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Reflection and transmission of internal solitary waves across a barrier
Wave Motion 22 ( 1995) 325-333
Reflection and transmission of internal solitary waves across a barrier A. Jeffrey *, M.P. Ramollo Department
of Engineering Maths, University ofNewcastle upon Tyne, Newcastle, UK Received 26 August 1994
Abstract Transmission and reflection of an internal solitary wave on a two-layer fluid as it encounters a barrier is considered. An edgelayer theory is used to derive the reduced boundary conditions relevant to the shallow-water equations. It is found that for a moderate height of the barrier the incident wave is not affected except by being phase shifted. On the other hand, for a relatively high barrier the incident wave splits into a transmitted wave and a reflected wave. The reflected wave evolves into a small soliton with decaying ripples, and the transmitted wave evolves into a larger soliton and an oscillating tail.
1. Introduction This paper investigates an interface solitary wave on a two-layer system when it crosses a submerged vertical wall of negligible thickness. A soliton on a single layer system when it crosses a barrier has been studied by Sugimuto, Hosokawa and Kakutani [ 11. Our results are checked by comparison with theirs by setting the density ratio o = 0. This means that the top fluid layer may be ignored. Internal solitary waves have been studied by several authors. See for example Keulegan [ 21, Benjamin [ 31, Miles [ 4,5], Long [ 63, Kakutani and Yamasaki [ 7 I, Dai and Jeffrey [ 81, Segur and Hammack [ 91 and Koop and Butler [ lo]. They exist when the density of a fluid changes continuously or when there is a discontinuity in density. We shall consider a solitary wave in a two-layer fluid as it encounters a barrier. It will be assumed that far away from the step the horizontal component of the flow dominates the vertical component. Hence shallow-water theory holds in the regions far away from the step (the shallow-water region). However, near to the obstacle (the edgelayer region) the vertical component cannot be ignored, and so a new scaling is then required. The horizontal coordinate is resealed in an appropriate manner and the boundary conditions across the barrier are derived and then used together with the shallow-water equations to investigate the behaviour of a soliton as it encounters a barrier. It is found that the K-dV equations govern the flow. The initial conditions make the analytical investigation difficult. So a numerical method is used. It should be noted that solitons arise in the K-dV equation when there is a balance between the steepening effect due to nonlinearity and the smoothing effect due to dispersion. In this paper we exclude the case for which the coefficient of the nonlinear term vanishes. Thus we do not consider the case of the so-called critical thickness ratio. * Corresponding author Elsevier Science B.V. SSDIO165-2125(95)00032-l
A. Jeffrey, M.P. Rarnollo / Wave Motion 22 (1995) 325-333
326
Unless stated otherwise, a superscript plus/minus sign has been used to denote quantities to the right/left, respectively, of the barrier (x = 0) , and a subscript i = 1 or 2 has been used to denote quantities in the lower or upper layer, respectively. Where equations apply to both sides of x = 0, the + / - signs have often been omitted. Hence, quantities like &,J, h, are understood to be $J;’ ,f’ , and h:, respectively.
2. Shallow-water
equations
We consider an irrotational and inviscid flow of two superimposed immiscible horizontal rigid boundaries. We shall use the following scales to non-dimensionalize
x_ I
x=
1’
z+, t- t’G1 ,
&f
hi=
ao,
I
g!, 0 0
(i’l,
fluids confined between the governing equations
2) ,
two
(1)
where the dimensional variables and quantities are denoted by a prime. The 4; (i = 1,2) are the velocity potentials for the lower and upper fluids, 77is the displacement of the interface (see Fig. 1) and h, is the depth of the lower layer. 1 and a, are the characteristic wavelength and the wave amplitude, respectively, and C,, is the characteristic phase velocity for linear waves
We then obtain the following /-@,~+~,ZZ=o, p242r*f42zz=0
7
with the boundary
governing
equations in the shallow-water
region
-l
(2)
q
(3)
conditions
&=O,
z= -1,
(4)
&=O,
z=R,
(5)
and the matching conditions 4iz = P*( r), + a7,Jx4;,)
at the interface z = “~7 ,
(6a,b)
i = 1, 2 ,
(7) z’
0
1
A r”,
X’ h
~
barrier
\
z’= -h 1
Edge-layer Shallow-water regnn
Shallow-water regmn
Fig. I. Two-layer
fluid with barrier.
