- Email: [email protected]
Reflection of interface solitary waves at a slope
WAVE M O T I O N 11 (1989) 463-479 NORTH-HOLLAND
463
R E F L E C T I O N O F I N T E R F A C E S O L I T A R Y W A V E S AT A S L O P E H.H. DAI and A. JEFFREY Department of Engineering Mathematics, The University of Newcastle upon Tyne, NE1 7RU, U.K. Received 17 May 1988
This paper deals with the problem of the reflection of interface solitary waves at a slope in a two-layer fluid system with rigid upper and lower boundaries. An edge-layer theory is developed to treat the region near to the slope in which shallow-water theory is not valid. We work to O(e 3/2) so that the effects of the surging movement at the shoreline are included. After deriving the 'reduced boundary conditions' relevant to the shallow-water equations, a perturbation procedure involving methods of strained coordinates and inner-outer expansions is employed to solve this problem. Analytical results, such as m a x i m u m run-up at the slope and the time when it occurs, are presented. For a plane slope with an angle of inclination ~, = ~r/4, we present some graphical results. It is found that, for the reflection problem, the properties of the depression mode differ from those of the elevation mode in m a n y ways. In Appendix A, we also present an alternative way of obtaining the second order solutions for interface solitary waves in an infinite region.
1. Introduction
Internal solitary waves in shallow water in an unbounded region have been studied by many authors. Keulegan [1] was the first to consider solitary waves in the interface between two superimposed fluids with slightly different densities when the fluids are contained between two horizontal rigid boundaries, while Long [2] considered the case of an arbitrary density ratio. Kakutani and Yamasaki [3] also considered a two-layer model but with a free upper surface. Segur and Hammack [4] and Koop and Butler [5] made some experimental investigations. Various other authors investigated continuous stratification cases; notably, Peters and Stoker [6], Benjamin [7], Benny [8] and Miles [9], [10]. Benjamin [7] established a theory for this class of long solitary waves with his effective variational integral method. Benny [8] worked with Eulerian coordinates and provided an alternative method. Miles [9], [ 10] (also Kakutani and Yamasaki [3]) considered cases in which the cubic and quadratic nonlinearity are of comparable significance. For second order solutions see Gear and Grimshaw [11] and Koop and Butler [5]. For a general review, see Miles [12]. Recently large scale internal solitary waves have been observed in the ocean (Osborne and Burch [13]; Osborne, Burch and Scarlet [ 14]) and these observations have generated additional interest in this problem. Unfortunately comparatively little progress has been made with the reflection problem, although reflection at a vertical wall has been studied (Mirie and Su [15]; Mirie and Su [16]) as it is equivalent to the problem of the collision of two identical solitary waves in an infinite region. Hence, in this paper, we will deal with the reflection of interface solitary waves incident upon a slope, which may be of an arbitrary shape. The problem is shown in Fig. 1. Away from the sloping region, it is reasonable to assume that the effect of the slope is negligible, and hence that the well-known weakly nonlinear dispersive wave theory (the shallow-water theory) is valid. However, close to the sloping region, as the boundary condition on the sloping surface requires the normal 0165-2125/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
464
H.H. Dai, A. Jeffrey / Interface solitary waves
velocity to be zero, the motion in the vertical direction becomes of primary importance, and so shallow-water theory is no longer valid. To overcome this difficulty, we use an edge-layer theory. The edge-layer theory was first introduced by Sugimoto and Kakutani [17] and Sugimoto et al. [18] to treat the reflection of a shallow-water soliton at a sloping beach. Jeffrey and Dai [19] extended the theory to a higher order approximation to include the effect of the surging movement and gave analytical results. Sugimoto et al. ([20], [21]) also applied this theory to the problems of an incident soliton over a step and an incident soliton over a vertical barrier. In our problem we divide the whole region into three parts, as shown in Fig. 1. One is the shallow-water region in which shallow-water theory is valid. Another is the edge-layer region in which two-dimensional motion is of primary importance. Between these two regions is a third region called a matching region. Using the matched-asymptotic-expansion method, we derive a matching condition. By applying Stokes' theorem in the edge-layer, two boundary conditions (corresponding to the upper and lower fluids) relevant to the governing equations in the shallow-water region are obtained, while the edge-layer solutions themselves are reduced to finding two irrotational flow-fields (corresponding to the upper and lower fluids) in the semi-channnel regions induced by a source and sink distribution on the slope. If v = "rr/m (m an integer), the edge-layer solutions can be expressed in integral form.
Edge Loyer z'
Matching Region
x'= Blz')
.
x
Fig. 1. Illustration of the problem. In Section 4, a perturbation procedure combining strained coordinates and inner-outer expresions is employed to solve the governing equations in the shallow-water region under the derived boundary conditions and imposed initial condition. Analytical results, such as maximum run-up at the slope and the time when maximum run-up is attained, are obtained. From our results, by letting m = 2, we recover the case of the collision of two identical interface solitary waves, and our results agree with those of Mirie and Su [15] (to order O(~2), in their paper). However, the time at which maximum run-up is attained for the collision of two identical interface solitary waves is new and is given for the first time in this paper. In Appendix A, starting from the shallow-water equations, and using the method of strained coordinates, we obtain the second order solutions for interface solitary waves in an infinite region. The phase velocity is accurate up to O(e2). The solution provides an initial condition for our problem.
2. Governing equations in the shallow-water region We shall consider the incompressible irrotational and inviscid flow of two superimposed immiscible fluids between two horizontal rigid boundaries. The coordinates are chosen as shown in Fig. 1.
H.H. Dai, A. Jeffrey / Interface solitary waves
465
We shall non-dimensionalize the equations in the shallow-water region using the following scales x = x ' / l,
z = z ' / hl,
t = ( t'/ l)Co,
*l = ~f / ao,
(i = 1, 2),
~, =~-~ ~I/(aoCo)
(2.1)
where @1 and ~2 are the velocity potentials for the lower and upper fluids respectively, 7/is the interface displacement, I the characteristic wave length, ao the characteristic wave amplitude and Co the characteristic phase velocity for linear waves. Thus 5 ( 1 - tr)R ghl, C°= V tr+R
(2.2)
tr = PE/Pl,
(2.3)
where R = hE~ha,
p2 and Pl, h 2 and h~ are the densities and thicknesses of the upper and lower fluids, respectively, tr and R are two important parameters. The wave is an elevation wave (or a depression wave) if tr is smaller (or greater) than R 2. In this paper we shall not consider the critical ease tr = R 2, which we leave for future investigation. The governing equations are as follows: 2 i~)1xx @ I~-)1zz ~---0 ,
1 < z < er/,
--
tz2~zxx + qb2= = 0,
(2.4) (2.5)
~'q
q0~z-- 0
at z = - 1 ,
(2.6)
q~2z= 0
at z = R ,
(2.7)
• i =/z2(rh + e~ix~x) 2[R(1-tr)
L ~--+)~- q'.+n
( i = 1,2) ]
at z = e~,
(2.8)
,R(1-tr)
42
tr+~(~2q~x+~z)
= t r { / 2 [ R ( 1 - t r ) ~ 2 , + ~7] q" , R ( a - t r )
tr+~-R--
}
2 o ' + ~ (/z2tp22x+~2z) '
(2.9)
where/.t = h~/l and ~ = ao/h are assumed to be small. However, the Ursell number Ur = ~//x 2 is assumed to be of order one. Next, we expand q~l and ~2 in powers of p2 (of. Whitham [22]) q~l = ~ ( - 1 ) " ,.=o
~2--,.=o
(z+l) TM a2mfl
2,.
ax2m p,
(2m)!
(2.10)
'
02mf2 O-~/~2m
(z-R)
(-1)" ~
(2.11)
It is easy to confirm that (2.4)-(2.7) are satisfied automatically. Further, from (2.8)-(2.9) up to O(/~4), we obtain 2 /Z
.-
_/z
4
2 .-(6)
n, + [(1 + "~/)Ax]x --g-J,x.xx-r ~-.~j ,
~/x
,
~.
---~- ~~7J,~,). = O,
(2.12)
H.H. Dai,A. Jeffrey/ Interfacesolitarywaves
466 ~ , + [ ( , r / - , aR~r 2 x J x1- - +# ~
( ~1 + )~
2
4
R3¢ # J2=xx-'~.
n _ (crf2 _A,) _ ~" (o.f~: _ f l ~: )
2
R5¢(6) ,/z J2-'-~--RZ(*lf2xxx)~=O,
(2.13)
# " (~R4f~xxx~,-f,xx~,) + #7: (~R2f~xx,-fl~x,)+T.,
+ "#2 (o'R:f~xx-f~xx) +--~,#2 (erR 2f2, f2x,~ - f ~ , f~,~) 2
Oz n (o'Rf2xx, +f~x,,) = 0.
(2.14)
Equations (2.12)-(2.14) are the equations we shall deal with. Second order interface solitary wave solutions were first obtained by Koop and Butler [5] and confirmed by Gear and Grimshaw [11]. In Appendix A, working from equations (2.12)-(2.14), we obtain second order solutions in a different fashion by using the method of strained coordinates. The surface displacement is
*l = sign(a) sech2{ C +
( l + "k,)[X- Xo+ ( l + " ~ ) t ] }
~ / 1 - ~ 2 (1 + ,k,)
Ix
-
1
• tanh2{~ll~b~(l+,k,)[X-Xo+(l+,~)t]}.
(2.15)
The coefficients a, b, k~, C are defined in Appendix A. For the phase velocity, we obtain a result accurate to O(~2). Koop and Butler [5] and Gear and Grimshaw [11] only gave the first order result. Hence equation (2.15) for Xo>>1, t = 0 provides an initial condition for the reflection problem for interface solitary waves at a slope.
3. The edge-layer theory For the single-layer fluid case, Sugimoto and Kakutani [7] argued that the edge-layer develops seaward to an order O ( # ) length scale. Actually, as the boundary condition on the sloping surface requires the normal velocity to be zero, the motion in the vertical direction is as important as that in the horizontal direction in the near-sloping region. Hence the non-dimensional scale factor along both directions should be the same. In the near-sloping region, the depth and width are of the same order. Thus it is reasonable to choose h~ as the non-dimensional scale factor. This is equivalent to introducing a new variable ¢ = x/#, corresponding to the length scale of the matching region being of the order O(#). Then, from equations (2.4)-(2.9), we obtain for the governing equations in the edge-layer region • 1~+ ~lzz = 0,
-1 < z<,'0,
(3.1)
~2~e+ ~2= = 0,
,~
(3.2)
~lz = 0
at z = - 1 ,
(3.3)
~2z=0
at z = R ,
(3.4)
~iz - #2rh - e ~ e ~ = 0
(i = 1, 2)
at z = "7,
(3.5)
H.H. Dai, A. Jeffrey / Interface solitary waves
467
r~ -~ (1 -- o')__17
=or[r/+(1-or)R~2,-t" or+R
~ (I_+~_~R(~22 + ~ ) 2#x2
]
atz=er/.
