Intern. J. Math. & Math. Sci.
Vol. 6 No. 2 (1983) 395-402
395
GENERALIZED FUNCTION TREATMENT OF THE
ALFVIN-GRAVITY WAVE PROBLEM
L. DEBNATH and K. VAJRAVELU
Department of Mathematics
East Carolina University
Greenville, N. C. 27834, U.S.A.
(Received September 30, 1982 and in revised form April 12, 1983)
ABSTRACT.
A study is made of the steady-state Alfvn-gravity waves in an inviscid
incompressible electrically conducting fluid with an interface due to a harmonically
oscillating pressure distribution acting on the interface.
The generalized function
method is employed to solve the problem in the fluid of infinite, finite and shallow
depth.
A unique solution of physical interest is derived by imposing the Sommerfeld
Several limiting cases of physical interest are ob-
radiation condition at infinity.
tained from the present analysis.
The physical significance of the solutions and
their limiting cases are discussed.
KEY I#ORS AND PHRASES. Alvdn-ravi waves, enerazed run,ions, FoYer transforms of dtribions, water ,v, dersion reons.
I80 MATHEATICS SUBJECT CLASSIFICATION CES. 76B15, 76(95.
i.
INTRODUCTION.
Steady-state surface wave problem in which the wave motion is simple harmonic in
time is one of the most fundamental problems in the theory of surface wave phenomena
in oceans.
This is, indeed, a key problem for discussion of a variety of results and
However, it is, in general, not possible to derive a
unique solution of the problem by imposing only bounded condition at infinity. In
observations in water waves.
fact, sharper conditions such as the Sommerfeld radiation condition are required to
obtain a unique solution of physical interest.
Several methods have been employed to solve the steady-state wave problem in order
to obtain a unique solution of physical interest.
fictitious damping force method
method
These methods include (i) Lamb’s
rl, (ii) Stoker’s complicated
[2], (iii) Lighthill’s method [37
complex variable
concerning an alternative way of applying
the radiation condition and (iv) Debnath’s generalized function method [47.
All
these methods have been aimed at deriving a unique solution of physical interest, and
in fact, led to the same solution.
An extensive usage of classical Fourier transform methods in
water wave problems
and other wave problems are well known and readily available in the existing
L. DEBNATH and K. VAJRAVELU
396
literature (Stoker
[27, Sneddon (57)
on the subiects.
However, Debnath F47 has
clearly pointed out certain inherent difficulties involved in the steady-state water
wave problems and in their method of solution by the classical Fourier analysis.
These difficulties are mainly related to the existence, uniqueness, and the real
A careful
singularies of the inverse Fourier integral solution of the problems.
study indicates that a new treatment of the steady-state surface wave problem by the
generalized function method [47 can not only eliminate the inherent difficulties of
the classical Fourier transform analysis, but also enable us to obtain unique solution
Thus it seems to be an extremely useful device from the view
of physical interest.
point of sufficient generality.
Another convincing point about this method is that
nor any
I7
obtained by Lighthill’s
there is neither any need for modification of the basic equations (Lamb
justification of the limit operation involved in the solution
method [37.
In spite of the tremendous progress on the steady-state surface wave problems in
non-conducting fluids, hardly any attention has been given to the steady-state
Alfvn-gravity wave problem in
The main purpose of
an electrically conductin fluid.
this paper is to make a steady-state investigation of the
Alfvn-gravity waves in
an
inviscid, incompressible electrically conducting liquid with an interface due to a
harmonically oscillating pressure acting on the interface.
A new generalized function
method will be employed to solve this problem in the fluid of infinite, finite and
shallow depth.
A unique solution of physical interest is derived by imposing the
Sommerfeld radiation condition at infinity.
been obtained from the present analysis.
and their limiting
Several limiting cases of interest have
The physical significance of the solutions
cases are discussed.
MATHEMATICAL FORMULATION.
2.
We consider the two-dimensional linearized steady-state Alfvn-gravity wave
problem in an inviscid, incompressible electrically conducting fluid of uniform density
In its undisturbed state, the field-free conducting
with a free surface.