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (I 995) 325-333
32-l
The ratios R, p, a and o are defined by P=
R= $
!!L 1 ’
I
an
a=
-, h,
o=
!?Z
(8)
PI
where p, and p2 are the densities of the upper and lower fluids, respectively. p2 and o are assumed to be small, but of a comparable order of magnitude. Thus we make the shallow-water approximation assumption and seek solutions to Eqs. (2) and (3) which satisfy the boundary conditions (4) and (5). We find that 4, (1, z, r) =f, (A t) - $
+2(x, G t) =fz(x, t) Substituting
( 1-
(z+I)&+
$
$ (z-R)*fh+
(9)
$ (z-R)~~,,+...
for 4, and $2 in Eqs. (2.6)-(2.7)
ct t7x- cl3f?flx, -fd
,
(z+I)4f,_+...
( 10)
yields
$ C;( uR2f,,
- ~G(~f~ft,-flxfi,> +
-flsx>
= Wd?
P4).
These are the equations which we shall solve in the edge-layer region (i.e. in the vicinity of the barrier). proceed by deriving the boundary conditions.
f 13) So we
3. The edge-layer conditions In the neighbourhood of the barrier (that is as 1x1 -0) we anticipate an appearance of an edge-layer in which the motion in the vertical direction is as important as that in the horizontal direction. Thus we renormalize x by h, and introduce a new variable (=x/p. Changing the variable from x to 5, Eqs. (2)-(7) are rewritten as +,&f&=0,
-I
(14)
42K+42u=0,
q
(15)
subject to the boundary
conditions
&=O,
z= - I,
(16)
&=O,
z=R,
(17)
4i7=p2~,+a4ig77s, i=1,2atz=crq, G4,,+ 77+ $
G%4:,+ 49 = u
(18)
{
G42,+77+ $
Gc4&+4~,>
>
atz=aq,
(19)
while at the barrier A~-=0 >
-l
(20)
328
A. Jeffrey. M.P. Ramollo/ Wave Motion 22 (1995) 325-333
where r is as shown in Fig. 1 (in terms of the z variable the height of the barrier is 1 - r) . Expanding Eqs. (9) and ( 10) about the origin x = 0 and setting x = p.$ gives (5*- (z+ 1>*>_L+ f
Am=f,(O, r) +/-&fix+ 5
(g-3-35(z+1)2)fi,+...
,
(21)
(~3-3&-R)*)f2,+...
,
(22)
2 422m=.f2(0,
r> +/.&ix+
:
f
(5*--(Z-~)*1fzxx+
where here, and hereafter in this section,A (i = 1, 2) and its derivatives are evaluated at x = 0, so that they depend on t alone. We also have rim= 5
I (ei-fir)
+ P5(dx,-.flxr)
+ ; (CT&-jy,>
+...
- $
k@ficx,--hxr,--
52b&.cfix*t)
>
.
(23)
>
The matching conditions require +i+++(i=1,2)
andv+q,as
IQ+“.
Hence d+can be written in the form +i=+ico+P$i(&
Z9
t)
(i=
1,2)
(24)
9
where$ri+Oas 151-+m. When Eqs. ( 14)-( 20) are expressed in terms of I,!J~ we obtain ~,~~+~lu=O. $22a+G2CIZzt=0
-l
(25) (26)
O
with $lz=O,
z= -1,
1&=0,
(i=1,2)
atz=R,
(@2-
Il/~>,,+o(ap)
(27) (28)
P2G
*i, = ~_a
~,~=-fi~+~(z+l)tfi_+O(~~~),at5=0,
77=7jm+ j$
/-4Q~*,--+d+
,
(i=
1, 2)
(29)
at z=O,
-l
~(a~~-~~)+a(~f~~2C-f1x~*5)+
{
(30)
;w+k,+... >
(31)
where the boundary conditions have been taken to apply at z = 0 (instead of at z = a~). Since there is no physical discontinuity between the regions on the right and the left of t= 0, we have ++-4;=Ci, where Ci are constants, at t= 0 we have
(i=1,2), where the plus and minus denote, respectively,
a quantity to the right and left of [= 0. Thus
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (199s) 325-333
*:-*F=-t--‘Cf:-f;+C*)+ *I+&=
ti; -
- cl:,-f,)
9%= -/--‘(if:
$&-$2>=
-f;
-V&-f,>
(z+ 1)2(f:_-f;_)
+c2) + ;
+
5
-r
(33)
+O(p3) , O
(34)
,
+O($)
(~-R)~(f&--f&)
(~-R)~Cff---f2ru)
(32)
-r
;(2+1)2CfB-f1,)+o(y”))
+ g
329
+W3)
,
O
Applying Green’s theorem to Eqs. (25) and (26), and making use of the boundary conditions and (35) gives
(35) (27)-(
30), (33)
0
(4
- Icr:)t,dt+
0
I
--m
+wqb (~$2 - 4; >,,d5 > z=o
p”>
9
(36) and
(37)
Eqs. (29) imply that, to order 0( CL),the two fluids can be regarded as if they are enclosed by rigid walls. Hence, if the I,$ are found up to order O(p), the interface and the free surface can be treated as rigid walls. Therefore, following Sugamoto et al. [ I], the following reduced boundary conditions at t= 0 can be derived f I: -f
I, =W,f
:,+o(P2)
7
(38)
where K,
=
2
log(sin(Tr12))
.
