599
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 599-606
GENERATIONS OF WAVES AT AN INERTIAL SURFACE DUE TO EXPLOSIONS
B.N. MANDAL and N.K. GHOSH
Department of Applied Mathematics
University of Calcutta
92, A.P.C. Road
700009
Calcutta
India
(Received March 25, 1988)
ABSTRACT.
This paper is concerned with the generation of surface waves by an axially
symmetric initial disturbance applied at an inertial surface in an ocean of finite
depth.
This
initial
disturbance
can be
either
in
form of an impulse or an
the
The depression of the inertial surface is obtained as an
elevation or depression.
infinite integral in each case.
The method of stationary phase is applied to evaluate
the integral for large values of time and distance.
KEY WORDS AND PHRASES.
Ocean of finite depth, surface distribution, inertial surface,
velocity potential, Laplace transform, Hankel transform, method of stationary phase.
1980 AMS SUBJECT CLASSIFICATION CODE. 76B15.
1.
INTRODUCTION.
Waves are generated by an explosion above or within an ocean.
The formulation of
the problems associated with the generation of these waves as an initial value problem
If the explosion occurs above
is based on the linear theory of surface water waves.
the ocean surface, the initial condition on the surface is taken as an initial impulse
distributed over a certain region while for the case when the explosion occurs at or
below the ocean surface,
depression
of
the
same
the initial condition is taken as an initial elevation or
surface.
[I]
Stoker
and
Lamb
[2]
the
considered
two
dimensional unsteady motion due to an initial surface disturbance concentrated at the
origin and gave the velocity potential as well as the asymptotic form of the free
surface for large values of time and distance.
Kranzer and Keller [3] considered the
three
arbitrary
dimensional
unsteady
motion
due
to
an
axially
symmetric
initial
surface disturbance in an ocean of uniform finite depth and gave explicit formula for
the surface elevation and compared the theory with experimental results.
deduced these results
considered
for an ocean of
infinite depth.
Chaudhuri
They also
[4] and Wen [5]
the case when the initial surface disturbance is a combination of both
surface impulse and surface elevation distributed over an arbitrary region of the
B. N. MANDAL AND N. K. GHOSH
600
surface of an ocean of uniform finite depth and obtained asymptotic results by using
the method of stationary phase.
In all these cases, the ocean was essentially bounded above by a free surface.
Recently there has been a considerable interest in various problems concerned with the
generation
of
surface waves
due
to different
types
of sources of arbitrary time-
dependent strengths present in a liquid covered by an inertial surface composed of a
thin but uniform distribution of noninteracting floating particles
floating on an ocean surface)
Mandal
[9,10]). It
is
(e.g. broken ice
(see Rhodes-Robinson [6], Mandal and Kundu [7,8] and
thus natural
to extend
the
problems due to initial surface
disturbances in an ocean with a free surface to the case when the ocean is covered by
an inertial surface.
Recently, Mandal [I0] considered the two dimensional unsteady
motion in a deep ocean covered by an inertial surface due to initial disturbances at
the inertial surface while Mandal and Mukherjee [II] considered the three dimensional
unsteady motion due to a prescribed axisymmetic initial surface disturbance.
In this paper we extend the problem considered in [II] for a deep ocean to an
ocean of uniform finite depth.
of
generation
considered
of
water
waves
This may also be viewed as an extension of the problem
the
at
free
surface
of
ocean
an
of
finite
depth
[3] to an ocean covered by an inertial surface.
by Kranzer and Keller
After formulating it within the framework of llnearized theory as an initial value
problem, it is reduced to a boundary value problem by taking the Laplace transform in
The
time.
Hankel
transform
is
then used
to
solve
this
boundary value
problem.
Finally, Laplace inversion gives the potential function from which the depression of
the inertial surface is obtained in terms of an infinite double integral.
By applying
the method of stationary phase twice in succession to this integral, the asymptotic
form of the inertial surface is obtained for large values of time and distance.