0
fluid of constant depth
occupies the region
h
-h _< z -< 0
in the rectangular
Cartisian coordinate system with the origin at the free surface, and the region
z > 0
is a vacuum under the action of constant magnetic field
B
(B0,0,0)
0
We investigate the generation and p$opagation of magnetohydrodynamic surface
waves in the fluid system due to a harmonically-oscillating pressure distribution
acting at the interface
0
z
in the form
p(x,t)
P
where
and
m
is a constant,
f(x)
Pf(x)e
imt
(2.1)
is a physically realistic arbitrary function of
x
is the fixed frequency.
In view of the fact that the motion is irrotational, there exists a velocity
potential
(x,z;t)
which satisfies the Laplace equation
V2
in
-h _< z -< 0
and
< x
<
+
0
(2.2)
ALFVEN-GRAVITY WAVE PROBLEM
397
Outside the fluid the vacuum magnetic field must be a potential field so that the
also satisfies the Laplace equation
magnetic potential
)2
x +
V2
in
z > 0
)z
(2.3)
0
< x <, with the boundary condition at infinity
and
Ivl
0
as
z+
.
(2.4)
The free su#face dynamic and kinematic conditions in the linearized form are
4)t +
gn
+
B0
z
p(x,t)
z
at
nt
n(x,t)
where
(2.5 ab)
0
4)z
is the vertical free surface elevation, and
g
is the acceleration
due to gravity.
The interfacial condition describes the constraint of frozen-in lines of force
in the fluid on the vacuum magnetic field so that the interface remains a magnetic
field line even in the disturbed state and is given by
Bn
BO x
z
(2.6)
0
z
on
In the case of infinitely deep fluid, the bottom boundary condition is given by
4)
0
(2.7)
z
as
For the case of fluid of finite depth, the boundary condition at the rigid bottom is
4)z
0
(2.8)
-h
z
on
In addition, an appropriate radiation condition has to be imposed to ensure the
uniqueness of the solution of the steady-state wave problem.
This will be done later
at appropriate places.
3.
SOLUTION OF THE PROBLEM IN A FLUID OF INFINITE DEPTH.
As a trial, we assume that functions
transform with respect to
x
4)(k,z;t)
4)
#
and
possess ordinary Fourier
defined by the integral
1
[
e
-ikx
4)(x,z:t)dx
(3.I)
Application of the Fourier transform method to equations (2.1)-(2.7) gives solution for
4)
@
and
n
as
pim
(k,z;t)
(k,z;t)
(k) e
0
imt
D(m,k)
PilklB O (k)
0
+
Iklz
(3.2)
e
D(m,k)
(3.3)
L. DEBNATH and K. VAJRAVELU
398
P
(k t)
(k)eimt
lkl0 D(m,k)
(3.4)
where
and
B / 410
a
is the
O
Alfvn
glk
m2
D(m,k)
a2k 2
(3.5)
wave velocity.
The use of the inverse ourier transformation gives the integral solutions for
and
in the form
e
ikx
(3.6 abc)
dk
It is noted that these inversion integrals are divergent because of the fact that
the integrands of (3.6 abc) have polar singularities at the zeros of
axis of integration.
and
D(m,k)
on the
Consequently, the original assumDtion that functions
possess the Fourier transform in the ordinary sense was invalid.
This is
logical because a function which has a Fourier transform must tend to infinity, but
and
we expect
to have wave-like behaviour, that is, bounded at infinity
but not tending to zero at infinity.
To overcome this inherent difficulty, we treat the functions
Lihthill 6]
generalized functions in the sense of
as
in which case they do have the
Thus (3.6 abc) represent valid inversion integrals in the gener-
Fourier transform.
alized sense.
and
So one can use the generalized function method to evaluate the inte-
gral (3.6 abc).
It would be sufficient for determination of the principal features
method, that is, if
P(k)
We next evaluate (3.6c)
(x,t)..
of the wave motions to evaluate the integral for
by using result (24) of Debnath and Resenblat [77
based on the generalized function
k
has a simple pole at
k
I
in
a < k
I
< b
then as
b
P (k)
e
ikx
dk
i
Sgn x
e
iklX
F (k)
(residue of
at
k
k
I)
+
0(i I).