(3.7)
The boundary conditions at the slope ~ = B ( z ) ( B ( z ) is assumed to be of order one) are -1
Ix (~l~-Bz~lz)=O, (1 + B2z)1/2
-~rs< ~?< 0 ,
(3.8)
B(R)< ~<-~s,
(3.9)
-1
(1 +izB 2) 1/2 ( ~ 2 ~ - B z ~ 2 z ) = O ,
where -~rs is the position of the interface on the slope. In the matching region, we have the following matching condition: the edge-layer region solutions ~l,2(x/Iz, z; t) with x fixed a n d / z ~ 0 shallow water region solution 4~1,2(tz~:,z; t), with ~r fixed and/.~-~0. Setting x =/z~: in (2.10), (2.11) and (2.14), and expanding around zero, we obtain
•
oo
oO
( - 1 ) - (z+ l) m (2m) !p!
oo
(3.10)
2m+pfpL~x p÷2" .L,=o'
oo
F 3p+2m ¢ "1
fl)2oo=m~=op~=o(__l)m(Z-R) 2m "~m+p/:pi
(2m)!p! #z
J'~| . Loxp+2. jx:o,
(3.11)
and
+!2 (4L -f~) -~- (orR~A~x,-f,~,) + . . .
•
(3.12)
Here, and hereafter, even if we do not emphasize it, f and its derivatives are evaluated at x = 0. Here we also stress that in (3.12) we need to retain terms up to order O(~ 4) and O ( ~ 2 ) , though for brevity we omit the details. The matching condition requires ~1 ''>
II~lco,
~2oo "~ ~2c~
and
r/--> r/~
as ~: -+ oo.
Thus we suppose that in the edge-layer region ~ , = ~ , ~ + / z V~(t) ~b;(se, t)
(i=1,2).
(3.13)
We introduce factors Vl(t) and V2(t) in order to make ~/'1 and ~b2 independent of t. The matching condition requires that
In fact ~bl, ~b: should tend to zero at a rate faster than an algebraic one. Afterwards, we shall show that I~tl, I]/2 are in fact exponentially small.
H.H. Dai, A. Jeffrey / Interface solitary waves
468
Substituting (3.10)-(3.12) into (3.5)-(3.7) and expanding z at Zo= ~7o(B(0), t), while retaining terms up to O(/z3), O(O~), we obtain
V'O'= -/z2 ~r( V2q&)" V'0')'I+o'/R - (
2(1 +cr/R)/Ze [ (~r2V202~-2 2
+ 2f, x ~( v2~,2),~ - ( v, ~,,),~ ~ 2 v,~,~(~Ax, - f,x,).] j=0 l+o'/R l +cr/R o'(V2,1,~),-(v,,l,,),+
~7 = ~7~+tz
~
~ 2
V , ~ , ¢2+ 2 o ' f 2 = V 2 ~ 2 ~ - 2 f ~ x V , ~ , e ) 2
(i= 1, 2)
at z = ~r/o,
(3.14)
2
2(1 +tr/R) [orV2'~2~-- VII//I~
l+o'/R
+ 2 crf2x V2ff=e - 2Ax 1/1 @, e + 2/~:(crf2x= V2 r/2e -f~x= V, 0, ¢) 2
2
2
2
+ trV2t~2z - V,~b,z+21z(trRf2=V202=+fl~V,~b,z)]
at z = Er/o.
(3.15)
From (3.1)-(3.3), and (3.8), we obtain ~blee+ ~1= = 0,
--1 < z < ~r/,
(3.16)
O2ee+ ~2= = 0,
eTq
(3.17)
0~z = 0
at z = - 1 ,
(3.18)
V,(Ol¢_BAb,=)=_f~_lxf~XX~z[(Z+l)B] +~
(z+l)2
2 - - J ~2 x x x
..[ ].g3
d
, 2fix==dzz d [ ( z + 1)B 2] -~/x
63
d
--6J~xxxx-~z [(z+l)3B]+-~-~--- flxxxx
[(z + 1)B3]"
(3.19)
Next we apply Stokes' theorem to (3.16) along the boundary Os~ of the region shown in Fig. 2(a) and to (3.17) along the boundary 0s2 of the region shown in Fig. 2(b). That is, we use the result V~
d~ Sl
(0,¢dz-~b,zd~:)=0
(i=1,2).
(3.20)
Substituting (3.18), (3.14) and (3.19) into (3.20), and noticing that ~ - > 0 as ~:->o~, we obtain
-f~x-I~B(O)flxx - elXrloB~(O)fl~-~Iz 1 2 B 2 (0)f~x+~/z 1 2f~x~ 3 3 2 fco o.(V21[12)u_(Vll]tl)tt + ~---6 B(0)f~ . . . . -~-----B3(O)f]xxxx-I 6 d, /,-~ ~ i,_,~o _ d~ = 0.
zt
z.,.E',
""/"
/ / /ez=R/
z:-I /
/
e"
z= er/~
/ / / /
(a)
(b)
Fig. 2. Geometry for the application of Stokes' theorem (a) the lower layer; (b) the upper layer.
(3.21)
H.H. Dai, A. Jeffrey / Interface solitary waves
469
I f we let V~ = 2fix and V2 = 2f2x and substitute (3.21) back to (3.19) up to O(/.0, we obtain qJle-B~qJlz = -
~{
1
d [(z+l)B(O)]}_~s<,=B(z)
1- B(0) dz
'
(3.22)
As fix is of the order O(/~), to be accurate up to O(~tz) or O(/~ 3) we only need ~0~ to be correct up to O(/z). From (3.16), (3.18), (3.14) and (3.22), we see that to this order qJa is independent of the time, and the problem of finding qq is equivalent to finding a potential field in a semi-infinite channel caused by a source and sink distribution on the slope. Similarly, we can obtain the result
--fz~--IxB(O)A,,x--tXenoB(O)fz~ -~/~ ' 2B 2(0)f2xxx+~/z , 2R 2f2x~ +~_y_R2B(O)f2x.x~y_Bs(O)fzxx~+lZ2 | o
o
a-e~
tr(V2~b2).-( cr + R
d~=O.
(3.23)
z=,,0
If the slope is a plane wall with an angle of inclination v = 1r/m (m an integer), an integral expression of ~0~ can be obtained. We omit the result here, but for the necessary details we refer to Sugimoto and Kakutani [17]. After a suitable transformation taken from Sugimoto and Kakutani's results, qJ2 can be expressed in a similar form. It can be proved that
~bl(~,z)~Ae-~ecos[~r(z+l)]+O(e
-2~¢ )
as ~ ,
where A is a constant. This agrees with the matching condition. A similar conclusion holds for q~2(~:,z). Here, we again emphasize that the expansions are carried out at z = ~7o(B(0), t) instead of z = 0. We argue that the edge-layer is so narrow, compared with the characteristic wave length, that the surface displacements at each point in edge-layer at the same time are almost identical. Thus, in this manner, we believe that the surging movement of the shoreline can be taken into account. Our asymptotic results, which will be presented in the next section, confirm this argument. As equations (3.21) and (3.23) are evaluated at x = 0 , they provide two boundary conditions for the governing equations in the shallow water region.
4. The perturbation procedure In this section we shall use a perturbation procedure to solve equations (2.12)-(2.14) under the initial condition (2.15) and the boundary conditions (3.21) and (3.23). The method is similar to that used by Jeffrey and Dai [19] for the one-layer fluid case, in that it combines the method of strained coordinates and the method of inner-outer expansions. First, we introduce the strained coordinates
X=(l+~yOx,
T=(l+~y,)(l+av,)t,
(4.1)
and let
f,x= F,,
f~x= F2.
(4.2)
Further, we suppose that '17 = T'](O)-[ - EI/2T'](1)"I - ~'l~(2)-J - ~3/2T](3)"~ O(~3/2),
F, = Fl°)+ e I/2F~ 1)+ eF~2) + e3/EFI3)+ o(e 3/2)
(4.3) ( i = 1,2).
470
H . H . D a i , A . J e f f r e y / I n t e r f a c e solitary w a v e s
Substituting (4.1)-(4.3) into (2.12)-(2.14), we obtain:
first term ~(T0)+ rr(0)F1X= 0 ,
~(T0)+ ,,u 101:7(°) 2X = 0 ,
(4.4a, b)
(l+cr/R)r/(xO)_, ~ 0 ",1-"(2 oT )-~- (aol T) ~! = O;
(4.4c)
third term /2
~7(T2) + ~(2) --,x _- v, @r°)+ (n(°)F~°))x
7 T(-
1
(r(0)
r(o)
1-' I X T T - - O ' I - ' 2 X T T 1
(4.5a)
= 0,
• 6(1 + or/R) e g 2 Dl[7(2) .a(o) ~-0, ,', n ~ ) + (n(°)F(?))x /z• 6(1 + cr/R)~~F(O) 2, - - ~) x 1XTT_ ~°I2XTT]--
(1 + cr/ R ) ~ x(u) -
(2) - -
(orF2T
(2)
(4.5b)
,(Or/"2T ,~(0)- ,~(0), I~IT)
FIT ) --/)1 2
, ~(o)r(o) /z 1 __l+(rR (cr~(°)--trrr- - ~(o) ~=0. --(or/~2 / ~ 2 X - - l~(o)~(o)~q l alX] J[1TTTJ
(4.5c)
• 2 l+cr/R
For the sake o f brevity we will not give the second and fourth terms. F r o m (4.4a)-(4.4c), we obtain
77(°) = G(°)(X + T) + H(°)(X - T),
(4.6a)
crF(2° ) - F~ °) = (1 + o ' / R ) [ G ( ° ) ( X + T) - H(°)(X - T)],
(4.6b)
F~O)+ RF(2O)= i(o)(T),
(4.6c)
where G (°), H (°) and I (°) are arbitrary functions of their arguments. Using the initial condition (2.24) we have
G(°)(X)+H(°)(X)=sign(a)sech2[,~)~2(l+ekl)(x-xo) V 12b/z
].
(4.7)
To determine G(°)(X + T), H ( ° ) ( X - T) and I(°)(T), we need two other conditions. However, we shall not use the b o u n d a r y conditions (3.21) and (3.23) directly. As argued before, in the region very near to x = 0, the length scale is O(/z), due to the effect o f the slope. Thus there is a ' b o u n d a r y layer' near to x = 0. We introduce the scaled coordinates.
= X/iz,
T = T,
F 1 = U1,
F 2 = U2,
77 N,
(4.8)
=
and s u p p o s e that N = N (°) + •I/2N(D + EN (2) + •3/2N(3) + O(•3/2), U / = U~ 0) Jr- • 1/2 U~I)._F •U~2>..~_ •3/2 U~3) q_ 0(•3/2)
(4.9)
(i = 1, 2).