In
the absence of the parameter characterizing the inertial surface, known results given
earlier in Kranzer and Keller [3] are recovered.
2.
STATEMENT AND FORMULATION OF THE PROBLEM.
We consider the unsteady three dimensional motion of an inviscid, incompressible,
homogeneous liquid of volume density p covered by an inertial surface composed of thin
but uniformly distributed disconnected floating matter of area density
p e (E
depth h.
0) under the action of gravity g only.
The liquid is of uniform finite
We assume that the
0 corresponds to a liquid with a free surface.
motion in the liquid is started at time t
0 from a state of rest by a prescribed
axfsymmetric initial disturbance at the inertial surface.
We choose a cylindrical
polar coordinate system (r,8,y) in which the y axis is taken vertically downwards and
is the axis of symmetry of the initial disturbance, the plane y
liquid.
0 is the position of
h is the bottom of the unlimited mass of the
the inertial surface at rest, and y
Since the motion starts from rest, it is irrotational and can be described by
a velocity potential
(r,y;t) for t
0.
Within the framework of linearized theory,
601
GENERATION OF WAVES AT AN INERTIAL SURFACE DUE TO EXPLOSIONS
it can be shown that
satisfies
+---
(r r)
O,
y2
,h
y
0,
0
t
with the inertial surface condltion
2
t
ay)
(0-
2
0, t
0 on y
>
(2.2)
0
while the boundary condition at the bottom is
0 on y
0y
(2.3)
O.
h, t
If the initial disturbance is due to an axlsymmetrlc displacement G(r) of the inertial
surface at a dstance r from the origin, then the initial condlttons are
0-e
--(
0, y
Oy
e
..)
y
0,
(2.4a)
O,
t
g G(r), y
(2.5a)
O.
0, t
We note that the initial displacement of the laert[al surface may be a depression or
elevation depending on G(r) being positive or negative.
However, if the initial
disturbance is due to an axially symmetric impulse of strength F(r) at a distance r
from the origin, then the initial conditions are
o-
t
3.
F(r)
Oy=
(- e
P
O,
on y
0 on y
y
0
(2.4b)
0
t
(2.5b)
O.
t
SOLUTION OF THE PROBLEM.
Let (r,y;p) denote the Laplace transform of (r,y;t) in time t defined as
f (r,y;t)e-Ptdt,
(r,y;p)
p
>
O.
o
Then (r,y;p) satisfies the boundary value problem
r
r
(r 3
--)
32
+-2-
O, 0
g
(3.1)
y g h,
g G(r) on y
2
p -(g+
and
Y
0 on y
h.
ep2 )(R)y
(3.2a)
0
or
-P F(r)
on y
0
(3.2b)
(3.3)
B. N. MANDAL AND N. K. GHOSH
602
(3.2a) holds for an initial surface displacement while (3.2b) holds for an initial
Let (k;y) be the Hankel transform of (r’y) defined as
surface impulse.
o
Then
satisfies
d2
dy
k
2
2
p V
?
and
Here
(3.4)
r O(r’v) J (kr) dr, k 7> O.
o
(k’y)
Y
(k)
and
(k)
2
0, 0
y
I
ep2 )Y
(g +
0 on y
(3.5)
h,
g G(k)
(3.6a)
or
(3.6b)
P
(3.7)
h.
C(r) and F(r) respectively.
are the Hankel transforms of
Then
coshk(h-)
g G(k)
(3.8a)
2
p D(k) + gk sinh kh
(k,y)
or
P
O
where D(k)
(k
coshk(h-y)
p2D(k)
(3.8b)
+ gk sinh kh
cosh kh + ek sinh kh.
(3.9)
Hence
(k)k
g
/
cosh k(h-y)J (kr)
o
D(k)(p2
o
(r,y;p)
+
ij2)
(3. tOe)
dk
or
pk F(k) cosh k(h-y)J (kr)
whe re IJ
2
dk
D(k)(p2+u 2)
o
(3.10b)
gk sinh kh
D(k)
Taking the Laplace inversion we obtain the potential function.
