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Forced capillary-gravity waves in a circular basin
Wave Motion 18 (1993) 401-412 Elsevier
401
Forced capillary-gravity waves in a circular basin M.C. Shen and S.M. Sun Department of Mathematics, University of Wisconsin, Madison, W153706, USA
D.Y. Hsieh Department of Mathematics, The Hong Kong University of Science & Technology, Clear Water Bay, Hong Kong Received 5 January 1993, Revised 14 July 1993
A linear theory is developed to study the forced capillary-gravity waves generated on water in a circular basin by the horizontal oscillations of its side wall. A Green's function method is used to construct a solution of the linearized equations subject to an edge condition at the contact line. The problem is reformulated in terms of an integral equation on the equilibrium free surface, and a uniqueness result is obtained by invoking the Fredholm's alternatives.
I. Introduction
The problem of forced capillary-gravity waves generated by the harmonic motion of part of the boundary of a liquid body has been extensively studied in the past. On the basis of linearized equations governing the liquid motion, the classical plane wave-maker problem in the absence of surface tension was first studied by Havelock [ 1 ]. Evans [2] included the effect of surface tension in his investigation of surface waves produced by a heaving circular cylinder. He pointed out that the main difficulty is not the complexity of the free surface condition, but rather the uncertainty over the condition at a contact line, the intersection of a free surface and rigid boundary of the liquid body. An edge condition was prescribed by him at the contact line so that the wave amplitude at infinity is uniquely determined. The Havelock' s problem including the capillary effect was investigated by Rhodes-Robinson [ 3 ]. Hocking [4] formulated an edge condition dealing with the dynamical variation of the contact angle at a contact line as well as its hysteresis. More recently Mandel & Bandyopadhyay [ 5 ] have employed Fourier transform to restudy the problem considered by Rhodes-Robinson [ 3 ]. Up to now the uniqueness of a solution in relation to the edge condition has not been completely resolved. The nonlinear theory of forced surface waves was essentially developed for the investigation of the subharmonic excitation of the so-called Faraday waves in a basin. We refer readers to the review article by Henderson & Miles [6] for more details. The work reported here was stimulated by the observation of surface waves excited by the horizontal oscillation of the side wall of a basin of antiquity. The apparatus, invented in ancient China for entertainment, has two handles symmetrically fixed at the rim of the basin half-filled with water. When the handles are rubbed with both hands, a standing wave first appears with a number of nodes around the circular wall. After the pace of rubbing the handles becomes faster, the free surface starts to break up and eventually the water in the basin spurts high into the air. The basin is called a "Fish Wash", because the unknown inventor engraved four fish on the bottom of the basin and it looks as if the water spurted out of each fish's mouth. The mechanical properties of the Fish Wash have been studied both experimentally and theoretically by Wang [7,8]. To simplify our mathematical model, we shall assume that 0165 -2125 /93 / $06.00 © 1993 - Elsevier Science Publishers B.V. All rights re served
402
M.C. Shen et al. / C a p i l l a ~ - g r a v i ~ waves
the harmonic motion of the circular wall of the basin is prescribed, and our objective of this work is to develop a linear theory to study the capillary-gravity waves excited by the prescribed motion. We shall construct a solution to the linearized equations by means of the Green's function method, and a prescribed edge condition at the contact line originally formulated by Evans [2] is seen to be necessary to obtain a complete solution. Later the problem will be reformulated in terms of an integral equation on the equilibrium free surface and it is shown that the edge condition together with other conditions is also sufficient to ensure a unique solution of the linearized equations. Therefore the uniqueness question for the present problem is completely resolved. Needless to say, the linear theory fails when the amplitude of the excited surface wave is too large. We shall defer the development of a nonlinear theory to a subsequent study. This paper is organized as follows. In Section 2 the problem in terms of the linearized equations is formulated, and the corresponding eigenvalue problem is studied. In Section 3 we construct a solution of a related problem and use it to make the boundary condition on the side wall homogeneous. The derivative of a Green's function is found by constructing a solution for the case of zero edge condition, and in terms of it we finally obtain a solution of our problem. Some numerical results are presented to describe the surface wave patterns based upon the linear theory. In Section 4 we construct two Green's functions subject to standard boundary conditions and use them to reduce the linearized equations to an integral equation on the equilibrium free surface. The Fredholm's alternatives for an integral equation with a symmetric weakly singular kernel are invoked to determine the uniqueness of a solution of the problem.
2. Formulation
We consider a circular basin V with a side wall M and base B (Fig. 1 ) filled with an incompressible, inviscid liquid of constant density and assume that the liquid motion is irrotational. Let q~ be the potential function and z = Z(r, ~9, t) be the free surface. Then the linearized equations governing the liquid motion are the following [ 2 ] : V~3qb=0
inV,
(1)
V
M
T h
,Y
Fig. 1. The circular basin.