(38)
In the same way it can be shown that for the upper layer f2:--f;=0(p2)
4. Transmission
,att=O.
(39)
and reflection
We now investigate the behaviour of a soliton when it is incident upon a barrier. Thus we solve Eqs. ( 1 l)-( 13) with the reduced boundary conditions (36)-( 39). Since we expect the presence of right going and/or the left going waves, we introduce the new time scales p *, and the long-space scale 7 as p+ =t-x,
p- =t+x,
7=ax,
(40)
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (1995) 325-333
330
where p + and p - denote the time measured, respectively, in the frames moving with unit velocity in the positive and negative directions of the x-axis. Introducing Eq. (40) into Eqs. ( 1l)-( 13) and retaining the first order terms we obtain the following relations;
;118*. f1p*p*=-77p*, f*zp*p*=
(41)
Here and hereafter the plus and minus signs are vertically ordered. From the second order terms we find that the elevation 17at the interface is governed by the K-dV equation 7771 Tar&*
T brlp*p*p* = 0,
(42)
where (Qb,c) Hence in this case the critical thickness ratio is R = &, though this critical case is not considered here. From the K-dV equations we identify two types of flow; namely, the case of an incident soliton propagating from the right, and the case when a soliton is incident from the left. However, since both sides of the barrier have equal depth these cases are identical. Consequently we shall consider only the case of a wave that is incident from the right-hand side of the barrier. Thus on the side t> 0 we have the incident and the reflected wave(s), while on the side t< 0 there is only the left-going transmitted wave(s). Hence we seek solutions of the form
f:,=IH+(/3+,7)-G+(p-,~11, fl,=--G-(/3-,7).
(43)
Then
fl:= -H+(j3+,r)-G+(P-,
fl,=-G-(P-,
r)+?+(t),
r)+(t),
(44)
where? * (t) are (so far) arbitrary functions of t. Thus from Eq. (44) we obtain
fii= +
f&= -$fL
[G-(P-97)+f-(01
, f;:= j [H+(P+, 7) -G+(P-,
7) +f+(f)l
(45)
where the S* are found by substituting Eq. (45) into Eq. ( 13), and as a result are given by g*(t) =
p(t)
The boundary conditions across t= 0 then become G+(t) =G-
+rG- , G- +H+
=G+
,
(46)
where
with the dot in the first equation signifying differentiation with respect to t (since x = 0). From the K-dV Eqs. (42) we see that the incident wave is given by G+ = sign(a)sech*
{J&3--
Y#-
(47)
A. Jeffrey, M.P. Rarnollo / Wave Motion 22 (1995) 325-333
331
Thus we have now arrived an initial value problem. Once the incident wave is known, the transmitted and reflected wave(s) may be found at T= 0. Hence the K-dV equations can be solved subject to the initial wave(s). Eqs. (46) have the same form as those which were derived by Sugimoto et al. [ 11. Hence we follow their argument and conclude that when r= 0( II) we have G+(r)
=G-(t-0
,
H+(r)=r&(t)
.
