Interlat. J. Math. & Math. Sci.
VOL. 19 NO. 2 (1996) 363-370
363
OBLIQUE
INTERFACE-WAVE DIFFRACTION BY A SMALL
BOTTOM DEFORMATION IN TWOSUPERPOSED FLUIDS
B. N.MANDAL
Physicaland EarthSoencesDvison Indian Statlsncal Institute 203,BarrackporeTrunk Road
Calcutta 700035, INDIA
u.
BASUDepartmentofAppliedMathematics Universityof Calcutta
92, APC Road Calcutta700009,INDIA
(ReceivedAugsut5, 1993and n revisedformDecember 12, 1994)
ABSTRACT. The problem of diffraction of oblique interface-waves by a small bottom deformation of the lower fluid intwosuperposedfluids has beeninvestigatedhereassuminglineartheory and invokingasimplified perturbation analysis First ordercorrectionstothe velocity potentials in the twofluids are obtainedby using theGreen’sintegraltheorem in a suitable manner The transmission and reflectioncoefficients areevaluatedapproximately These reducetothe known resultsforasinglefluid inthe absenceoftheupperfluid
KEY WORDS AND PHRASES. Oblique interface-wave diffraction, transmission and reflection coefficients,waterwaves
1991AMSSUBJECTCLASSIFICATIONCODE. 76C10,76D33 1. INTRODUCTION.
A time-harmonicprogressive wave train propagatingon the surface ofan ocean experiences no reflection iftheocean isofuniform finitedepth However,ifthebottom ofthe oceanhasadeformation, thenthe wave train ispartially reflected by and partiallytransmitted overthe bottom deformation Miles obtainedapproximately the transmission and reflection coefficients foroblique surface-waveswhen the bottomhasa small deformation in the form ofalong cylinderinthe lateral direction Mandal and Basu [2]extended thisproblemtoincludesurfacetensioneffectatthefree surface
In the present paper the oblique surface-wavediffraction problem considered in for a single fluid is generalizedto two superposedfluids wherein theupperfluid extendsinfinitely upwardsand the lower fluid is of finite but nonuniform depth below the mean interface and its bottom has a small deformation in the form of a longcylinderinthe lateral direction Utilizinga simplified perturbational analysis directlyto the governing partial differential equation and theboundary and other conditions describingthephysical problem,theoriginal boundaryvalueproblem(BVP)isreduceduptofirstorder toanother BVP SolutionofthisBVPisthen obtainedbyanappropriateuseofGreen’sintegraltheorem to the potential functions describing the BVP and source potentials given in [3] The first order corrections tothe reflection and transmission coefficients are then evaluated from the requirementsat infinity Itis verifiedthatin theabsenceof theupper fluid,known resultsforasinglefluid arerecovered
study of interface-waves in two superposed fluids is rather limited In the present paper, the two- dimensional source potentials ineach oftwo superposed fluids obtained earlier in [3] have been used suitably in the Green’s integral theoremto obtain representations for the first order corrections to the velocity potentialsineachof thetwo fluids
2. FORMULATION OF
THE
PROBLEMWeconsider twosuperposedimmiscible, nonviscous, incompressible andhomogeneousfluids with lowerfluidof densitypl occupying the region0
<_
y<_
h+ ec(x)
and upperfluid of densityoccupying the region y
_<
0Here,
theplaney 0 isthe position ofthe.