M.C. Shen et al. /Capillary-gravity waves qOt+gZ=TV~2Z)
onS,
(2)
z , = q~z
q0z=0
403
(3)
onB,
tier = f ( z ) (eimOq- e -imO) e iw¢ on M ,
(4)
C1)rz = ot( eima--}-e -imO) ei°,t
(5)
/",
where V~3 and V~2 denote respectively the three-dimensional and the two-dimensional Laplacian, g is the constant gravitational acceleration, pT is the constant surface tension coefficient where p is the water density, m is a positive integer, f ( z ) is a prescribed smooth function, a is a constant, S is the equilibrium free surface z = h , a n d / " i s the constant line, a circle with radius a. The condition on M is obtained by differentiation of the prescribed oscillatory displacement of the circular wall. The commonly observed value o f m is 2 [7]. The condition at I ' i s called the edge condition, the form of which is motivated by the one used in [2] and [3], and plays an important role in the uniqueness of a solution to the problem. It is obtained by prescribing Zr at/"in the same form as ~rz except a phase difference of xr/2 and then by differentiation. Since ( 1 ) to (5) are linear, we may just consider qbr = f ( z ) e i(+mO÷tot)
on M,
qbrz = ae i( + m a+ ~)
at F ,
to replace (4) and (5). Let
t~= ei( +m°+ °x)t]~(r, Z)
Z = e i( +ma+°x)rl( r ) .
We now measure r, z, Z and ~/in units of h, t in units of ( h / g ) 1/2, qb and ~b in units of gh 3/2, to in units of ( g / h ) l / a , T in units of gh 2, f ( z ) in units of (gh)l/2 and a in units of g l/2h-i/2. In terms of ~b and ~?, (1) to (5) become Y24~=
~r 2 + ---rOr - ( r e ~ r ) 2 +
-to~4,+4,~;r~,4,~=
4~=0
in V,
(6)
~ + rOr -(m/r)~ 4'z onS,
(7)
n= (ito)-~4~ z
~6z = 0 on B ,
(8)
~br = f ( z ) on M ,
(9)
~brz= a at F .
(10)
We first study the natural frequencies of the liquid in the basin and consider the eigenvalue problem .~2~b=0
inV,
-122qb+qbz=T.ZPl~bz
(11) on S,
(12)
~bn=0
on B U M ,
(13)
~)rz=O
at/'.
(14)
Condition (12), by using ( 11 ), can be rewritten as - 0 2 4'+ 4~z = - r4~=z .
(15)
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M . C . S h e n et al. / C a p i l l a ~ - g r a v i O, w a v e s
We remark here that although ( 15 ) is much more convenient to use than (12), it masks the importance of the edge condition (14) in relation to the operator ~t~b. in (12). Let
q~=R(z)Z(z) , and (11), (13) to (15) imply
R"+r-lR'-r-2m2R+~r2R=O R'(a) ---0
O<~r
Z " - cr2Z=0
0
-,(22Z+Z'=-TZ"
atz=l,
Z'=0atz=0.
It is easy to find that the eigenfunctions are given by
d~m. =Jm(crm. r) cosh Crm.Z, n = 1, 2 . . . . .
(16)
where Jm(x) is the mth order Bessel function of the first kind and O'm,,aare the zeros of J~,,(~ra) = 0. The eigenvalues are given by O~,. = O'm,, tanh o-.,.( 1 + T~r2,..) .
(17)
Next we construct a function ff satisfying ~2~=0
inV,
on S U B ,
3.-=0
ffr=f(z)
onM.
By separation of variables again, it is not difficult to obtain
¢=Aorm+
y" A,,lm(n~r) cos n'rrz,
(18)
n--I
where I
Ao=(mam-i)
i I f ( z ) dz, *¢
0 I
A,, = 2 I f ( z )
cos nTrz d z / ( n T r l , ' ( n w a ) ) ,
n= 1, 2 ....
0
and Ira(X) is the ruth order modified Bessel function of the first kind. Now we let
4,= q,+,~. Then (6) to (10) become 5LY2~0=0 i n V ,
(19)
- oJ2q,+ q,~ - T_~, q~ = o~2,~} r/= (ion) -~(qJ~ + ~.-) ~0n = 0
onMUB,
qJr~ = c~ a t F ,
_
on S ,
(20) (21) (22)
M.C. Shen et al. / Capillary-gravity waves
405
where we assume f(z) is sufficiently smooth so that termwise differentiation of the series in (18) is permissible.