(48)
/ Reflected
-T
-T
= 2.25
“I,/ ,,,*,’ l\_,,’
= 3.25
I
Fig. 2. (a) The initial waves shown at x= 0. (b) Spatial evolution of the transmitted
wave. (c) Spatial evolution of the reflected wave
332
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (I 995) 325-333
That is, for a moderate height of the barrier, the only effect on the incident wave is a phase shift, while the reflected wave is of order 0( CL). On the other hand, for a relatively high barrier (i.e. r= 0( 1)) the equation that relates the initial waves (46) can be written as the differential equation dGa -
+G-
=sign(a)sech2x,
where
(49b) It is not easy to solve Eq. (42)) subject to initial conditions (46) and (49)) analytically. Accordingly, we have used the finite difference scheme proposed by Zabusky and Krnskal [ 121, Vliegenthart [ 131 and Johnson [ 141. The initial waves were found by using the DOlAMF subroutine (NAG library, Newcastle University) to integrate Eqs. (49). Several cases are considered. For r = 0.05 a system with density ratio u = 0.5 is investigated, with results shown in Fig. 2. The calculations were performed using A r= 0.0005 and A p = 0.09 so that the stability condition derived by Vliegenthart is satisfied.
5. Discussion of results It is seen from the numerical results that the transmitted wave evolves into a soliton and an oscillating tail. Zabusky [ 121 and Vliegenthart [ 131 found that due to an accumulation of error in the scheme used, oscillations occur behind the main solutions. However the oscillating tails which arise from this error vanish as the time step approaches zero. The tails in the present results remain the same even for a smaller time step (than those used here), which implies that these tails are not associated with the accumulation error. Indeed, from the form of the initial waves we should expect these tails which result from a non-zero continuous spectrum. Johnson [ 141 reported similar tails. The initial reflected wave has a negative part and the positive part. The negative component evolves into decaying ripples, and the positive component evolves into a soliton. Segur [ 151 showed that a wave with positive area evolves necessarily into, at least, one soliton, while a wave with negative area does not evolve into solitons. The amplitude of the initial transmitted wave decreases as the density ratio increases. For example, when R = 1.2 and r = 0.05 it is 0.8209 (initial wave) for u = 0.2, and it decreases to 0.6929 for u = 0.9. We recover the one-layer results by setting u =0 in Eqs. (42), (46) and (49). These are in agreement with Sugimuto et al. [ 1] which we regard as a test of our results. We have shown results for the elevation mode only. This is because of Eq. (49) which imposes a constraint on the parameter q. In order for the initial wave given by Eq. (49) to have the same mode as the soliton solutions of the K-dV Eq. (42) we require that q > 0, and thus R > CT.From this it is obvious that far away from the critical thickness ratio, R = G, there are no transmitted depression solitons. It would be interesting to study these waves in the neighbourhood of the critical thickness ratio.
Acknowledgement The second author (M.P.R.) was sponsored by the Association for a Ph.D. at the University of Newcastle upon Tyne.
of Commonwealth
Universities
while studying
A. Jeffrey. M.P. Ram&o / Wave Motion 22 (1995) 325-333
References [ I I N. Sugimoto, K. Hosakawa and T. Kakutani, J. Phys. Sot. Jpn. 56 ( 1987) 2744. [2] G.H. Keulegan, J. Res. Nat. Bur. Stand. 51 (1953) 131.
131 T.B. Benjamin, J. Fluid Mech. 25 ( 1966) 241. [4] J.W. Miles. Tellus 31 (1979) 456. 15 1J.W. Miles, Tellus 33 (1981) 397. [ 6 1R.R. Long, Tellus 17 ( 1965) 46. 17 1R.R. Long, Tellus 5 ( 1953) 42. [ 8 I T. Takutani and N. Yamasaki, J. Phys. Sot. Jpn. 45 (1978) 674. 19 1H.H. Dai and A. Jeffrey, Wave Motion 11 ( 1989) 463. I 101 H. Segur and J.L. Hammack, J. Fluid Mech. 118 (1982) 285. Ill] C.G. Koop and G. Butler, J. Fluid Mech. 122 (1981) 225. [ 121 N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240. [ 131 AC. Vliegenthart, J. Engg. Math. 5 (1971) 137. [ 141 R.S. Johnson, J. Fluid Mech. 54 (1972) 81. 1151 H. Segur, J. Fluid Mech. 59 (1973) 721.