For the case of an
initial axially symmetric displacement we use (3.10a) to obtain
(r,y;t)
-(k)J o (kr)
f
0
sin
Ut coshk(h-y)
s i nhkh
dk
(3.11a)
603
GENERATION OF WAVES AT AN INERTIAL SURFACE DUE TO EXPLOSIONS
For the case of Inltlal axially symmetric impulse we obtain from (3.10b)
2
F(k)
1___
O(r,y;t)
Pg
cost coshk(h-y) Jo (kr)
dk.
sinhkh
0
(3.11b)
In these cases the inertial surface depression (r,t) is given by
(r,t)
=
[
(-
y--0
so that
(k)
k
=
(r,t)
(3.12a)
J (kr) cost dk
o
0
or
0f
9(k)
k
(3.12b)
slnvt J o (kr) dk.
The results given in Kranzer and Keller [3] are recovered by making c
making h
4.
0.
Also, by
0, results given in Stoker [I] are recovered.
and setting c
ASYMPTOTIC EXPANSIONS.
The method of stationary phase is now applied to obtain an asymptotic form of
(r,t) for the integral in equation (3.12a) for large
t and r but finite ratio
Using the result
J (kr)
o
=
2
cos(kr cosB)dS,
0
(3.12a) is written as
_ 1 Re 0f
/ k (k)
[exp{it(g + k
cos )}
0
+ exp{it(g
k
r
cos)}] d dk.
By using the method of stationary phase first to the -Integral, we get
E(r t)
where
(2__)2
r
.
k
2
(k)
0
-
[exp{it
+ kr
Pl(k
P2(k)
I
Now
P_(k)
Q(k)
+ exp{it
P2(k)}]
dk
(4.2)
I
4t
t
and
PI(R)}
(4.3)
kr + w
4t
r
r/t.
604
B. N. MANDAL AND N. K. GHOSH
=-
where Q(k)
I/2
.(
[--{D(k) }3/2
-n--k- )I/2
kh
sinh kh.I/2
+ cosh kh
(---)
P(k)
It
obvious
that
P(k) <
decreasing function in 0
and
<
-
P(k)
(gh)
P(k)
r
k
0 for 0
<
I/2
<
<
k
P(k)
Hence
m.
is
strictly
a
<
(gh) 1/2,
there
r
-as
as k
is
a
0
k
+
(R).
single
<
r/t
(gh)
O.
P1/2(k)
I/2
a
(say), where
(gh)I/2,
When r/t
in 0
point
stationary
exists only one stationary point at k
the equation
monotonic
Also
m.
These show that there Is no stationary point in the Integral for r/t
for r/t
(4.4)
)]
Q’(k).
so that
is
(gh)
<
k
<
> (gh) t/2
so
that
while
there
a is the real positive root of
it gives a smaller contribution than the
case so that contribution from it may be neglected.
Also
Pl(k)
has no
stationary point in the range of integration. The stationary phase method for the k-
integral when applied to (4.2) leads to
()
(r,t)
(a)
r
cos (t
I/2
(a,Q(oO
-Q’(a)
P2 (a)
cos((a)t
4
at)
where
(l+ea tanhah )1/2
ga tanhah
(a)
In partlcular, when the displacement is concentrated at the origin, G^
In this case
(r,t)
Without
solving
for a from
(,Q,Q(a)
(a))
a
the
I/2
cos((a)t
equation
P(a)
(4.6)
at).
0,
we
2--
consider
the
following
special cases.
CASE I. Shallow ocean (h small).
-
For small values of h, we can approximate
[gk2h(l
k2h2
2
1/2
ek h)]
as
(4.7)
two
GENERATION OF WAVES AT AN INERTIAL SURFACE DUE TO EXPLOSIONS
(r)2
3 {gh
I/2
--
a=[
so that
2
Q’()
P"(a)
2
(t
(_h)
I12
[2 (1
2 .3/2
2h2
2
(4.8)
()2
h {4gh + 9Eg
Jh
{I
605
a2h2 ){I-
2h(9e+4h)
(4.9)
(ge + 4h)}]
Using (4.8) and (4.9) in (4.5) we obtain (r,t).