interface atrestand y-axisis taken vertically downwardsintothe lower fluid Thereis asmall deformationatthebottom of the lower fluid inthe lateral direction andisdescribedbyy h+ ec(x)
wherec(x)
is abounded andcontinuous function of compact support so thatc(x)
0 asIx[
oo, and e is a small positive number characterizing the smallness of the deformation Far awayfrom the deformation, the lower fluid isof uniform finite depth h The motion in eachfluid isassumed tobe small and irrotational sothatit is describedbythe velocity potentialsRe{oh(x,
y,z)e
-zt}
andRe{(x,
y,z)e
-zt}
inthelower and upper fluidsrespectively,abeing thefrequencyof the incomingtrainof progressivewavesatthe interface and thetime dependence e-wt being dropped throughout the analysis. Assuming lineartheory,satisfy the followingcoupled BVP
7
2
0 inthe region 0<
y<
h+ ec(x),
(2 1)X7 0 inthe region
<
0(2
2)where 7 isthethree-dimensionalLaplaceoperator,
u
u
on y 0, (2.3)K
+ I,u s(K + u)
on y 0wheres p2/pl,
K a2/g,
gbeing the acceleration duetogravity,bn
0 on y h+ c(z)
wherendenotesthe inward drawn normaltothebottom,
0
as yoo.(24)
(2
5)
(2.6)
A train of progressive interface-waves represented by the velocity potentials o(x,y)e ando(x,
y)e inthe lower and upperfluidsrespectively wherecosha(h y) (x’ Y)
sinhah e
’ (2
7)0(z,
y) eu+’" (28)isobliquelyincidentupon the bottomdeformationfrom negativeinfinity. Hereaisthe unique positive zeroof A
(k)
whereA(k)
Kcosh kh+ {s(K + k) k}
sinhkh, (2 9)and v asin0,# acos0 where O characterizes the oblique incidence ofthe wave train 0 0 correspondstonormalincidence This wave train ispartially reflectedby and partiallytransmitted over the bottomdeformation sothat q and satisfy the infinity requirements
OBLIQUE INTERFACE-WAVE DIFFRACTION PROBLEM 365
e
+ R
e as x x (2 11)whereTandRdenote respectivelythe unknown transmission and reflectioncoefficients
Assuming tobe very small,the bottom condition(25)canbe expressedinapproximate form[2]
as
, + e{c’(x) c(:r)y,} + 0(e 2)
0. (2 12)Inview ofthe geometry of the problemswe can assume
,(,
y,)= (, )’
,(,
y,) (z,
y) so that(x,
y) and(x,
y) satisfythefollowingBVP27
u2)
0 in 0<
y< h+ec(z),
(2 14)(V u2)
0 in y<
0where V isthetwo-dimensionalLaplace operator,
(2 15)
Cu ’u
on y 0, (2 16)K +u s(K+ Cu)
on y 0 (2 17){’ }
+ (c()) , + o(d) o
on y h (2 18)since satisfies(2.14),and
V-O
as y -oo.(2.19)
Also
,
/, satisfy the infinity requirements TCo(X,
y),o(z,y)
[o(z,y)]
o(z,y)as x oo,
(2 20)
+R o(
x,y)(2 21)
o(
x,y) as x oo3. METHOD OFSOLUTION
Inviewof the approximate bottom condition
(2
18)coupledwith thefact that aninterface-wave trainexperiencesno reflection ifthe lowerfluid has auniformbottom, we canassume aperturbation expansion for, ,T
andR
intermsofeaso + + O(d),
o
q-eel
q-O(e2),
(3 1) T 1
+
(T+ O(e2),
R eR, +O(e 2)
Using the expansions
(3.1)
in(2 14)to(221)
wefindthat ql,/31 satisfy thefollowing BVP(27 u2)1
0 in 0<
y<
h (3 2)(34)
KqS
+
s(KqS_,+ 2)
on 5’ 0 (35)where
qS
q(m) on 5’- hq(x) d
sinh oh #
-z (c(x)e
’"’)c(x)e
wAlso,ql,
1
satisfythefollowing infinity requirements,o(, ) ]
TI
1 ’0 (’ 5’) J
R1 0(Z Y)
as x oo, (37)
as x c (3 8)
(36)
Again, we apply the Green’s integral theorem to l(X,y) and H(x,y;
, rl)
in the region bounded externally bythe lines y=0(-X_<x<_X), x=+X(-Y_<y_<0),
y=Y(-X_<x<X) and ultimatelymakeX,
Y c HerewenotethatH(x,
y;, rl)
has no singularityintheregion Then we find0
[q21Hv- Hq21y]v=odx.