Substituting (4.8) and (4.9) into (2.21), (2.23), (3.26) and (3.29), we obtain:
first term u(O) , e -_0
N(eO)= 0
(i = 1, 2),
(4.10a)
with -- u~O)
- B ( O ) U ~(o) e - ~ B1
2
(o)
1
(o)
1
(o)
(O)Ule~+gUI~+gU1~e((B(O)-B3(O))=O
_U(20) - ~,t,,J l ~ / ~ lr~,2¢ r(0) -2,-, I_I.)2[FY~I?(O) ..l_1102 ¥ r(o) ~,,J ,-" 2e~6,- ,-,2ee+~U(2°e)ee(R2B(O) - a3(0)) = 0
at so=0, at ~: = 0;
(4.10b) (4.10c)
H.H. Dai, A. Jeffrey/Interface solitary waves
471
third term It
U ( 2 ) 2_
/L/, ,~ T--/Ti N(~)+ ( N (°) U~°))~:
-°rr(z)+ it N(l)+tN(°)U(°)~'¢-itT~. ~-'-'2e el/2 ( l + o . / R ) N T ) _ _ _Eil /t2,
2
r r(o) ~ t r(o) ( t..~ 1 6 T T - - ,a, ,...."2 6 T T / ---~ 0 ,
(4.11a)
2 R2 (IT(O) o'If(O) \ E 6 ( l + o ' / R ) ~,-'levr- ,-~2~TT)=0,
2 ls
( O r,,(i) U2T-
1 6(1 + t y / R )
(4.11b)
U~))_(o.U(2o)rr(O)_~,2eU~°)1r(°)x0 , v ~, e-
(4.1 lc)
with - U ~ z) - B ( O ) r•~le r( 2 ) _
(2)
+gUIeee(B(O) -
U ~ 2) -
1
B(0)
(2)
_
i r(o) ylB(0),-,l¢
-
. .(o) r r ( 2 ) _~ ! r r ( 2 ) B~(0)'OoUle -½B2(0) i--Jl~:sc-6t-~l~e
2
2
2trD2 ,,(ol +It 2D1 rr(O) - 0 E l + t r / R u2rr ~ l + o ' / R , - , 2 r r -
3 B (0))
It
ate:=0,
(4.11d)
ate=0,
(4.11e)
t r(o) _ (o) r(2) "-'1,~ Bz(O)noU2e - ½ B 2 ( 0 ) ~ -t ' r(2) 2 ¢ 6 --a-6 ! o, . 2 lr'-"26~
~2~
r r(2) _ Yl B(0)
2
+gU2~*e(R B(0)-B3(0))
it_2 2o'D2 rr(O) _LIt2 2D1 tr(O) - n ~ tr+R ,-'2rr- E l + t r / R ~ ' r r - "
where D 1=
f
~bl(~, 0) ds¢,
D2 =
~s
Lo02(s¢, 0) ds¢.
(4.12)
G
We shall call (4.4a)-(4.5c) the outer equations and (4.10a)-(4.11e) the inner equations. The basic procedure is now as follows. From the outer equations we can determine ~/u), F~) and F(2°(i = 0, 1, 2, 3) in terms of three arbitrary functions G~(X + T), H")(X - T) and IU)(T) (i = 0, 1, 2, 3). The initial condition provides a relation between G u) and H u). From the inner expansions, U~°, U~') can be determined, but the N u) (i = 0, 1, 2, 3) are given in terms of the arbitrary functions jU)(T) (i = 0, 1, 2, 3). The matching conditions between F~~) and U~°, F~° and U(2°, 7/") and N "), that is i terms (inner) matching i terms (outer), will give the other three conditions. Thus G °), H u), I u) and j , ) can be uniquely determined. From (4.10a)-(4.10e), we obtain UI°) = 0
(i = 1,2),
N(°)=J(°)(T).
(4.13)
The matching conditions between (o'F(2°) - F] °)) and (crU~°) - ~--'ll r(o)]], ~--lt~(°)+ RF~O)) and ( U]°)+ RU~ °)) give G(°)(T) - H(°)( - T) = 0,
j(o)(T) = 0.
(4.14a, b)
From (4.7) and (4.14a), taking note of the fact that Xo>>1, we obtain
G(°)(X)=sign(a)sech2[~l~b--~E2(X-Xo) ] ,
r/%
H(°)(X)=sign(a)sech2L~[ l - ~ 2 ( X + X o )
],
(4.15a)
yl=k,,
(4.15b, c)
where
Xo= (1+ ~k,)xo.
(4.16)
The matching condition between ~(o) and N (°) then gives j(O)(T) = 2 sign(a) sech 2 O,
(4.17)
H.H. Dai,A. Jeffrey/ Interfacesolitarywaves
472 where
[ lal" ,.. (-~o-
0= ~ / ~
T).
From (4.5a)-(4.5c), eliminating F~2) and F~22), we obtain
(4.18)
H.H. Dai, A. Jeffrey / Interface solitary waves
473
where
/ 01 = B ( 0 ) q
lal 12b'
03=sign(a)
02=
sign(a) 1 - - o ' / R 2 4 l+cr/R '
-2B(0) l+o'/R
(4.25a, b)
CB(O)+7~ sign(a) B(0)
R)D 2 + D, B(0)] " + 2Bz(O)- 12 ( t r /i--+ ~---'~
(4.25c)
We point out that when or = 0, we recover the one-layer fluid case. The present results agree with those obtained by Jeffrey and Dai [19]. If we let B ( 0 ) = 0 , corresponding to a vertical wall, then up to O(~) (scaled by hi) our results agree with those given by Mirie and Su [15] who considered the collision of two interface solitary waves. However, Mirie and Su did not give the time at which maximum run-up occurs. Here we emphasize that the first two terms on the right hand side of equation (4.23) are exactly the same as the corresponding results in the vertical wall case. Thus the third term, which represents the effect of the slope, plays an important role. It may be worth considering cases in which the slope has different forms and seeing the effect on the maximum run-up and the time at which maximum run-up is attained. We leave consideration for the future.
5. Discussion of results
First, consider the reflection of a solitary wave at a vertical wall, or the collision of two identical solitary waves. In this case B(0) = 0. Then, from equations (4.26a)-(4.26c), we see that 01 = 03 = 0, and for both the elevation mode (1 - o,2/R > 0) and the depression mode (1 - t r / R 2 < 0), 02 < 0. Thus maximum run-up only occurs after a time delay. From Appendix B, in the case B ( 0 ) = 0 we have 1 - t r / R 2 _½ ~7= sign(a) [sech 2 LI + sech 2 R1] + ~ 1 + o-/R [ sech 2 L~ tanh Ll(tanh R1 - 1) -½ sech 2 R1 tanh R~ (tanh L~ + 1) + 2 sech 2 L~ sech 2 R1] + C sech 2 L1 tanh 2 L I + C sech 2 R~ tanh 2 R l } .
(5.1)
Letting t ~ o0, and using Taylor's theorem, we obtain rI = sign(a) sech 2 Ri + eC sech 2 Ri + O(e2),
(5.2)
where
Ri Ax
--
~f -~~ e /xe
(1 + , 7 , ) [Lx + X o - (1 + ~ ) t + d x ]
1/2 s i g 2 ( a ) ~
(5.3)
1 - or/R 2
Vlal l+tr/R"
(5.4)
This shows that the reflected wave has an extra phase shift. To the order O(~), our results agree with those obtained by Mirie and Su [15].
H.H. Dai, A. Jeffrey / Interface solitary waves
474
If we consider the reflection at an overhanging boundary ~ = e W 2 B ( Z ) ( > 0), then in a manner similar to that used in Sections 3 and 4, we obtain the following result fl -o'/R 2 r/= sign(a) [sech 2 L~ + sech 2 R~] + ~'[] ~ [-½ sech 2 L~ tanh L, (tanh R ~ - 1) -½ sech 2 R~ tanh R1 (tanh L, + 1)+ 2 sech 2 L1 sech 2 R~] + C sech 2 L~ tanh 2 L~ + C sech 2 R1 tanh 2 R 1 } + , sign(a)4B(0)~/ll-~b sech2 R1 tanh Rl.
(5.5)
Similarly, we have r/= sign(a) sech 2/~, + ~C sech z/~, tanh 2/~1,
t ~ co,
(5.6)
where (5.7)
1¢ 12b/z sign(a) . / ] ~
a~=~'/2
2
1-o'/R 2
V lal I+~/R
(5.8)
~'/22B(0).
Thus, if we let
B(0)
sign(a) , / ~ 1 - ~r/R 2 ~/ - ~ l + tr / R '
(5.9)
to order O(¢), the reflected wave will be exactly the same as the incident wave, although it propagates in the opposite direction. This represents a real 'elastic' collision. We now give some graphical results for a plane slope with an angle of inclination v--'rr/4. From Fig. 3(a), we see that, for the elevation mode, the value of maximum run-up decreases as tr becomes larger. For the depression mode, the tendency is opposite for tr in the range 0.8 to 0.9, although the value of the maximum run-up increases by a very small amount. In both cases the maximum run-up 1.0
-1.0
R=2"5
o"
0.8.
•
0.6
R=0.48
0.0
-0.8
oi!
- 0.6
O"
.90 0.85
O8O 0.4
- 0.4
0.2 ¸
- 0.2"
0
0
0.1
0.2
0.3
0
0
OJ
0.2
Fig. 3. The m a x i m u m r u n - u p at a p l a n e slope with a n a n g l e o f i n c l i n a t i o n ~, = ,rr/4 fo r different wave a m p l i t u d e s . (a) The e l e v a t i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for tr = 0.8, (b) the d e p r e s s i o n mode; the b r o k e n line c o r r e s p o n d s to the vertical wall case for cr = 0.9.
14.1-1. Dai, A. Jeffrey / Interface solitary waves
475
4.5
4.5
R=0.48
R=2.5
CY
r
4.0 . . ¢ ~ - - . ~
/
" ' - ~ ' - . "-... -..
0.8 0.6 0.4 0.2 0.0
4.0.
o"
~
0 . 8 0
0.85 0.90 %
/ 3.50
_~_ 0.1
~ 0.2
~ 0.3
(a~ 3 . 5 - 0
-
,0.1
. • ........ 0.2
(b)
Fig. 4. The t i m e w h e n m a x i m u m r u n - u p at a p l a n e s l o p e with a n a n g l e o f i n c l i n a t i o n ~, = ~ / 4 is a t t a i n e d for different w a v e a m p l i t u d e s . (a) The e l e v a t i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for ~ = 0.8, (b) the d e p r e s s i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for (7 = 0.9.
~0.1
0
1
2
3
4
5
" T:3.6
,4,,
3
i T:3.6
T:4.0 /
]0.1
T=O
T~2.5
5
,
6
io.t
T~4.4
,L
/~. . . . . . . . . . . . . . . T=3.9 "T=4.0
I0.1
T:4.1
/ / _ ,f,.,
T=4.8
T=5.5
1o.1
T=8.0
T0.1
T:4.4 T=5.5 T:8.0 ,0
1
2
3
4
5
.......
(~
Fig. 5. The w h o l e process o f reflection o f an i n c i d e n t solitary w a v e initially at x o = 4 with a m p l i t u d e ~ = 0.1. (a) The e l e v a t i o n m o d e w i t h R = 2.5, ~ = 0.8, (b) the d e p r e s s i o n m o d e with R = 0.48, cr = 0.9.
H.H. Dai, A. Jeffrey / Interface solitary waves
476
at the slope is larger than that at a vertical wall, as would be expected. Comparing the two modes, the case of the depression m o d e is seen to give a slightly larger maximum run-up. Consideration of when the time of m a x i m u m run-up is attained shows that the value of the time decreases as or becomes larger in the case of the elevation mode. For the depression mode, the opposite occurs. In both modes this time becomes smaller as the initial wave amplitude increases. This conclusion seems to hold only for a slope with a small inclination. For the case of a gentle slope (say, ~ = ~r/3) the tendency may be the opposite, as reported by Jeffrey and Dai [19] for the one-layer case. Comparing the two modes, the case of the depression mode gives the larger value of the time. However, in both modes, the case of a vertical wall gives a much smaller value of the time. In Fig. 5, we show the whole process of reflection. In both cases, we find that, when the incident wave gets close to the slope, before run-up begins, the wave amplitude became smaller compared with the incident wave amplitudes. These results are shown in Figs. 5(a) (t = 3.2), 5(b) (t = 3.6). In the case of the elevation mode, the m a x i m u m run-up is 2.10 times the incident wave amplitude, which is attained at t = 3.99. In the case of the depression mode, the maximum run-up is 2.30 times the incident wave amplitude, which is attained at t = 3.92. In both cases, the reflected waves are found to have experienced distortion, relative to the incident waves. For the elevation mode the distortion is small. The amplitude of the reflected wave is 1.05 times the amplitude of the incident wave. However, for the depression mode, the distortion is found to be large. The amplitude of the reflected wave is 1.5 times the amplitude of the incident wave. Further, a small elevation wave train is found in front of the main depression wave. In both cases, the reflected waves move faster than the incident waves. In conclusion, it seems that, at least for the problem of the reflection of interface solitary waves at a slope, the properties of the depression mode differ from those of the elevation mode in many ways.