333
Reflection and transmission of internal solitary waves across a barrier A. Jeffrey *, M.P. Ramollo Department
of Engineering Maths, University ofNewcastle upon Tyne, Newcastle, UK Received 26 August 1994
Abstract Transmission and reflection of an internal solitary wave on a two-layer fluid as it encounters a barrier is considered. An edgelayer theory is used to derive the reduced boundary conditions relevant to the shallow-water equations. It is found that for a moderate height of the barrier the incident wave is not affected except by being phase shifted. On the other hand, for a relatively high barrier the incident wave splits into a transmitted wave and a reflected wave. The reflected wave evolves into a small soliton with decaying ripples, and the transmitted wave evolves into a larger soliton and an oscillating tail.
1. Introduction This paper investigates an interface solitary wave on a two-layer system when it crosses a submerged vertical wall of negligible thickness. A soliton on a single layer system when it crosses a barrier has been studied by Sugimuto, Hosokawa and Kakutani [ 11. Our results are checked by comparison with theirs by setting the density ratio o = 0. This means that the top fluid layer may be ignored. Internal solitary waves have been studied by several authors. See for example Keulegan [ 21, Benjamin [ 31, Miles [ 4,5], Long [ 63, Kakutani and Yamasaki [ 7 I, Dai and Jeffrey [ 81, Segur and Hammack [ 91 and Koop and Butler [ lo]. They exist when the density of a fluid changes continuously or when there is a discontinuity in density. We shall consider a solitary wave in a two-layer fluid as it encounters a barrier. It will be assumed that far away from the step the horizontal component of the flow dominates the vertical component. Hence shallow-water theory holds in the regions far away from the step (the shallow-water region). However, near to the obstacle (the edgelayer region) the vertical component cannot be ignored, and so a new scaling is then required. The horizontal coordinate is resealed in an appropriate manner and the boundary conditions across the barrier are derived and then used together with the shallow-water equations to investigate the behaviour of a soliton as it encounters a barrier. It is found that the K-dV equations govern the flow. The initial conditions make the analytical investigation difficult. So a numerical method is used. It should be noted that solitons arise in the K-dV equation when there is a balance between the steepening effect due to nonlinearity and the smoothing effect due to dispersion. In this paper we exclude the case for which the coefficient of the nonlinear term vanishes. Thus we do not consider the case of the so-called critical thickness ratio. * Corresponding author Elsevier Science B.V. SSDIO165-2125(95)00032-l
A. Jeffrey, M.P. Rarnollo / Wave Motion 22 (1995) 325-333
326
Unless stated otherwise, a superscript plus/minus sign has been used to denote quantities to the right/left, respectively, of the barrier (x = 0) , and a subscript i = 1 or 2 has been used to denote quantities in the lower or upper layer, respectively. Where equations apply to both sides of x = 0, the + / - signs have often been omitted. Hence, quantities like &,J, h, are understood to be $J;’ ,f’ , and h:, respectively.
2. Shallow-water
equations
We consider an irrotational and inviscid flow of two superimposed immiscible horizontal rigid boundaries. We shall use the following scales to non-dimensionalize
x_ I
x=
1’
z+, t- t’G1 ,
&f
hi=
ao,
I
g!, 0 0
(i’l,
fluids confined between the governing equations
2) ,
two
(1)
where the dimensional variables and quantities are denoted by a prime. The 4; (i = 1,2) are the velocity potentials for the lower and upper fluids, 77is the displacement of the interface (see Fig. 1) and h, is the depth of the lower layer. 1 and a, are the characteristic wavelength and the wave amplitude, respectively, and C,, is the characteristic phase velocity for linear waves
We then obtain the following /-@,~+~,ZZ=o, p242r*f42zz=0
7
with the boundary
governing
equations in the shallow-water
region
-l
(2)
q
(3)
conditions
&=O,
z= -1,
(4)
&=O,
z=R,
(5)
and the matching conditions 4iz = P*( r), + a7,Jx4;,)
at the interface z = “~7 ,
(6a,b)
i = 1, 2 ,
(7) z’
0
1
A r”,
X’ h
~
barrier
\
z’= -h 1
Edge-layer Shallow-water regnn
Shallow-water regmn
Fig. I. Two-layer
fluid with barrier.