(3 9b)Multiplying(3 9b)bysandsubtracting from(3 9a)wefind
2r1(,
r/)= q(z)G(z,h;,7)dz+ [(lGy- Gly)- s(7,Hu- gl)]v=od:c
Usingthe conditions(3 4)and(3 5)for4)1 and
1
and theconditions on5’ 0forGandHgiveninthe Appendix,wefind that on5’ 0,To solve the above coupled
BVP,
we need two-dimensional source potentials for the modified Helmholtz’sequation duetoaline sourcesubmergedin either of twosuperposedfluids wherein thelower fluid is of uniform finite depthbelow the mean interface 5’ 0 and the upperfluid extends infinitely upwards When the source is submergedinthe lower fluid at((,r/)(0 <
r/<h),
letG(x,
5’;(,r/)
and H(z,
5’;(,r/)
denote the sourcepotentialsinthelower andupperfluidsrespectively, andwhen the source is submerged in the upper fluid at(, r/)(r/< 0),
letG(x,
y;, 7)
andH(x,
y;, 7)
denote the source potentials inthe lower and upperfluidsrespectively Expressions for thesesourcepotentials and their asymptotic behaviors as[x- c]
oo are given in [3] and are reproduced in the Appendix after correctingthemisprintsTofind
qa (, T])(0 <
f]< h)
weapply Green’s integral theoremtoql(X,
y) andG(x,
y;, rl)
inthe region bounded externally by the lines y=0(-X_<x_<X), x=+X(0_<5’_< h),
y
h(-
X_<
x<_ X)
and internally by the circleC with center at((, r/)
andradius6, andultimately makeX oeand 6 0 Wethen obtain27r1 (, r/) q(x) G(x,
h;, rl)dx + [lGy Gly]y=o
dx(3 9a)
OBLIQUEINTERFACE-WAVE DIFFRACTION PROBLEM 367
K
Thus thetermwthinthe square bracketinthe secondintegralvanishesidentically Henceweobtain
(, )
q(z)a(z,h;, v)dz
0< <
h. (3 0) To findif31 (,/’])(T] <
0) we apply Green’s integral theorem to@1 (x,
y) andH(x,
y;(, r/) in the region bounded externally by the lines y=0(-X_<z_<X), x= -t-X(-Y_<y<_0), yY(
X_<
z_< X)
and internallybythecircleC’
ofradius6with center at((, r/)
and ultimately letX,
Y ooand 6 0 Wethenfind2r(,r) [- b]=odz.
(3 lla)Again, weapplythe Green’sintegral theoremto
(x,
y) andGx,
y;(,r/) (r/< 0)
in theregion bounded externally by the linesy=0(-X<x_<X),
x=-I-X(0_<y_<h),
y=h(-X_<x_<X) and ultimately makeX
oo Noting thatGhasnosingularityinthe regionwefind0
[qly qlyV---]
y=Odx+ q(x)-(x,
h", 7)dx.
(3 lb) Multiplying(3 la)bysandaddingwith(3 lb)weobtain2T831 (, T]) [(qly qly) 8(31y )ly)]
y__0dx+
q(x)G(x,
h", 7)dx
(3 12) The term in the square bracket of the second integral vanishes because of the conditions satisfied by 1, andG,
Haty 0 Thuswefind(,) q(z)a(z,
h;,)az, v <
O.(3
3)4. EVALUATION OF
TIAND R1
T1
andRa
can be evaluated from the behavior of1(, r/)
orbl ((, r/)
as ( oe and -oo respectivelyin(3.10)
or(3 13).
TofindTa
we notefrom(3.7)
that1 (, ) T1o(, r/)
as cx.Also from(3.10)after using(A3)wefindas c sinh ah ql
(, ?’])
#(h
-’]-]- sinh2ah) q(x)dx o(, 7)
Thus
sinh ah
#( + L sinh2ah)
e q(x)dxia__ s_e_cO c(z)dx.
h
+ L
sinh2oh (4 1)Itisverifiedthat the sameexpression for
T1
is also obtainedby notingthe behaviorof31 (, T])
as( o in(3 7)and(313).