(3.7)
a
Making use of the result, we evaluate integral (3.6c) written explicitly in the
form
P e
(x,t)
imt
kl
0
(x,t)
to obtain the solution for
P i e
n(x,t)
0 N/2
+
imt
a
2
sgn x
(k2-k 11
k
I
e
ikx
dk
D(m,k)
Ixl
as
[
(k)
(3.81
in the form
x
(kl e-iklX k2(k2)e _ik2
kl(kl)eiklX- k2(k2)eik2xl
+
0[,,]
(3.9)
ALFVEN-GRAVITY WAVE PROBLEM
k
where
I
k
and
-
399
are the real roots of the dispersion equation
2
D(,k)
glk
2
a2k 2
(3.10)
0
The solution (3.9) for the free surface elevation
n(x,t)
contains four
Alfvn-gravity
two of them correspond to outgoing dispersive
exponential terms
waves
from the source of disturbance and the other two represent incoming waves from in-
finity.
In order to achieve unique solution of physical interest, we impose the Sommerfeld
-
incominE wave terms present in the solution (3.9).
(x,t) in the form
radiation condition to eliminate the
This leads to the unique solution for
P i e
n(x,t)
0a
2
[k (kl)e-ikllxl k2(k2)eik2’x’]
it
I
(k2_kl)
Alfvn-gravity
This result corresponds to dispersive
as
c
-
km
Cpm
C
+
I
+ 2a2k
/gk + a2k 2
2
In the absence of the external magnetic field
E
(x)
the Dirac delta distribution so that
pm
and the group velo-
(3.13)
B
.
all these results reduce to
-
In particular, when
(k)
we also recover
Debnath’s result for the steady-sace gravity waves from (3.11) in the limit a
4.
(3 11)
(3.12)
those for pure gravity wave solution due to Debnath [4
f(x)
C
a2)1/2
(k
3k
gm
lllj
waves characterized by the
dispersion relation (3.10) which gives the phase velocity,
city
+ 0
SOLUTION OF THE PROBLEM IN A FLUID OF FINITE DEPTH.
0
Application of the generalized Fourier transform to (2.1)-(2.6) combined with
$
bottom boundary condition (2.8) gives solutions for
(k
z’t)
e
f(k)e
0
D,k)
Pi
(k,z;t)
imt
+/-m
Ikl
B
0
(z
C_osh[_k_l
Cosh|klh
(k)
0
P
(k,t)
eimt
(k)
0
tanhlklh
and
+h)
exp(imt-
as
(4.1)
Iklz)
(4.2)
D, (re,k)
Ik! tanhlk, lh
D,(m, k)
(4.3)
where
2
D,(m,k)
(glk + a2k 2)
tanhlklh
(4.4)
The inverse Fourier transformation enables us to obtain an explicit integral
solution for
(x,t)
as
L. DEBNATH and K. VAJRAVELU
400
P e
(x,t)
imt
I_
ikltanhlkl h (.+
2Oo
where
a
(k)
is given by
-
)(k)eikxdk
(4.5)
(glkl + a2k2)tanhlklh
2(k)
Making use of result (3.7), we can evaluate the wave integral (4.5).
out that the significant contribution to the solution for
from the poles of the integrand of (4.5).
a(k)
+/-
m
a(k)
0
n(x,t)
as
Ixl
It turns
comes
/
These poles are the roots of the equation
and are readily found from the points of intersection of the curves
with the straight lines
+/-
From the graphical representation of
m
these curves, it turns out that real roots of the above equations are at
k
+/-
k
+ k*
2
Using the same formula (3.7), the wave integral (4.5) can easily be evaluated
to obtain the solution as
P i e
(x, t)
imt
sg.n x
(kl*)
2po/’2T
F(k2*)
{e ikl, x + e
{e ik2*x + e -ik2*x
]
-ikl*x
+
0
(4.7)
171
where
P(k) =[(k) k tanh
kh]/[k
(4.8)
As in the case of infinitely deep fluid, the solution (4.7) consists of both
outgoing waves from the source of the disturbance and incoming waves from infinity.
In order to eliminate the incoming solution, we require to impose the Sommerfeld
radiation condition.
This leads to the unique solution of physical interest in the
form
P i e
n(x,t)
it
[
(kl*)e _ikl,x
This solution also represents dispersive
the dispersion relation
of the wave motions.
a2(k
by Debnath [4].
(B 0
Alfvn-gravity
which gives the phase and
O)
(3.11).