The asymptotic form of the inertial surface given in (4.5) can also be expressed
in the form
(4.1o)
T
where
2
2
=-a
and %
(4.11)
For the case of an initial axially symmetric displacement, we obtain by a similar
analysis the asymptotic form
(r,t) of the
a
(r,t)
(rg)i/2
(l+ea
inertial surface given by
()
anhch 1/2 sin( u( a)t-r)
tanhh)l/2 (tlp,(a)i)
F(o)
Cl+ea tanhah) 1/2
a
1/2r
1/2 s
tanh
(,Q():Q;
in[_2(t -)r I
(4.12)
where a is given by (4.8) and Q’(a) is given by (4.9).
When
the
impulse
concentrated
is
at
the
^
In this case,
origin F
the
asymptotic form of the inertial surface has the form given by
(r,t)
a
2pg
1/2 (1
r
tanh )I/2
e2h) (.Q() _;)
sin
L2 ,( )
-
t
(4.13)
The result given in (4.5) and (4.12) are in complete agreement with the corresponding
results given by Kranzer and Keller [3] if we put :
0 and
CASE II. Deep ocean (h is infinite)
p
we get
kl/2(l
1/2
(l+ek) and from the equation
gk
+ k) 3/2 fig
1/2 t
2r
P(k)
0
B. N. MANDAL AND N. K. GHOSH
606
In this case the asymptotic form of the inertial surface can be obtained by a slmltar
analysis and the corresponding results given by Mandal and MukherJee [II] are easily
recovered.
5.
DISCUSSION.
.
From equation (4.10) and (4.12) we see that the surface wave amplitude, the
The amplitude
as well as
period T, and the wavelength A all depend upon
in both cases is zero if
> 1. The factor sin 2[t/T- r/A] or
I/2
both the wavelength
If r
(gh)
cos 2w[t/T -r/A] describes the actual waves.
r/(gh)I/2t
r/(gh)I/2t
and period will be infinite in wave motion and both decrease for fixed t or as t
increases for fixed r.
These observations were also arrived at for the corresponding
problem in water with a free surface (cf [3]).
Thus the qualitative behavior of the
solutions remains unaltered due to the presence of an inertial surface.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
STOKER, J.J., Water waves, Intersclence, New York, 1957.
LAMB, H., Hydrodynamics, Dover, New York, 1945.
KRANZER, H.C. and KELLER, J.B., Water waves produced by exploslon, J. Appl.
Phys. 30 (1959), 398-407.
CHAUDNURI, K., Waves in shallow water due to arbitrary surface disturbances,
Appl. Scl. Res. 19 (1968), 274-284.
WEN, S.L., A note on water waves created by surface disturbances. Int. J. Math.
Edu. Scl. Technol. 13 (1982).
RHODES-ROBINSON, P.F., On the generation of water waves at an inertlal surface.
J. Austral. Math. Soc. Set. B.25 (1984), 366-383.
MANDAL, B.N. and KUNDU, K., A note on the singularltles in the theory of water
waves with an inertial surface. J. Austral. Math. Soc. Set B. 28 (1986), 271278.
8.
9.
MANDAL, B.N., Ring source potential in a llquld with an Inertlal surface
Mech. Res. Comm. 13 (1986), 335-339.
MANDAL, B.N. and KUNDU, K., Ring source potential in a liquid with an inertial
surface in the presence of surface tension, int. J. Engg. Sci. 25 (1987).
I0. MANDAL, B.N., Water waves generated by disturbance at an interlal surface.
Appl. Sci. Res., (1987).
II. MANDAL, B.N. and MUKHERJEE, S., Water waves generated at an inertial surface by
an axisymmetrlc initial surface distribution. Int. J. Math. Edu. Scl.
Technol., (1987, accepted).