Again,toobtainR1,wenotefrom(3 8)
using(A3)in(3 10) find sinh ah
#(h_(_l- si--n-ah) /
eWq(x)dx0( , )
as Thussinh ah
RI #(h_l sinh2ah)
q(x)dxiasecOcos20
c(z)eUd:r
h
+ k
sinh ah (4 2)Itisagain verified that thesameexpression for
R1
isalsoobtainedbynoting thebehaviorof@1 (, 7)
as ( oin(3 8)and(3
13)Itmay be noted thatinthe absenceof the upperfluid
(s 0),
the results of forasinglefluid are recovered Inthat caseaisthe uniquerealpositivezeroof/(k)
Kcoshah k sinh kh Theresults for normal incidence of the wave train are obtained by putting 0 0 For 0r/4, R1
vanishesindependently of the bottomdeformation This wasalso observed byMiles forasinglefluid Also, oncethefunctionalform of
c(z)
isknown,T1
andR1
canbeobtainedexplicitlyAPPENDIX
(a)
G(x,
y;, r/)
andH(x,
y;, ) G(x,
y;, rl)
andH(x,
y;, rl)
satisfyV
vg.)G
0 in 0_<
y<
hexceptat(, r/)(0 <
r/<h),
a K0()
as(( ) + (U ’))
/ 0,(V2 u2)H
0in y_<
0,Gu= Hyon
y 0,KG
+ Gy s(KH + Hu)on
y 0,Gu=Oony=h,
7H 0as y--
-cx,G, H
have outgoing natureasIx [ .
ThenG(x,
y;, )
dH(x,
y;, )
egivenby(cf
[3]) after coectingsomespfintsa(,
u;, n) Ko()
1--8Ko(’)
++J ()
e-kh(sinhkr?
q-scoshkr/)
] cos{ (k 2( k- __U2)1/2 (Xu2)
1/2)}
x
coshk(h-
y)+fi
silky dk ,(A1)H
(x,
y;, 7)
2Ko(vr) +
2f
sinhkr/+8coshkr?
e-khl+s
J,
coshkhe-h{s(K + k) k}(sinh kr/+ scoshk7)sechkh k(1 s)e -’
sinhkhi
(A)
OBLIQUEINTERFACE-WAVE DIFFRACTIONPROBLEM 369
where
r’= {(x- ()’2 +
(y+ r)2),’.’
and the contour in each integral is indented below the pole at k-a to ensure the outgoing behavior ofG and H asx-(I
c From(AI) and (A2) it can be shown that as x ( ccosha(h r/)cosha(h
) e’’ ’’ ’
G(x,
y;(, ) 27vi (A3)h
+ -:
sinh ah(a2
/22)lcosha(h
r/)sinhahe"u e....
--,,/i-tH(z,
y;(,r)
27rz(b)
G(x,
V;,
rl)andH(x,
y;,
rl) h+ -sinhah (2 v"_)/2
(A4)
G(x,
V;,
7)andH(x,
y;, rl)
satisfy(7 2-v2)=0in0<y<
h,u2)
0iny_<
0exceptat(, r/) (r/< 0), H Ko(ur)
asr 0,Gj H on y 0,
KG
+ Gy s(KH + Hv)
on y 0,Gy=Oony=h,
7H
0 as yG, Hhave outgoingnatureas
Ix 1
oo ThenG(x,
y;, rl)
andH(x,
y;, r/)
aregiven by(cf[3])
aftercorrecting the misprints
G(x,
y;, )
1
+
s/k(k)
coshk(h
y)e-h } cos((k ,)l/( )}
+ c
sinhky e(k Z i
dk (AS)(A6) where again the contour in each integral is indented below the pole k a to ensure the outgoing behaviorofGandHas
Ix (I
oo. From(A5)
and(A6)
itcanbe shown thatasIx 1
oo(z,
y;, r)
2sTrie’’sinh ahcosha(h
y) e’(2-")1/1x-1 h+ L
sillh a(a2 v2)
1/2(A7)
e’(n+v)sinh ah e
n(x,
y;, rl)
2sTrih
+ .L
sinh ah(a v)
/ (A8)[2]
[4]
[5]
(1981), 121-123
MANDAL, BN and BASU, U, A note on oblique water wave diffraction by a cylindrical deformationofthe bottomin thepresenceof surfacetension,Arch. Mech. 42(1990),723-727
MANDAL, B N and CHAKRABARTI, RN, Two-dimensional, source potentialsin atwo-fluid mediumfor themodifiedHelmholtz’s equation, Internat.J.Math.
&
Math.Sc. 9(1986), 175-184GORGUI, M A and KASSEM, S
A,
Basic singularities in the theory ofinternal waves, Q..ll.Mech.Appl.Math. 31(1978),31-48.
RHODES-ROBINSON, PF, On wavesat an interface between twoliquids, Math. Proc. Camb.
Phil.Soc. 38(1980), 183-191