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (I 995) 325-333
32-l
The ratios R, p, a and o are defined by P=
R= $
!!L 1 ’
I
an
a=
-, h,
o=
!?Z
(8)
PI
where p, and p2 are the densities of the upper and lower fluids, respectively. p2 and o are assumed to be small, but of a comparable order of magnitude. Thus we make the shallow-water approximation assumption and seek solutions to Eqs. (2) and (3) which satisfy the boundary conditions (4) and (5). We find that 4, (1, z, r) =f, (A t) - $
+2(x, G t) =fz(x, t) Substituting
( 1-
(z+I)&+
$
$ (z-R)*fh+
(9)
$ (z-R)~~,,+...
for 4, and $2 in Eqs. (2.6)-(2.7)
ct t7x- cl3f?flx, -fd
,
(z+I)4f,_+...
( 10)
yields
$ C;( uR2f,,
- ~G(~f~ft,-flxfi,> +
-flsx>
= Wd?
P4).
These are the equations which we shall solve in the edge-layer region (i.e. in the vicinity of the barrier). proceed by deriving the boundary conditions.
f 13) So we
3. The edge-layer conditions In the neighbourhood of the barrier (that is as 1x1 -0) we anticipate an appearance of an edge-layer in which the motion in the vertical direction is as important as that in the horizontal direction. Thus we renormalize x by h, and introduce a new variable (=x/p. Changing the variable from x to 5, Eqs. (2)-(7) are rewritten as +,&f&=0,
-I
(14)
42K+42u=0,
q
(15)
subject to the boundary
conditions
&=O,
z= - I,
(16)
&=O,
z=R,
(17)
4i7=p2~,+a4ig77s, i=1,2atz=crq, G4,,+ 77+ $
G%4:,+ 49 = u
(18)
{
G42,+77+ $
Gc4&+4~,>
>
atz=aq,
(19)
while at the barrier A~-=0 >
-l
(20)
328
A. Jeffrey. M.P. Ramollo/ Wave Motion 22 (1995) 325-333
where r is as shown in Fig. 1 (in terms of the z variable the height of the barrier is 1 - r) . Expanding Eqs. (9) and ( 10) about the origin x = 0 and setting x = p.$ gives (5*- (z+ 1>*>_L+ f
Am=f,(O, r) +/-&fix+ 5
(g-3-35(z+1)2)fi,+...
,
(21)
(~3-3&-R)*)f2,+...
,
(22)
2 422m=.f2(0,
r> +/.&ix+
:
f
(5*--(Z-~)*1fzxx+
where here, and hereafter in this section,A (i = 1, 2) and its derivatives are evaluated at x = 0, so that they depend on t alone. We also have rim= 5
I (ei-fir)
+ P5(dx,-.flxr)
+ ; (CT&-jy,>
+...
- $
k@ficx,--hxr,--
52b&.cfix*t)
>
.
(23)
>
The matching conditions require +i+++(i=1,2)
andv+q,as
IQ+“.
Hence d+can be written in the form +i=+ico+P$i(&
Z9
t)
(i=
1,2)
(24)
9
where$ri+Oas 151-+m. When Eqs. ( 14)-( 20) are expressed in terms of I,!J~ we obtain ~,~~+~lu=O. $22a+G2CIZzt=0
-l
(25) (26)
O
with $lz=O,
z= -1,
1&=0,
(i=1,2)
atz=R,
(@2-
Il/~>,,+o(ap)
(27) (28)
P2G
*i, = ~_a
~,~=-fi~+~(z+l)tfi_+O(~~~),at5=0,
77=7jm+ j$
/-4Q~*,--+d+
,
(i=
1, 2)
(29)
at z=O,
-l
~(a~~-~~)+a(~f~~2C-f1x~*5)+
{
(30)
;w+k,+... >
(31)
where the boundary conditions have been taken to apply at z = 0 (instead of at z = a~). Since there is no physical discontinuity between the regions on the right and the left of t= 0, we have ++-4;=Ci, where Ci are constants, at t= 0 we have
(i=1,2), where the plus and minus denote, respectively,
a quantity to the right and left of [= 0. Thus
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (199s) 325-333
*:-*F=-t--‘Cf:-f;+C*)+ *I+&=
ti; -
- cl:,-f,)
9%= -/--‘(if:
$&-$2>=
-f;
-V&-f,>
(z+ 1)2(f:_-f;_)
+c2) + ;
+
5
-r
(33)
+O(p3) , O
(34)
,
+O($)
(~-R)~(f&--f&)
(~-R)~Cff---f2ru)
(32)
-r
;(2+1)2CfB-f1,)+o(y”))
+ g
329
+W3)
,
O
Applying Green’s theorem to Eqs. (25) and (26), and making use of the boundary conditions and (35) gives
(35) (27)-(
30), (33)
0
(4
- Icr:)t,dt+
0
I
--m
+wqb (~$2 - 4; >,,d5 > z=o
p”>
9
(36) and
(37)
Eqs. (29) imply that, to order 0( CL),the two fluids can be regarded as if they are enclosed by rigid walls. Hence, if the I,$ are found up to order O(p), the interface and the free surface can be treated as rigid walls. Therefore, following Sugamoto et al. [ I], the following reduced boundary conditions at t= 0 can be derived f I: -f
I, =W,f
:,+o(P2)
7
(38)
where K,
=
2
log(sin(Tr12))
.