Acknowledgment The first author (H.H. Dai) is being sponsored by K.C. Wong Education Foundation Ltd., while studying for this Ph.D. at the University of Newcastle upon Tyne.
Appendix A. The second order solutions of interface solitary waves We introduce the strained coordinates 151/2
~:=
be
~3/2
K ( x - t),
r=
ge
/.L
Ktot,
-
A=~-~TiA,
].£
-
f2=~-i-~f2,
(A.1)
and suppose that
p = p(O)+ ep(l)+
~2p(2)+o(e2)
( P = ~:,fl ,f2, K, to).
(A.2)
Substituting (A.1) and (A.2) into (2.21)-(2.23), and retaining only the first term, we obtain Mo
0 W (°)
of
=0,
(A.3)
H.H. Dai, A. Jeffrey / Interface solitary waves
477
where -1
1
-1
0
Mo =
1-tr
).]
n(')
-OR
1-tr
o'(1-cr
l + tr/ R
-~-~J
w.,_-
'
(A.4)
(i--0,1,2).
The left and right eigenvectors of Mo are respectively Le=
[.1.n
,[
J
Re-
1,1,-
(A.5)
.
Retaining terms up to the second order, we obtain Mo
c3W
(1)
a¢
SI=
+ Sl T/~ ) + e~"2 _(o)_(o) ~ c _(o) ~ $4~(o) = O, ~I ~I~ - - ~-~3'1 ~(:t~- -
1
r ,
$2 =
(1- or)(1- cr/R2] ' l+cr/R
S3-
(A.6)
l
-1/6
,
I(1--~)(1-~/R)-[ L 2(l+tr/g)
s.=
(A.7)
J
To eliminate the secular terms we multiply (A.6) from the left, by Le to obtain 7(o)--
(o)~/~(o) -r a~7(o)~7~ ~',~¢ = 0,
(A.8)
where
Le" $2 3 1 - c r / R 2 a - L e " S-----]l-2 l+o~/R'
Le" $3
1 l+trR
(A.9, 10)
b=Le" S-------~l-6l+cr/g"
Equation (A.8) is the well-known KdV equation, which has the steady solitary wave solution ~7(°)= sign(a) se
(A.11)
Similarly, retaining terms up to third order, we find that the elimination of secular terms leads to 2(1+ o-/R) 7?(~>+ 3 (1 - or/R2)~?(°),/~1>+ 3(1 -o-/R2)n(~°)~la)+½(l+o-/R)~)~ • (o) - I(1 - ~/R2)~/(°)7/7 ) -½(1 + o'R)~/¢e~+(°)[ _ 6 ( l + t r / R 3) +2tOl(1 + o'/R)*l~ 3(1-tr/R2)2"], 42 l + t r / g j t ~ _(o)_(o)_I(1
(0),,2
(0)+2
],/~
(0)
[
l
l+~rR }(1-o-) ]
kl~(l+trg)~/,¢e+ ~(1-tr/R2) l+o./-----~
`` (o) ( o ) _ / ' 1
1
1\
+ e r R )'r/666~¢=0.
(A.12)
Substituting (A.11) into (A.12), to eliminate the secular terms, gives kl sign(a)=
1 I + ~ / R 3 1 1-cr/R 2 1 1 - t r / R 2 1 (1-tr/R2)(l+trR 3) + 2 1 - t r / R 2 24 l+tr/R 24 l + t r R 8 (l+orR) 2
(A.13)
478
H.H. Dai, A. Jeffrey/Interface solitary waves
and
I + o ' / R 3 2 1 - c r / R 2 1 1 - ~ r / R 2 1 ( 1 - t r / R 2 ) ( l + ~ R 3) 1 - o ' / R 2 t 3 l + t r / R +12 l + t r R 20 (l+o'R) 2
2to, sign(a)=
(A.14)
It then follows that
•,,1, = c sech:r, F
L V 12b
tanh: F , F
L V 12b
'
(A.15)
where
l+tr/R 3 1 1-tr/R 2 1 1-tr/R 2 1-tr C-I-o'/R2 4 l + c r / R +4 l+crR l+trR
3 ( 1 - t r / R 2 ) ( l + o ' R 3) 4 (l+trR) 2
(A.16)
If we let ~r=0 (i.e. for a one-layer fluid), then kl = _5, to1 = _ 3 , C = _3, which agree with the results given by Laitone [23]. Koop and Butler [5] who worked from Laplace's equations gave analytical expressions for the second order solutions, but they did not give the value of to1.
Appendix B. The asymptotic expression for the interface displacement First we define the following functions
Ho(x)=sechEx,
Hl(x)=tanhx,
H2(x)=sech4x,
Ha(x)=sechExtanh2x,
(B.1)
and the variables
Li = ~ f-~, / 1 - ~ 2 ( X - X o - T ),
R, = ~ /Ia[, / 1 - ~ 2 (x + X o - T ),
where X, Xo, T are defined in Section 4. The asymptotic expressions of interest are as follows:
first term 7 (0) = sign(a) (Ho(L,) + Ho(R0),
N (°) = sign(a) 2Ho(O);
(B.2a, b)
second term ~(1)= -sign(a) 2B(0) ~ / 1 ~ b H~(R,),
N ( ' = -sign(a) 2B(0) ~ f l ~ b H~)(O);
(B.3a, b)
third term r/(2) = 1 - o'/ R 2 [1H ~( L,)( HI( R, ) _ l) + ~H ~( R1)( H~( Lx) + 1) + ½Ho(L,) Ho( R,) ] 1 +cr/R
+ C[H3(L,) + H3(R1)] + 1 - crlR_~2 3 B2(O)H~)(R,)" l+cr/R 2
(B.4a)
N(2) - 1 - o ' / R___2____[~H3( O)+½H2( O)+ I H~( O)]+ 2CH3( O).+ 1 - c r / R 2 3B2(0._.....~)Hg( O) l+tr/R l+o'/R 2 1
131"/R 2
_3H~(O)[~:2 - 2B(0)se]; -~ l + o ' / R 4
(B.4b)
H.H. Dai, A. Jeffrey / Interface solitary waves
479
fourth term 7 (3) ~_~
•
/
.
l
a
l
1 - ~ / R__2 B(O)[-Ho(L1)H~(R~) - 2 H ~ ( R I ) ( H ~ ( L I ) + 1) -½H~(L~)Ho(RI)]
12b l + ~ r / R
+ ~V~ 1 2 b 11-+ O'/o./~ R2 B ( O ) [ - 3 H ~ ( R O H 2 ( R O + 2 H ~ ( R 1 ) - ~H~(R~)] - ~ a ~12b . v 2CH~(R~)
_dial
V12b
1-tr/R2 1 + o'/~
f ~ B3(O)Hg'(R~) - ~[ ~ zy~ sign(a)B(O)H~(gl)
_ ~ / ~ 2 ~b 2 B z ( O , 7 % s i g n ( a ) H ~ ( R 1 ) _ . t / lal tr/ R O 2 + D~ 3B(O)Hg,(RI,, V12b l+tr/R
(B.5a)
N(3) = .~/~a~ 1 - t r / R 2 Hg(O)[~B2(O)e_~B(O)e2]
V12b
+ .~ ~
l+cr/R
1 - t r l R 2 B(O)[-aH~(O1)H~(Oo)+2H~(O,)-2H3(O)+H2(O)-½H~(0)]
V 12b l + c r / R
- ~ ] 1 ~ b 2 C B ( O ) . ' 3 ( O ) - - l l V 1-]a[ I-°/REB3(O)H~(O)Eb 1 +tr/~ -
~ f l ~ b 270 s i g n ( a ) B z ( 0 ) H ~ ( 0 ) -
~f-~2v , sign(a)B(O)H~(O)
~/1-~
~/-lal ~. ~ / 1 2 b ~ / 1R+ ~DE - - ~+ D1 ~. .t.~. (. .O ) n o,,,.(v).
(a.5b)
References
[1] [2] [3] [4] [ 5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
G.H. Keulegan, "Characteristics of internal solitary waves," J. Rears. Nat. Bur. Stand. 51, 133-140 (1953). R.R. Long, "Solitary waves in the one and two fluid systems," TeUus 8, 460-471 (1956). T. Kakutani and N. Yamasaki, "Solitary waves on a two-layer fluid," J. Phys. Soc. Japan 45, 674-679 (1978). H. Segur and J.L. Hammack, "Soliton models for long internal waves," J. Fluid Mech. 118, 285-304 (1982). C.G. Koop and G. Butler, "An investigation of internal solitary waves in a two-fluid system," J. Fluid Mech. 112, 225-251 (1981). A.S. Peters and J.-J. Stoker, "Solitary waves in liquids having non-constant density," Comm. PureAppl. Math. 13, 115-164 (1960). T.B. Benjamin, "Internal waves of finite amplitude and permanent form," J, Fluid Mech., 25, 97-116 (1966). D.J. Benney, "Long non-linear waves in fluid flows," J. Math. and Phys. 45, 52-63 (1966). J.W. Miles, "Internal solitary waves I," Tellus 31, 456-462 (1979). J.W. Miles, "Internal solitary waves II," Tellus 33, 397-401 (1981). J.A. Gear and R. Grimshaw, "A second-order theory for solitary waves in shallow fluids," Phys. Fluids 26, 14-29 (1983). J.W. Miles, "Solitary waves," Ann. Rev. Fluid Mech. 12, 11-43 (1980). A.R, Osborne and T.L. Burch, "Internal solitons in the Andama Sea," Science 208, 451-460 (1980). A.R. Osborne, T.L. Burch and R. Scarlet, "The influence of internal waves on deep water drilling," J. Petroleum Tech., 1497-1503 (1978). R.M. Mirie and C.H. Su, "Internal solitary waves and their head-on collision, Part I", J. Fluid Mech. 147, 213-231 (1984). R.M. Mirie and C.H. Su, "Internal solitary waves and their head-on collision, Part II", Phys. Fluids 29, 31-37 (1986). N. Sugimoto and T. Kakutani., "Reflection of a shallow-water soliton, Part I. Edge layer for shallow-water waves," J. Fluid Mech. 146, 369-382 (1984). N. Sugimoto, Y. Kursaka and T. Kakutani, "Reflection of a shallow-water soliton, Part 2. Numerical evaluation," J. Fluid Mech. 178, 99-117 (1987). A. Jeffrey and H.H. Dai, "On the reflection of a solitary wave at a sloping beach: analytical results," Wave Motion 10, 375-389 (1988). N. Sugimoto, W. Nakajina and T. Kakutani, "Edge-layer theory for shallow-water waves over a stelr--reflection and transmission of a soliton," J. Phys. Soc. Japan 56, 1717-1730 (1987). N. Sugimoto, K. Hosokawa and T. Kakutani, "Reflection and transmission of a shallow-water soliton." G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). E.V. Laitone, "The second approximation to cnoidal and solitary waves," J. Fluid Mech. 9, 430-444 (1960).
