives
/cðR;y;tÞ ¼PðR;yÞ þ Z 1
QðR;y;kÞð1ÿcosX0tÞdk: ð3:28Þ For time-harmonicsource strengthmðtÞ ¼sinrt and in this case (3.26) produces
/ðR;y;tÞ ¼sinrtPðR;yÞ þ Z 1
X20ÿr2 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 X20ÿr2 i.e.
D0ðkÞ, in the integral in (3.29). Then as t! 1, we find /!sinrt PðR;yÞ
2lhð1ÿKþDl4Þ þ ð1ÿKþ5Dl4Þsinh 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:
ÿgðy;g;l1ÞH0ð2Þðl1RÞg ÿ4p2agðy;g;lÞfsinrtY0ðlRÞ þcosrtJ0ðlRÞg ð3:31Þ
ð1ÿKþ5Dk4Þsinh 2khþ ð1ÿKþDk4Þ2kh ð3:32Þ and the second term in (3.31) is real. For large R, we find that, ast! 1
/! ÿ8p2alð1ÿKþDl4ÞcoshlðhÿyÞcoshlðhÿgÞJ0ðlaÞ ð1ÿKþ5Dl4Þsinh 2lhþ ð1ÿKþDl4Þ2lh
2 plR 1=2
ð3:33Þ This shows that/represents progressive outgoing waves as R! 1.
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  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 . The effect of surface tension at the ice-cover can be incorporated in the above results.
This work is supported by CSIR, New Delhi.
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