(38)
In the same way it can be shown that for the upper layer f2:--f;=0(p2)
4. Transmission
,att=O.
(39)
and reflection
We now investigate the behaviour of a soliton when it is incident upon a barrier. Thus we solve Eqs. ( 1 l)-( 13) with the reduced boundary conditions (36)-( 39). Since we expect the presence of right going and/or the left going waves, we introduce the new time scales p *, and the long-space scale 7 as p+ =t-x,
p- =t+x,
7=ax,
(40)
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (1995) 325-333
330
where p + and p - denote the time measured, respectively, in the frames moving with unit velocity in the positive and negative directions of the x-axis. Introducing Eq. (40) into Eqs. ( 1l)-( 13) and retaining the first order terms we obtain the following relations;
;118*. f1p*p*=-77p*, f*zp*p*=
(41)
Here and hereafter the plus and minus signs are vertically ordered. From the second order terms we find that the elevation 17at the interface is governed by the K-dV equation 7771 Tar&*
T brlp*p*p* = 0,
(42)
where (Qb,c) Hence in this case the critical thickness ratio is R = &, though this critical case is not considered here. From the K-dV equations we identify two types of flow; namely, the case of an incident soliton propagating from the right, and the case when a soliton is incident from the left. However, since both sides of the barrier have equal depth these cases are identical. Consequently we shall consider only the case of a wave that is incident from the right-hand side of the barrier. Thus on the side t> 0 we have the incident and the reflected wave(s), while on the side t< 0 there is only the left-going transmitted wave(s). Hence we seek solutions of the form
f:,=IH+(/3+,7)-G+(p-,~11, fl,=--G-(/3-,7).
(43)
Then
fl:= -H+(j3+,r)-G+(P-,
fl,=-G-(P-,
r)+?+(t),
r)+(t),
(44)
where? * (t) are (so far) arbitrary functions of t. Thus from Eq. (44) we obtain
fii= +
f&= -$fL
[G-(P-97)+f-(01
, f;:= j [H+(P+, 7) -G+(P-,
7) +f+(f)l
(45)
where the S* are found by substituting Eq. (45) into Eq. ( 13), and as a result are given by g*(t) =
p(t)
The boundary conditions across t= 0 then become G+(t) =G-
+rG- , G- +H+
=G+
,
(46)
where
with the dot in the first equation signifying differentiation with respect to t (since x = 0). From the K-dV Eqs. (42) we see that the incident wave is given by G+ = sign(a)sech*
{J&3--
Y#-
(47)
A. Jeffrey, M.P. Rarnollo / Wave Motion 22 (1995) 325-333
331
Thus we have now arrived an initial value problem. Once the incident wave is known, the transmitted and reflected wave(s) may be found at T= 0. Hence the K-dV equations can be solved subject to the initial wave(s). Eqs. (46) have the same form as those which were derived by Sugimoto et al. [ 11. Hence we follow their argument and conclude that when r= 0( II) we have G+(r)
=G-(t-0
,
H+(r)=r&(t)
.