463
R E F L E C T I O N O F I N T E R F A C E S O L I T A R Y W A V E S AT A S L O P E H.H. DAI and A. JEFFREY Department of Engineering Mathematics, The University of Newcastle upon Tyne, NE1 7RU, U.K. Received 17 May 1988
This paper deals with the problem of the reflection of interface solitary waves at a slope in a two-layer fluid system with rigid upper and lower boundaries. An edge-layer theory is developed to treat the region near to the slope in which shallow-water theory is not valid. We work to O(e 3/2) so that the effects of the surging movement at the shoreline are included. After deriving the 'reduced boundary conditions' relevant to the shallow-water equations, a perturbation procedure involving methods of strained coordinates and inner-outer expansions is employed to solve this problem. Analytical results, such as m a x i m u m run-up at the slope and the time when it occurs, are presented. For a plane slope with an angle of inclination ~, = ~r/4, we present some graphical results. It is found that, for the reflection problem, the properties of the depression mode differ from those of the elevation mode in m a n y ways. In Appendix A, we also present an alternative way of obtaining the second order solutions for interface solitary waves in an infinite region.
1. Introduction
Internal solitary waves in shallow water in an unbounded region have been studied by many authors. Keulegan [1] was the first to consider solitary waves in the interface between two superimposed fluids with slightly different densities when the fluids are contained between two horizontal rigid boundaries, while Long [2] considered the case of an arbitrary density ratio. Kakutani and Yamasaki [3] also considered a two-layer model but with a free upper surface. Segur and Hammack [4] and Koop and Butler [5] made some experimental investigations. Various other authors investigated continuous stratification cases; notably, Peters and Stoker [6], Benjamin [7], Benny [8] and Miles [9], [10]. Benjamin [7] established a theory for this class of long solitary waves with his effective variational integral method. Benny [8] worked with Eulerian coordinates and provided an alternative method. Miles [9], [ 10] (also Kakutani and Yamasaki [3]) considered cases in which the cubic and quadratic nonlinearity are of comparable significance. For second order solutions see Gear and Grimshaw [11] and Koop and Butler [5]. For a general review, see Miles [12]. Recently large scale internal solitary waves have been observed in the ocean (Osborne and Burch [13]; Osborne, Burch and Scarlet [ 14]) and these observations have generated additional interest in this problem. Unfortunately comparatively little progress has been made with the reflection problem, although reflection at a vertical wall has been studied (Mirie and Su [15]; Mirie and Su [16]) as it is equivalent to the problem of the collision of two identical solitary waves in an infinite region. Hence, in this paper, we will deal with the reflection of interface solitary waves incident upon a slope, which may be of an arbitrary shape. The problem is shown in Fig. 1. Away from the sloping region, it is reasonable to assume that the effect of the slope is negligible, and hence that the well-known weakly nonlinear dispersive wave theory (the shallow-water theory) is valid. However, close to the sloping region, as the boundary condition on the sloping surface requires the normal 0165-2125/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
464
H.H. Dai, A. Jeffrey / Interface solitary waves
velocity to be zero, the motion in the vertical direction becomes of primary importance, and so shallow-water theory is no longer valid. To overcome this difficulty, we use an edge-layer theory. The edge-layer theory was first introduced by Sugimoto and Kakutani [17] and Sugimoto et al. [18] to treat the reflection of a shallow-water soliton at a sloping beach. Jeffrey and Dai [19] extended the theory to a higher order approximation to include the effect of the surging movement and gave analytical results. Sugimoto et al. ([20], [21]) also applied this theory to the problems of an incident soliton over a step and an incident soliton over a vertical barrier. In our problem we divide the whole region into three parts, as shown in Fig. 1. One is the shallow-water region in which shallow-water theory is valid. Another is the edge-layer region in which two-dimensional motion is of primary importance. Between these two regions is a third region called a matching region. Using the matched-asymptotic-expansion method, we derive a matching condition. By applying Stokes' theorem in the edge-layer, two boundary conditions (corresponding to the upper and lower fluids) relevant to the governing equations in the shallow-water region are obtained, while the edge-layer solutions themselves are reduced to finding two irrotational flow-fields (corresponding to the upper and lower fluids) in the semi-channnel regions induced by a source and sink distribution on the slope. If v = "rr/m (m an integer), the edge-layer solutions can be expressed in integral form.
Edge Loyer z'
Matching Region
x'= Blz')
.
x
Fig. 1. Illustration of the problem. In Section 4, a perturbation procedure combining strained coordinates and inner-outer expresions is employed to solve the governing equations in the shallow-water region under the derived boundary conditions and imposed initial condition. Analytical results, such as maximum run-up at the slope and the time when maximum run-up is attained, are obtained. From our results, by letting m = 2, we recover the case of the collision of two identical interface solitary waves, and our results agree with those of Mirie and Su [15] (to order O(~2), in their paper). However, the time at which maximum run-up is attained for the collision of two identical interface solitary waves is new and is given for the first time in this paper. In Appendix A, starting from the shallow-water equations, and using the method of strained coordinates, we obtain the second order solutions for interface solitary waves in an infinite region. The phase velocity is accurate up to O(e2). The solution provides an initial condition for our problem.
2. Governing equations in the shallow-water region We shall consider the incompressible irrotational and inviscid flow of two superimposed immiscible fluids between two horizontal rigid boundaries. The coordinates are chosen as shown in Fig. 1.
H.H. Dai, A. Jeffrey / Interface solitary waves
465
We shall non-dimensionalize the equations in the shallow-water region using the following scales x = x ' / l,
z = z ' / hl,
t = ( t'/ l)Co,
*l = ~f / ao,
(i = 1, 2),
~, =~-~ ~I/(aoCo)
(2.1)
where @1 and ~2 are the velocity potentials for the lower and upper fluids respectively, 7/is the interface displacement, I the characteristic wave length, ao the characteristic wave amplitude and Co the characteristic phase velocity for linear waves. Thus 5 ( 1 - tr)R ghl, C°= V tr+R
(2.2)
tr = PE/Pl,
(2.3)
where R = hE~ha,
p2 and Pl, h 2 and h~ are the densities and thicknesses of the upper and lower fluids, respectively, tr and R are two important parameters. The wave is an elevation wave (or a depression wave) if tr is smaller (or greater) than R 2. In this paper we shall not consider the critical ease tr = R 2, which we leave for future investigation. The governing equations are as follows: 2 i~)1xx @ I~-)1zz ~---0 ,
1 < z < er/,
--
tz2~zxx + qb2= = 0,
(2.4) (2.5)
~'q
q0~z-- 0
at z = - 1 ,
(2.6)
q~2z= 0
at z = R ,
(2.7)
• i =/z2(rh + e~ix~x) 2[R(1-tr)
L ~--+)~- q'.+n
( i = 1,2) ]
at z = e~,
(2.8)
,R(1-tr)
42
tr+~(~2q~x+~z)
= t r { / 2 [ R ( 1 - t r ) ~ 2 , + ~7] q" , R ( a - t r )
tr+~-R--
}
2 o ' + ~ (/z2tp22x+~2z) '
(2.9)
where/.t = h~/l and ~ = ao/h are assumed to be small. However, the Ursell number Ur = ~//x 2 is assumed to be of order one. Next, we expand q~l and ~2 in powers of p2 (of. Whitham [22]) q~l = ~ ( - 1 ) " ,.=o
~2--,.=o
(z+l) TM a2mfl
2,.
ax2m p,
(2m)!
(2.10)
'
02mf2 O-~/~2m
(z-R)
(-1)" ~
(2.11)
It is easy to confirm that (2.4)-(2.7) are satisfied automatically. Further, from (2.8)-(2.9) up to O(/~4), we obtain 2 /Z
.-
_/z
4
2 .-(6)
n, + [(1 + "~/)Ax]x --g-J,x.xx-r ~-.~j ,
~/x
,
~.
---~- ~~7J,~,). = O,
(2.12)
H.H. Dai,A. Jeffrey/ Interfacesolitarywaves
466 ~ , + [ ( , r / - , aR~r 2 x J x1- - +# ~
( ~1 + )~
2
4
R3¢ # J2=xx-'~.
n _ (crf2 _A,) _ ~" (o.f~: _ f l ~: )
2
R5¢(6) ,/z J2-'-~--RZ(*lf2xxx)~=O,
(2.13)
# " (~R4f~xxx~,-f,xx~,) + #7: (~R2f~xx,-fl~x,)+T.,
+ "#2 (o'R:f~xx-f~xx) +--~,#2 (erR 2f2, f2x,~ - f ~ , f~,~) 2
Oz n (o'Rf2xx, +f~x,,) = 0.
(2.14)
Equations (2.12)-(2.14) are the equations we shall deal with. Second order interface solitary wave solutions were first obtained by Koop and Butler [5] and confirmed by Gear and Grimshaw [11]. In Appendix A, working from equations (2.12)-(2.14), we obtain second order solutions in a different fashion by using the method of strained coordinates. The surface displacement is
*l = sign(a) sech2{ C +
( l + "k,)[X- Xo+ ( l + " ~ ) t ] }
~ / 1 - ~ 2 (1 + ,k,)
Ix
-
1
• tanh2{~ll~b~(l+,k,)[X-Xo+(l+,~)t]}.
(2.15)
The coefficients a, b, k~, C are defined in Appendix A. For the phase velocity, we obtain a result accurate to O(~2). Koop and Butler [5] and Gear and Grimshaw [11] only gave the first order result. Hence equation (2.15) for Xo>>1, t = 0 provides an initial condition for the reflection problem for interface solitary waves at a slope.
3. The edge-layer theory For the single-layer fluid case, Sugimoto and Kakutani [7] argued that the edge-layer develops seaward to an order O ( # ) length scale. Actually, as the boundary condition on the sloping surface requires the normal velocity to be zero, the motion in the vertical direction is as important as that in the horizontal direction in the near-sloping region. Hence the non-dimensional scale factor along both directions should be the same. In the near-sloping region, the depth and width are of the same order. Thus it is reasonable to choose h~ as the non-dimensional scale factor. This is equivalent to introducing a new variable ¢ = x/#, corresponding to the length scale of the matching region being of the order O(#). Then, from equations (2.4)-(2.9), we obtain for the governing equations in the edge-layer region • 1~+ ~lzz = 0,
-1 < z<,'0,
(3.1)
~2~e+ ~2= = 0,
,~
(3.2)
~lz = 0
at z = - 1 ,
(3.3)
~2z=0
at z = R ,
(3.4)
~iz - #2rh - e ~ e ~ = 0
(i = 1, 2)
at z = "7,
(3.5)
H.H. Dai, A. Jeffrey / Interface solitary waves
467
r~ -~ (1 -- o')__17
=or[r/+(1-or)R~2,-t" or+R
~ (I_+~_~R(~22 + ~ ) 2#x2
]
atz=er/.