(4.9)
waves characterized by
te
In the limiting case of infinitely deep fluid
solution (4.9) reduces to result
magnetic field
m2
]
(k2,) eik2,x
group velocities
(kh
=)
On the other hand, when there is no external
these findings are in agreement with those obtained earlier
ALFVEN-GRAVITY WAVE PROBLEM
5.
401
-IISPERSIVE WAVES IN A VERY SHALLOW CONDUCTING FLUID.
(kh
In this case
c2
gh
tanhlklh
0)
Iklh
e2(k)
so that
c2k 2 +
a2hlklk 2
Consequently, the integral solution (4.5) can easily be simplified to
(x,t)
obtain an explicit representation for
P0
(x, t)
eimt
a2 2
!
k 23+Bk2+D(k)
ikx
(e
+
e
-ikx)
(5. i)
dk
where
c2
B
ha--/
> 0
oj
D
and
2
ha--/
(5.2 ab)
< 0
The significant contribution to the solution for
singularities of the integrand in (5.1).
are the roots of the cubic equation for
k
Since the constant form in
k
comes from real polar
in the form
+ Bk 2 + D
(5.3)
n(x,t)
The singularities of the integrand in (5.1)
(5.3)
0
is negative, this cubic equation must, there-
fore, allow at least one positive real root,
k
k
This lead to factorize (5.3)
0
in the form
k0)(k2
(k-
+ 2sk + 8)
(5.4)
0
By comparison with equation (5.3), it turns out that
D
2k0
> 0
and
8
D
-;---K0 >
(5.5 ab)
0
It can then readily be shown that the two remaining roots of (5.4) are either real
and negative or complex conjugate with negative real parts.
A simple inspection reveals that the significant contribution to the solution
for
(x,t)
comes from the positive real poles of the cubic (5.4).
Following the
same procedure used earlier, the integral (5.1) can easily be evaluated to obtain the
solution in_the form
(x, t
P e
it
0
(i sgn x)
a2k/-
(k02
k02(k0)
+ 2ek0 +
8)
.eik0x + e-ik0x
(5.6)
By imposing the radiation condition, a unique solution of physical interest has
the form
(x,t)
Pni
0
sgn
a2k/
k02(k0)
(k02 + 2ak0 +
This solution represents the dispersive
8)
e
i(mt-k
01xl)
(5.7)
Alfve’n-gravity waves characterized by the
dispersion equation (5.3).
readily be calculated.
Hence, the phase and the group velocities of the waves can
In the absence of the magnetic field (B
0 and a
0)
0
the abmve result (5.7) reduces to the corresponding solution in a shallow
non-condmcth/g
liquid, and shows a striking contrast to the non-dispersive shallow water waves.
L. DEBNATH and K. VAJRAVELU
402
6.
CLOSING REMARKS.
The above entire analysis has been carried out for an arbitrary function
involved in the applied pressure given by (2.1).
f(x)
f(x)
Some simple functional form for
can easily be chosen to illustrate the general theory presented in this paper.
However, the present study has revealed the principal features of the dispersive
Alfven-gravlty wave motions in a conducting liquid.
ACKNOWLEDGEMENT.
This work was partially supported by a research grant from East
Carolina University.
REFERENCES
i.
2.
3.
4.
LAMB, H. Hydrodynamics, Cambridge University Press, London, 1932.
STOKER, J. J. Water Waves, Interscience Publishers, New York, 1957.
LIGHTHILL, M. J. Studies on Magnetohydrodynamic Waves and Other Anisotropic
Wave Motions, Phil. Trans. Roy. Soc. A252 (1960) 397-430.
DEBNATH, L.
A New Treatment of the Steady-State Wave Problem
(1969) 502-506.
I,
Proc. Nat.
Institute Sci., India A35
1952.
5.
SNEDDON, I.N.
6.
LIGHTHILL, M. J. An Introduction to Fourier Analysis and Generalized Functions,
Cambridge University Press, 1958.
7.
DEBNATH, L., and ROSENBLAT, S.
Fourier Transforms, McGraw Hill Publishing Company,
The Ultimate Approach to the Steady-State iN the
Jour. Mech and Appl. Math.
Generation of Waves on a Running Stream,
22 (1969) 221-231.
Ouart.