(48)
/ Reflected
-T
-T
= 2.25
“I,/ ,,,*,’ l\_,,’
= 3.25
I
Fig. 2. (a) The initial waves shown at x= 0. (b) Spatial evolution of the transmitted
wave. (c) Spatial evolution of the reflected wave
332
A. Jeffrey, M.P. Ramollo / Wave Motion 22 (I 995) 325-333
That is, for a moderate height of the barrier, the only effect on the incident wave is a phase shift, while the reflected wave is of order 0( CL). On the other hand, for a relatively high barrier (i.e. r= 0( 1)) the equation that relates the initial waves (46) can be written as the differential equation dGa -
+G-
=sign(a)sech2x,
where
(49b) It is not easy to solve Eq. (42)) subject to initial conditions (46) and (49)) analytically. Accordingly, we have used the finite difference scheme proposed by Zabusky and Krnskal [ 121, Vliegenthart [ 131 and Johnson [ 141. The initial waves were found by using the DOlAMF subroutine (NAG library, Newcastle University) to integrate Eqs. (49). Several cases are considered. For r = 0.05 a system with density ratio u = 0.5 is investigated, with results shown in Fig. 2. The calculations were performed using A r= 0.0005 and A p = 0.09 so that the stability condition derived by Vliegenthart is satisfied.
5. Discussion of results It is seen from the numerical results that the transmitted wave evolves into a soliton and an oscillating tail. Zabusky [ 121 and Vliegenthart [ 131 found that due to an accumulation of error in the scheme used, oscillations occur behind the main solutions. However the oscillating tails which arise from this error vanish as the time step approaches zero. The tails in the present results remain the same even for a smaller time step (than those used here), which implies that these tails are not associated with the accumulation error. Indeed, from the form of the initial waves we should expect these tails which result from a non-zero continuous spectrum. Johnson [ 141 reported similar tails. The initial reflected wave has a negative part and the positive part. The negative component evolves into decaying ripples, and the positive component evolves into a soliton. Segur [ 151 showed that a wave with positive area evolves necessarily into, at least, one soliton, while a wave with negative area does not evolve into solitons. The amplitude of the initial transmitted wave decreases as the density ratio increases. For example, when R = 1.2 and r = 0.05 it is 0.8209 (initial wave) for u = 0.2, and it decreases to 0.6929 for u = 0.9. We recover the one-layer results by setting u =0 in Eqs. (42), (46) and (49). These are in agreement with Sugimuto et al. [ 1] which we regard as a test of our results. We have shown results for the elevation mode only. This is because of Eq. (49) which imposes a constraint on the parameter q. In order for the initial wave given by Eq. (49) to have the same mode as the soliton solutions of the K-dV Eq. (42) we require that q > 0, and thus R > CT.From this it is obvious that far away from the critical thickness ratio, R = G, there are no transmitted depression solitons. It would be interesting to study these waves in the neighbourhood of the critical thickness ratio.
Acknowledgement The second author (M.P.R.) was sponsored by the Association for a Ph.D. at the University of Newcastle upon Tyne.
of Commonwealth
Universities
while studying
A. Jeffrey. M.P. Ram&o / Wave Motion 22 (1995) 325-333
References [ I I N. Sugimoto, K. Hosakawa and T. Kakutani, J. Phys. Sot. Jpn. 56 ( 1987) 2744. [2] G.H. Keulegan, J. Res. Nat. Bur. Stand. 51 (1953) 131.
131 T.B. Benjamin, J. Fluid Mech. 25 ( 1966) 241. [4] J.W. Miles. Tellus 31 (1979) 456. 15 1J.W. Miles, Tellus 33 (1981) 397. [ 6 1R.R. Long, Tellus 17 ( 1965) 46. 17 1R.R. Long, Tellus 5 ( 1953) 42. [ 8 I T. Takutani and N. Yamasaki, J. Phys. Sot. Jpn. 45 (1978) 674. 19 1H.H. Dai and A. Jeffrey, Wave Motion 11 ( 1989) 463. I 101 H. Segur and J.L. Hammack, J. Fluid Mech. 118 (1982) 285. Ill] C.G. Koop and G. Butler, J. Fluid Mech. 122 (1981) 225. [ 121 N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240. [ 131 AC. Vliegenthart, J. Engg. Math. 5 (1971) 137. [ 141 R.S. Johnson, J. Fluid Mech. 54 (1972) 81. 1151 H. Segur, J. Fluid Mech. 59 (1973) 721.
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