(3.7)
The boundary conditions at the slope ~ = B ( z ) ( B ( z ) is assumed to be of order one) are -1
Ix (~l~-Bz~lz)=O, (1 + B2z)1/2
-~rs< ~?< 0 ,
(3.8)
B(R)< ~<-~s,
(3.9)
-1
(1 +izB 2) 1/2 ( ~ 2 ~ - B z ~ 2 z ) = O ,
where -~rs is the position of the interface on the slope. In the matching region, we have the following matching condition: the edge-layer region solutions ~l,2(x/Iz, z; t) with x fixed a n d / z ~ 0 shallow water region solution 4~1,2(tz~:,z; t), with ~r fixed and/.~-~0. Setting x =/z~: in (2.10), (2.11) and (2.14), and expanding around zero, we obtain
•
oo
oO
( - 1 ) - (z+ l) m (2m) !p!
oo
(3.10)
2m+pfpL~x p÷2" .L,=o'
oo
F 3p+2m ¢ "1
fl)2oo=m~=op~=o(__l)m(Z-R) 2m "~m+p/:pi
(2m)!p! #z
J'~| . Loxp+2. jx:o,
(3.11)
and
+!2 (4L -f~) -~- (orR~A~x,-f,~,) + . . .
•
(3.12)
Here, and hereafter, even if we do not emphasize it, f and its derivatives are evaluated at x = 0. Here we also stress that in (3.12) we need to retain terms up to order O(~ 4) and O ( ~ 2 ) , though for brevity we omit the details. The matching condition requires ~1 ''>
II~lco,
~2oo "~ ~2c~
and
r/--> r/~
as ~: -+ oo.
Thus we suppose that in the edge-layer region ~ , = ~ , ~ + / z V~(t) ~b;(se, t)
(i=1,2).
(3.13)
We introduce factors Vl(t) and V2(t) in order to make ~/'1 and ~b2 independent of t. The matching condition requires that
In fact ~bl, ~b: should tend to zero at a rate faster than an algebraic one. Afterwards, we shall show that I~tl, I]/2 are in fact exponentially small.
H.H. Dai, A. Jeffrey / Interface solitary waves
468
Substituting (3.10)-(3.12) into (3.5)-(3.7) and expanding z at Zo= ~7o(B(0), t), while retaining terms up to O(/z3), O(O~), we obtain
V'O'= -/z2 ~r( V2q&)" V'0')'I+o'/R - (
2(1 +cr/R)/Ze [ (~r2V202~-2 2
+ 2f, x ~( v2~,2),~ - ( v, ~,,),~ ~ 2 v,~,~(~Ax, - f,x,).] j=0 l+o'/R l +cr/R o'(V2,1,~),-(v,,l,,),+
~7 = ~7~+tz
~
~ 2
V , ~ , ¢2+ 2 o ' f 2 = V 2 ~ 2 ~ - 2 f ~ x V , ~ , e ) 2
(i= 1, 2)
at z = ~r/o,
(3.14)
2
2(1 +tr/R) [orV2'~2~-- VII//I~
l+o'/R
+ 2 crf2x V2ff=e - 2Ax 1/1 @, e + 2/~:(crf2x= V2 r/2e -f~x= V, 0, ¢) 2
2
2
2
+ trV2t~2z - V,~b,z+21z(trRf2=V202=+fl~V,~b,z)]
at z = Er/o.
(3.15)
From (3.1)-(3.3), and (3.8), we obtain ~blee+ ~1= = 0,
--1 < z < ~r/,
(3.16)
O2ee+ ~2= = 0,
eTq
(3.17)
0~z = 0
at z = - 1 ,
(3.18)
V,(Ol¢_BAb,=)=_f~_lxf~XX~z[(Z+l)B] +~
(z+l)2
2 - - J ~2 x x x
..[ ].g3
d
, 2fix==dzz d [ ( z + 1)B 2] -~/x
63
d
--6J~xxxx-~z [(z+l)3B]+-~-~--- flxxxx
[(z + 1)B3]"
(3.19)
Next we apply Stokes' theorem to (3.16) along the boundary Os~ of the region shown in Fig. 2(a) and to (3.17) along the boundary 0s2 of the region shown in Fig. 2(b). That is, we use the result V~
d~ Sl
(0,¢dz-~b,zd~:)=0
(i=1,2).
(3.20)
Substituting (3.18), (3.14) and (3.19) into (3.20), and noticing that ~ - > 0 as ~:->o~, we obtain
-f~x-I~B(O)flxx - elXrloB~(O)fl~-~Iz 1 2 B 2 (0)f~x+~/z 1 2f~x~ 3 3 2 fco o.(V21[12)u_(Vll]tl)tt + ~---6 B(0)f~ . . . . -~-----B3(O)f]xxxx-I 6 d, /,-~ ~ i,_,~o _ d~ = 0.
zt
z.,.E',
""/"
/ / /ez=R/
z:-I /
/
e"
z= er/~
/ / / /
(a)
(b)
Fig. 2. Geometry for the application of Stokes' theorem (a) the lower layer; (b) the upper layer.
(3.21)
H.H. Dai, A. Jeffrey / Interface solitary waves
469
I f we let V~ = 2fix and V2 = 2f2x and substitute (3.21) back to (3.19) up to O(/.0, we obtain qJle-B~qJlz = -
~{
1
d [(z+l)B(O)]}_~s<,=B(z)
1- B(0) dz
'
(3.22)
As fix is of the order O(/~), to be accurate up to O(~tz) or O(/~ 3) we only need ~0~ to be correct up to O(/z). From (3.16), (3.18), (3.14) and (3.22), we see that to this order qJa is independent of the time, and the problem of finding qq is equivalent to finding a potential field in a semi-infinite channel caused by a source and sink distribution on the slope. Similarly, we can obtain the result
--fz~--IxB(O)A,,x--tXenoB(O)fz~ -~/~ ' 2B 2(0)f2xxx+~/z , 2R 2f2x~ +~_y_R2B(O)f2x.x~y_Bs(O)fzxx~+lZ2 | o
o
a-e~
tr(V2~b2).-( cr + R
d~=O.
(3.23)
z=,,0
If the slope is a plane wall with an angle of inclination v = 1r/m (m an integer), an integral expression of ~0~ can be obtained. We omit the result here, but for the necessary details we refer to Sugimoto and Kakutani [17]. After a suitable transformation taken from Sugimoto and Kakutani's results, qJ2 can be expressed in a similar form. It can be proved that
~bl(~,z)~Ae-~ecos[~r(z+l)]+O(e
-2~¢ )
as ~ ,
where A is a constant. This agrees with the matching condition. A similar conclusion holds for q~2(~:,z). Here, we again emphasize that the expansions are carried out at z = ~7o(B(0), t) instead of z = 0. We argue that the edge-layer is so narrow, compared with the characteristic wave length, that the surface displacements at each point in edge-layer at the same time are almost identical. Thus, in this manner, we believe that the surging movement of the shoreline can be taken into account. Our asymptotic results, which will be presented in the next section, confirm this argument. As equations (3.21) and (3.23) are evaluated at x = 0 , they provide two boundary conditions for the governing equations in the shallow water region.
4. The perturbation procedure In this section we shall use a perturbation procedure to solve equations (2.12)-(2.14) under the initial condition (2.15) and the boundary conditions (3.21) and (3.23). The method is similar to that used by Jeffrey and Dai [19] for the one-layer fluid case, in that it combines the method of strained coordinates and the method of inner-outer expansions. First, we introduce the strained coordinates
X=(l+~yOx,
T=(l+~y,)(l+av,)t,
(4.1)
and let
f,x= F,,
f~x= F2.
(4.2)
Further, we suppose that '17 = T'](O)-[ - EI/2T'](1)"I - ~'l~(2)-J - ~3/2T](3)"~ O(~3/2),
F, = Fl°)+ e I/2F~ 1)+ eF~2) + e3/EFI3)+ o(e 3/2)
(4.3) ( i = 1,2).
470
H . H . D a i , A . J e f f r e y / I n t e r f a c e solitary w a v e s
Substituting (4.1)-(4.3) into (2.12)-(2.14), we obtain:
first term ~(T0)+ rr(0)F1X= 0 ,
~(T0)+ ,,u 101:7(°) 2X = 0 ,
(4.4a, b)
(l+cr/R)r/(xO)_, ~ 0 ",1-"(2 oT )-~- (aol T) ~! = O;
(4.4c)
third term /2
~7(T2) + ~(2) --,x _- v, @r°)+ (n(°)F~°))x
7 T(-
1
(r(0)
r(o)
1-' I X T T - - O ' I - ' 2 X T T 1
(4.5a)
= 0,
• 6(1 + or/R) e g 2 Dl[7(2) .a(o) ~-0, ,', n ~ ) + (n(°)F(?))x /z• 6(1 + cr/R)~~F(O) 2, - - ~) x 1XTT_ ~°I2XTT]--
(1 + cr/ R ) ~ x(u) -
(2) - -
(orF2T
(2)
(4.5b)
,(Or/"2T ,~(0)- ,~(0), I~IT)
FIT ) --/)1 2
, ~(o)r(o) /z 1 __l+(rR (cr~(°)--trrr- - ~(o) ~=0. --(or/~2 / ~ 2 X - - l~(o)~(o)~q l alX] J[1TTTJ
(4.5c)
• 2 l+cr/R
For the sake o f brevity we will not give the second and fourth terms. F r o m (4.4a)-(4.4c), we obtain
77(°) = G(°)(X + T) + H(°)(X - T),
(4.6a)
crF(2° ) - F~ °) = (1 + o ' / R ) [ G ( ° ) ( X + T) - H(°)(X - T)],
(4.6b)
F~O)+ RF(2O)= i(o)(T),
(4.6c)
where G (°), H (°) and I (°) are arbitrary functions of their arguments. Using the initial condition (2.24) we have
G(°)(X)+H(°)(X)=sign(a)sech2[,~)~2(l+ekl)(x-xo) V 12b/z
].
(4.7)
To determine G(°)(X + T), H ( ° ) ( X - T) and I(°)(T), we need two other conditions. However, we shall not use the b o u n d a r y conditions (3.21) and (3.23) directly. As argued before, in the region very near to x = 0, the length scale is O(/z), due to the effect o f the slope. Thus there is a ' b o u n d a r y layer' near to x = 0. We introduce the scaled coordinates.
= X/iz,
T = T,
F 1 = U1,
F 2 = U2,
77 N,
(4.8)
=
and s u p p o s e that N = N (°) + •I/2N(D + EN (2) + •3/2N(3) + O(•3/2), U / = U~ 0) Jr- • 1/2 U~I)._F •U~2>..~_ •3/2 U~3) q_ 0(•3/2)
(4.9)
(i = 1, 2).
Substituting (4.8) and (4.9) into (2.21), (2.23), (3.26) and (3.29), we obtain:
first term u(O) , e -_0
N(eO)= 0
(i = 1, 2),
(4.10a)
with -- u~O)
- B ( O ) U ~(o) e - ~ B1
2
(o)
1
(o)
1
(o)
(O)Ule~+gUI~+gU1~e((B(O)-B3(O))=O
_U(20) - ~,t,,J l ~ / ~ lr~,2¢ r(0) -2,-, I_I.)2[FY~I?(O) ..l_1102 ¥ r(o) ~,,J ,-" 2e~6,- ,-,2ee+~U(2°e)ee(R2B(O) - a3(0)) = 0
at so=0, at ~: = 0;
(4.10b) (4.10c)
H.H. Dai, A. Jeffrey/Interface solitary waves
471
third term It
U ( 2 ) 2_
/L/, ,~ T--/Ti N(~)+ ( N (°) U~°))~:
-°rr(z)+ it N(l)+tN(°)U(°)~'¢-itT~. ~-'-'2e el/2 ( l + o . / R ) N T ) _ _ _Eil /t2,
2
r r(o) ~ t r(o) ( t..~ 1 6 T T - - ,a, ,...."2 6 T T / ---~ 0 ,
(4.11a)
2 R2 (IT(O) o'If(O) \ E 6 ( l + o ' / R ) ~,-'levr- ,-~2~TT)=0,
2 ls
( O r,,(i) U2T-
1 6(1 + t y / R )
(4.11b)
U~))_(o.U(2o)rr(O)_~,2eU~°)1r(°)x0 , v ~, e-
(4.1 lc)
with - U ~ z) - B ( O ) r•~le r( 2 ) _
(2)
+gUIeee(B(O) -
U ~ 2) -
1
B(0)
(2)
_
i r(o) ylB(0),-,l¢
-
. .(o) r r ( 2 ) _~ ! r r ( 2 ) B~(0)'OoUle -½B2(0) i--Jl~:sc-6t-~l~e
2
2
2trD2 ,,(ol +It 2D1 rr(O) - 0 E l + t r / R u2rr ~ l + o ' / R , - , 2 r r -
3 B (0))
It
ate:=0,
(4.11d)
ate=0,
(4.11e)
t r(o) _ (o) r(2) "-'1,~ Bz(O)noU2e - ½ B 2 ( 0 ) ~ -t ' r(2) 2 ¢ 6 --a-6 ! o, . 2 lr'-"26~
~2~
r r(2) _ Yl B(0)
2
+gU2~*e(R B(0)-B3(0))
it_2 2o'D2 rr(O) _LIt2 2D1 tr(O) - n ~ tr+R ,-'2rr- E l + t r / R ~ ' r r - "
where D 1=
f
~bl(~, 0) ds¢,
D2 =
~s
Lo02(s¢, 0) ds¢.
(4.12)
G
We shall call (4.4a)-(4.5c) the outer equations and (4.10a)-(4.11e) the inner equations. The basic procedure is now as follows. From the outer equations we can determine ~/u), F~) and F(2°(i = 0, 1, 2, 3) in terms of three arbitrary functions G~(X + T), H")(X - T) and IU)(T) (i = 0, 1, 2, 3). The initial condition provides a relation between G u) and H u). From the inner expansions, U~°, U~') can be determined, but the N u) (i = 0, 1, 2, 3) are given in terms of the arbitrary functions jU)(T) (i = 0, 1, 2, 3). The matching conditions between F~~) and U~°, F~° and U(2°, 7/") and N "), that is i terms (inner) matching i terms (outer), will give the other three conditions. Thus G °), H u), I u) and j , ) can be uniquely determined. From (4.10a)-(4.10e), we obtain UI°) = 0
(i = 1,2),
N(°)=J(°)(T).
(4.13)
The matching conditions between (o'F(2°) - F] °)) and (crU~°) - ~--'ll r(o)]], ~--lt~(°)+ RF~O)) and ( U]°)+ RU~ °)) give G(°)(T) - H(°)( - T) = 0,
j(o)(T) = 0.
(4.14a, b)
From (4.7) and (4.14a), taking note of the fact that Xo>>1, we obtain
G(°)(X)=sign(a)sech2[~l~b--~E2(X-Xo) ] ,
r/%
H(°)(X)=sign(a)sech2L~[ l - ~ 2 ( X + X o )
],
(4.15a)
yl=k,,
(4.15b, c)
where
Xo= (1+ ~k,)xo.
(4.16)
The matching condition between ~(o) and N (°) then gives j(O)(T) = 2 sign(a) sech 2 O,
(4.17)
H.H. Dai,A. Jeffrey/ Interfacesolitarywaves
472 where
[ lal" ,.. (-~o-
0= ~ / ~
T).
From (4.5a)-(4.5c), eliminating F~2) and F~22), we obtain
(4.18)
H.H. Dai, A. Jeffrey / Interface solitary waves
473
where
/ 01 = B ( 0 ) q
lal 12b'
03=sign(a)
02=
sign(a) 1 - - o ' / R 2 4 l+cr/R '
-2B(0) l+o'/R
(4.25a, b)
CB(O)+7~ sign(a) B(0)
R)D 2 + D, B(0)] " + 2Bz(O)- 12 ( t r /i--+ ~---'~
(4.25c)
We point out that when or = 0, we recover the one-layer fluid case. The present results agree with those obtained by Jeffrey and Dai [19]. If we let B ( 0 ) = 0 , corresponding to a vertical wall, then up to O(~) (scaled by hi) our results agree with those given by Mirie and Su [15] who considered the collision of two interface solitary waves. However, Mirie and Su did not give the time at which maximum run-up occurs. Here we emphasize that the first two terms on the right hand side of equation (4.23) are exactly the same as the corresponding results in the vertical wall case. Thus the third term, which represents the effect of the slope, plays an important role. It may be worth considering cases in which the slope has different forms and seeing the effect on the maximum run-up and the time at which maximum run-up is attained. We leave consideration for the future.
5. Discussion of results
First, consider the reflection of a solitary wave at a vertical wall, or the collision of two identical solitary waves. In this case B(0) = 0. Then, from equations (4.26a)-(4.26c), we see that 01 = 03 = 0, and for both the elevation mode (1 - o,2/R > 0) and the depression mode (1 - t r / R 2 < 0), 02 < 0. Thus maximum run-up only occurs after a time delay. From Appendix B, in the case B ( 0 ) = 0 we have 1 - t r / R 2 _½ ~7= sign(a) [sech 2 LI + sech 2 R1] + ~ 1 + o-/R [ sech 2 L~ tanh Ll(tanh R1 - 1) -½ sech 2 R1 tanh R~ (tanh L~ + 1) + 2 sech 2 L~ sech 2 R1] + C sech 2 L1 tanh 2 L I + C sech 2 R~ tanh 2 R l } .
(5.1)
Letting t ~ o0, and using Taylor's theorem, we obtain rI = sign(a) sech 2 Ri + eC sech 2 Ri + O(e2),
(5.2)
where
Ri Ax
--
~f -~~ e /xe
(1 + , 7 , ) [Lx + X o - (1 + ~ ) t + d x ]
1/2 s i g 2 ( a ) ~
(5.3)
1 - or/R 2
Vlal l+tr/R"
(5.4)
This shows that the reflected wave has an extra phase shift. To the order O(~), our results agree with those obtained by Mirie and Su [15].
H.H. Dai, A. Jeffrey / Interface solitary waves
474
If we consider the reflection at an overhanging boundary ~ = e W 2 B ( Z ) ( > 0), then in a manner similar to that used in Sections 3 and 4, we obtain the following result fl -o'/R 2 r/= sign(a) [sech 2 L~ + sech 2 R~] + ~'[] ~ [-½ sech 2 L~ tanh L, (tanh R ~ - 1) -½ sech 2 R~ tanh R1 (tanh L, + 1)+ 2 sech 2 L1 sech 2 R~] + C sech 2 L~ tanh 2 L~ + C sech 2 R1 tanh 2 R 1 } + , sign(a)4B(0)~/ll-~b sech2 R1 tanh Rl.
(5.5)
Similarly, we have r/= sign(a) sech 2/~, + ~C sech z/~, tanh 2/~1,
t ~ co,
(5.6)
where (5.7)
1¢ 12b/z sign(a) . / ] ~
a~=~'/2
2
1-o'/R 2
V lal I+~/R
(5.8)
~'/22B(0).
Thus, if we let
B(0)
sign(a) , / ~ 1 - ~r/R 2 ~/ - ~ l + tr / R '
(5.9)
to order O(¢), the reflected wave will be exactly the same as the incident wave, although it propagates in the opposite direction. This represents a real 'elastic' collision. We now give some graphical results for a plane slope with an angle of inclination v--'rr/4. From Fig. 3(a), we see that, for the elevation mode, the value of maximum run-up decreases as tr becomes larger. For the depression mode, the tendency is opposite for tr in the range 0.8 to 0.9, although the value of the maximum run-up increases by a very small amount. In both cases the maximum run-up 1.0
-1.0
R=2"5
o"
0.8.
•
0.6
R=0.48
0.0
-0.8
oi!
- 0.6
O"
.90 0.85
O8O 0.4
- 0.4
0.2 ¸
- 0.2"
0
0
0.1
0.2
0.3
0
0
OJ
0.2
Fig. 3. The m a x i m u m r u n - u p at a p l a n e slope with a n a n g l e o f i n c l i n a t i o n ~, = ,rr/4 fo r different wave a m p l i t u d e s . (a) The e l e v a t i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for tr = 0.8, (b) the d e p r e s s i o n mode; the b r o k e n line c o r r e s p o n d s to the vertical wall case for cr = 0.9.
14.1-1. Dai, A. Jeffrey / Interface solitary waves
475
4.5
4.5
R=0.48
R=2.5
CY
r
4.0 . . ¢ ~ - - . ~
/
" ' - ~ ' - . "-... -..
0.8 0.6 0.4 0.2 0.0
4.0.
o"
~
0 . 8 0
0.85 0.90 %
/ 3.50
_~_ 0.1
~ 0.2
~ 0.3
(a~ 3 . 5 - 0
-
,0.1
. • ........ 0.2
(b)
Fig. 4. The t i m e w h e n m a x i m u m r u n - u p at a p l a n e s l o p e with a n a n g l e o f i n c l i n a t i o n ~, = ~ / 4 is a t t a i n e d for different w a v e a m p l i t u d e s . (a) The e l e v a t i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for ~ = 0.8, (b) the d e p r e s s i o n m o d e ; the b r o k e n line c o r r e s p o n d s to the vertical wall case for (7 = 0.9.
~0.1
0
1
2
3
4
5
" T:3.6
,4,,
3
i T:3.6
T:4.0 /
]0.1
T=O
T~2.5
5
,
6
io.t
T~4.4
,L
/~. . . . . . . . . . . . . . . T=3.9 "T=4.0
I0.1
T:4.1
/ / _ ,f,.,
T=4.8
T=5.5
1o.1
T=8.0
T0.1
T:4.4 T=5.5 T:8.0 ,0
1
2
3
4
5
.......
(~
Fig. 5. The w h o l e process o f reflection o f an i n c i d e n t solitary w a v e initially at x o = 4 with a m p l i t u d e ~ = 0.1. (a) The e l e v a t i o n m o d e w i t h R = 2.5, ~ = 0.8, (b) the d e p r e s s i o n m o d e with R = 0.48, cr = 0.9.
H.H. Dai, A. Jeffrey / Interface solitary waves
476
at the slope is larger than that at a vertical wall, as would be expected. Comparing the two modes, the case of the depression m o d e is seen to give a slightly larger maximum run-up. Consideration of when the time of m a x i m u m run-up is attained shows that the value of the time decreases as or becomes larger in the case of the elevation mode. For the depression mode, the opposite occurs. In both modes this time becomes smaller as the initial wave amplitude increases. This conclusion seems to hold only for a slope with a small inclination. For the case of a gentle slope (say, ~ = ~r/3) the tendency may be the opposite, as reported by Jeffrey and Dai [19] for the one-layer case. Comparing the two modes, the case of the depression mode gives the larger value of the time. However, in both modes, the case of a vertical wall gives a much smaller value of the time. In Fig. 5, we show the whole process of reflection. In both cases, we find that, when the incident wave gets close to the slope, before run-up begins, the wave amplitude became smaller compared with the incident wave amplitudes. These results are shown in Figs. 5(a) (t = 3.2), 5(b) (t = 3.6). In the case of the elevation mode, the m a x i m u m run-up is 2.10 times the incident wave amplitude, which is attained at t = 3.99. In the case of the depression mode, the maximum run-up is 2.30 times the incident wave amplitude, which is attained at t = 3.92. In both cases, the reflected waves are found to have experienced distortion, relative to the incident waves. For the elevation mode the distortion is small. The amplitude of the reflected wave is 1.05 times the amplitude of the incident wave. However, for the depression mode, the distortion is found to be large. The amplitude of the reflected wave is 1.5 times the amplitude of the incident wave. Further, a small elevation wave train is found in front of the main depression wave. In both cases, the reflected waves move faster than the incident waves. In conclusion, it seems that, at least for the problem of the reflection of interface solitary waves at a slope, the properties of the depression mode differ from those of the elevation mode in many ways.
Acknowledgment The first author (H.H. Dai) is being sponsored by K.C. Wong Education Foundation Ltd., while studying for this Ph.D. at the University of Newcastle upon Tyne.
Appendix A. The second order solutions of interface solitary waves We introduce the strained coordinates 151/2
~:=
be
~3/2
K ( x - t),
r=
ge
/.L
Ktot,
-
A=~-~TiA,
].£
-
f2=~-i-~f2,
(A.1)
and suppose that
p = p(O)+ ep(l)+
~2p(2)+o(e2)
( P = ~:,fl ,f2, K, to).
(A.2)
Substituting (A.1) and (A.2) into (2.21)-(2.23), and retaining only the first term, we obtain Mo
0 W (°)
of
=0,
(A.3)
H.H. Dai, A. Jeffrey / Interface solitary waves
477
where -1
1
-1
0
Mo =
1-tr
).]
n(')
-OR
1-tr
o'(1-cr
l + tr/ R
-~-~J
w.,_-
'
(A.4)
(i--0,1,2).
The left and right eigenvectors of Mo are respectively Le=
[.1.n
,[
J
Re-
1,1,-
(A.5)
.
Retaining terms up to the second order, we obtain Mo
c3W
(1)
a¢
SI=
+ Sl T/~ ) + e~"2 _(o)_(o) ~ c _(o) ~ $4~(o) = O, ~I ~I~ - - ~-~3'1 ~(:t~- -
1
r ,
$2 =
(1- or)(1- cr/R2] ' l+cr/R
S3-
(A.6)
l
-1/6
,
I(1--~)(1-~/R)-[ L 2(l+tr/g)
s.=
(A.7)
J
To eliminate the secular terms we multiply (A.6) from the left, by Le to obtain 7(o)--
(o)~/~(o) -r a~7(o)~7~ ~',~¢ = 0,
(A.8)
where
Le" $2 3 1 - c r / R 2 a - L e " S-----]l-2 l+o~/R'
Le" $3
1 l+trR
(A.9, 10)
b=Le" S-------~l-6l+cr/g"
Equation (A.8) is the well-known KdV equation, which has the steady solitary wave solution ~7(°)= sign(a) se
(A.11)
Similarly, retaining terms up to third order, we find that the elimination of secular terms leads to 2(1+ o-/R) 7?(~>+ 3 (1 - or/R2)~?(°),/~1>+ 3(1 -o-/R2)n(~°)~la)+½(l+o-/R)~)~ • (o) - I(1 - ~/R2)~/(°)7/7 ) -½(1 + o'R)~/¢e~+(°)[ _ 6 ( l + t r / R 3) +2tOl(1 + o'/R)*l~ 3(1-tr/R2)2"], 42 l + t r / g j t ~ _(o)_(o)_I(1
(0),,2
(0)+2
],/~
(0)
[
l
l+~rR }(1-o-) ]
kl~(l+trg)~/,¢e+ ~(1-tr/R2) l+o./-----~
`` (o) ( o ) _ / ' 1
1
1\
+ e r R )'r/666~¢=0.
(A.12)
Substituting (A.11) into (A.12), to eliminate the secular terms, gives kl sign(a)=
1 I + ~ / R 3 1 1-cr/R 2 1 1 - t r / R 2 1 (1-tr/R2)(l+trR 3) + 2 1 - t r / R 2 24 l+tr/R 24 l + t r R 8 (l+orR) 2
(A.13)
478
H.H. Dai, A. Jeffrey/Interface solitary waves
and
I + o ' / R 3 2 1 - c r / R 2 1 1 - ~ r / R 2 1 ( 1 - t r / R 2 ) ( l + ~ R 3) 1 - o ' / R 2 t 3 l + t r / R +12 l + t r R 20 (l+o'R) 2
2to, sign(a)=
(A.14)
It then follows that
•,,1, = c sech:r, F
L V 12b
tanh: F , F
L V 12b
'
(A.15)
where
l+tr/R 3 1 1-tr/R 2 1 1-tr/R 2 1-tr C-I-o'/R2 4 l + c r / R +4 l+crR l+trR
3 ( 1 - t r / R 2 ) ( l + o ' R 3) 4 (l+trR) 2
(A.16)
If we let ~r=0 (i.e. for a one-layer fluid), then kl = _5, to1 = _ 3 , C = _3, which agree with the results given by Laitone [23]. Koop and Butler [5] who worked from Laplace's equations gave analytical expressions for the second order solutions, but they did not give the value of to1.
Appendix B. The asymptotic expression for the interface displacement First we define the following functions
Ho(x)=sechEx,
Hl(x)=tanhx,
H2(x)=sech4x,
Ha(x)=sechExtanh2x,
(B.1)
and the variables
Li = ~ f-~, / 1 - ~ 2 ( X - X o - T ),
R, = ~ /Ia[, / 1 - ~ 2 (x + X o - T ),
where X, Xo, T are defined in Section 4. The asymptotic expressions of interest are as follows:
first term 7 (0) = sign(a) (Ho(L,) + Ho(R0),
N (°) = sign(a) 2Ho(O);
(B.2a, b)
second term ~(1)= -sign(a) 2B(0) ~ / 1 ~ b H~(R,),
N ( ' = -sign(a) 2B(0) ~ f l ~ b H~)(O);
(B.3a, b)
third term r/(2) = 1 - o'/ R 2 [1H ~( L,)( HI( R, ) _ l) + ~H ~( R1)( H~( Lx) + 1) + ½Ho(L,) Ho( R,) ] 1 +cr/R
+ C[H3(L,) + H3(R1)] + 1 - crlR_~2 3 B2(O)H~)(R,)" l+cr/R 2
(B.4a)
N(2) - 1 - o ' / R___2____[~H3( O)+½H2( O)+ I H~( O)]+ 2CH3( O).+ 1 - c r / R 2 3B2(0._.....~)Hg( O) l+tr/R l+o'/R 2 1
131"/R 2
_3H~(O)[~:2 - 2B(0)se]; -~ l + o ' / R 4
(B.4b)
H.H. Dai, A. Jeffrey / Interface solitary waves
479
fourth term 7 (3) ~_~
•
/
.
l
a
l
1 - ~ / R__2 B(O)[-Ho(L1)H~(R~) - 2 H ~ ( R I ) ( H ~ ( L I ) + 1) -½H~(L~)Ho(RI)]
12b l + ~ r / R
+ ~V~ 1 2 b 11-+ O'/o./~ R2 B ( O ) [ - 3 H ~ ( R O H 2 ( R O + 2 H ~ ( R 1 ) - ~H~(R~)] - ~ a ~12b . v 2CH~(R~)
_dial
V12b
1-tr/R2 1 + o'/~
f ~ B3(O)Hg'(R~) - ~[ ~ zy~ sign(a)B(O)H~(gl)
_ ~ / ~ 2 ~b 2 B z ( O , 7 % s i g n ( a ) H ~ ( R 1 ) _ . t / lal tr/ R O 2 + D~ 3B(O)Hg,(RI,, V12b l+tr/R
(B.5a)
N(3) = .~/~a~ 1 - t r / R 2 Hg(O)[~B2(O)e_~B(O)e2]
V12b
+ .~ ~
l+cr/R
1 - t r l R 2 B(O)[-aH~(O1)H~(Oo)+2H~(O,)-2H3(O)+H2(O)-½H~(0)]
V 12b l + c r / R
- ~ ] 1 ~ b 2 C B ( O ) . ' 3 ( O ) - - l l V 1-]a[ I-°/REB3(O)H~(O)Eb 1 +tr/~ -
~ f l ~ b 270 s i g n ( a ) B z ( 0 ) H ~ ( 0 ) -
~f-~2v , sign(a)B(O)H~(O)
~/1-~
~/-lal ~. ~ / 1 2 b ~ / 1R+ ~DE - - ~+ D1 ~. .t.~. (. .O ) n o,,,.(v).
(a.5b)
References
[1] [2] [3] [4] [ 5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
G.H. Keulegan, "Characteristics of internal solitary waves," J. Rears. Nat. Bur. Stand. 51, 133-140 (1953). R.R. Long, "Solitary waves in the one and two fluid systems," TeUus 8, 460-471 (1956). T. Kakutani and N. Yamasaki, "Solitary waves on a two-layer fluid," J. Phys. Soc. Japan 45, 674-679 (1978). H. Segur and J.L. Hammack, "Soliton models for long internal waves," J. Fluid Mech. 118, 285-304 (1982). C.G. Koop and G. Butler, "An investigation of internal solitary waves in a two-fluid system," J. Fluid Mech. 112, 225-251 (1981). A.S. Peters and J.-J. Stoker, "Solitary waves in liquids having non-constant density," Comm. PureAppl. Math. 13, 115-164 (1960). T.B. Benjamin, "Internal waves of finite amplitude and permanent form," J, Fluid Mech., 25, 97-116 (1966). D.J. Benney, "Long non-linear waves in fluid flows," J. Math. and Phys. 45, 52-63 (1966). J.W. Miles, "Internal solitary waves I," Tellus 31, 456-462 (1979). J.W. Miles, "Internal solitary waves II," Tellus 33, 397-401 (1981). J.A. Gear and R. Grimshaw, "A second-order theory for solitary waves in shallow fluids," Phys. Fluids 26, 14-29 (1983). J.W. Miles, "Solitary waves," Ann. Rev. Fluid Mech. 12, 11-43 (1980). A.R, Osborne and T.L. Burch, "Internal solitons in the Andama Sea," Science 208, 451-460 (1980). A.R. Osborne, T.L. Burch and R. Scarlet, "The influence of internal waves on deep water drilling," J. Petroleum Tech., 1497-1503 (1978). R.M. Mirie and C.H. Su, "Internal solitary waves and their head-on collision, Part I", J. Fluid Mech. 147, 213-231 (1984). R.M. Mirie and C.H. Su, "Internal solitary waves and their head-on collision, Part II", Phys. Fluids 29, 31-37 (1986). N. Sugimoto and T. Kakutani., "Reflection of a shallow-water soliton, Part I. Edge layer for shallow-water waves," J. Fluid Mech. 146, 369-382 (1984). N. Sugimoto, Y. Kursaka and T. Kakutani, "Reflection of a shallow-water soliton, Part 2. Numerical evaluation," J. Fluid Mech. 178, 99-117 (1987). A. Jeffrey and H.H. Dai, "On the reflection of a solitary wave at a sloping beach: analytical results," Wave Motion 10, 375-389 (1988). N. Sugimoto, W. Nakajina and T. Kakutani, "Edge-layer theory for shallow-water waves over a stelr--reflection and transmission of a soliton," J. Phys. Soc. Japan 56, 1717-1730 (1987). N. Sugimoto, K. Hosokawa and T. Kakutani, "Reflection and transmission of a shallow-water soliton." G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). E.V. Laitone, "The second approximation to cnoidal and solitary waves," J. Fluid Mech. 9, 430-444 (1960).