3. G r e e n ' s function and solution of the p r o b l e m
We first construct a Green's function satisfying .~2G = - r - 1 6 ( r - r ' ) 6 ( z - z
-oo2G+Gz- T.~,G~=O
Assume
')
inV,
(23)
onS,
(24)
Gn=0
onMUB,
(25)
Grz = 0
at F .
(26)
G(r, z; r', z') exists. We interchange the roles of (r, z) and (r', z')
and form the integral
f I f (GS~;q'-q'S~'~G)r' dr' dz' dO=2~rqJ(r,z)= f l(GqJz,-qtGz,)r' dr' dO V
S
=t°-2T f I jr
(Gz'S~]qlz'-OeZ~]Gz')r' dr' dO+
s
(oGz,r'dr' dO
((
l.t
s
=to-2T2~raGz,(r,z,r'=a,z'=l)a+ ff(oGz,r'dr'dO
(27)
S
where we have used (19) to (26) and 02
2~= ~
1
0
+ r'Or'
02
( m / r ' ) 2q- 0 z , 2 ,
As seen from (27), we need only to know problem with a = 0. .ZF2t~=0
¢ --~1-
02
1
0
0r,2 + r'Or'
(m/r')2
Gz, on S in order to construct ~0. Hence we look for a solution t~ of the
in V,
-o)2(O+~z-T(Ozz~=~o2~
(28)
(29) onS,
(30)
~.=0
on M U B ,
(31)
~z = 0
at F .
(32)
~=
b. .]m(O'mnr) c o s h ( O ' m n Z ) ,
(33)
Assume E tl=l
where B. are constants to be determined. Apparently ~ satisfies (29), ( 31 ) and (32). We substitute (33) for ~ in (30) to obtain
~B.J,.(tr.~r)( - t o 2 cosh 0-.,. +trm~ sinh 0-,.. + To-3n sinh Ormn) -,~-t02~.
M.C. Shen et al. / Capillary-gravity waves
406
Assume
amn = ~'~2,n --0)2:=/:0.
Then by (17) and the orthogonality property of Jm(°',,,.r), we find
B. = 0)2 i JffJ.,(~r.,.r)r dr/ ( Am,,IIJm(O'm,,r)II2c°sh
(34)
o'.,,,)
0
where a
II&.( cr.,.r) II2= f j2m( O%nr)r d r = ( 6r],,,a2- mZ)j2 ( O'm.a ) ( 2~r2,,,) -' 0 By using (34) for B. in (33) and assuming that ~b on S is sufficiently smooth, we can interchange integration and summation to obtain
(0= f f G:,(r,z; r',z'=l)¢r' dr' dO, S
where
G.,( r, z; r', z' = 1) = (2ax)
1o)2
Jm( cr..nr)J,,,( O-m,,r') cosh O-m,,Z/( Am,, IIJ,,,(~r..,r) II2c°sh O-ran) •
(35)
n=l Finally we gather all the results and obtain a solution of ( 1 ) to (5) : @P= ( ~ + ~ ) ( e i m ° ± e -im°) e i°~,
~=Aor'"+
~_~ A.lm(n'rrr) cos nrrz, tt= l
I
Ao=(ma
m I)ff(z)dz, o
I
An=2 f f(z) cosn~rzdz/(nTrl',(nrra)),
n=l, 2.....
o
~t
~=oe0) 2 T a G z , ( r , z , r ' = a , z ' = l ) +
f ~(r',z'=l)G.,(r,z;r',z'=l)r'dr
',
0
G:,(r, z; r', z '= 1) is given by (35) and r/is determined by (7). For the case o f f ( z ) = K = c o n s t a n t , an approximate expression for ~b may be obtained if I,a.,~l << l, but ( 0)21Am~[ ) - I remains finite. In other words, 0) is near the natural frequency J2mk. NOW (o=(ma m l)-lKrm
and
M.C. Shen et al. /Capillary-gravitywaves
407
a
f Jm(Ormkr)~(r)r dr=KaZ(mo',~k) -IJm + l ( Ormka ) 0
dp~ ( Totto2Jm( trmka) + Ka2( mtr,~k) - IJm+ l(O'mka) ) X
Jm(Ormkr) cosh(trmkZ) / (toz Amk(Cosh Ormk) IIJm(trmkr) II)"
Some numerical computations for ~7cos 20 in the case o f f ( z ) = K = constant are presented in Fig. 2 to Fig. 6, which indicate rather interesting results. Figs. 2 and 3 show a wave pattern similar to the initial stage o f the capillarygravity waves generated by rubbing the handles of the basin. In the numerical experiments as shown in these two figures, the forced frequency of the side wall is near the lowest natural frequency O2~ = 1.5861. In Fig. 3, the amplitude K of the forced oscillations is five times larger than that in Fig. 2. However, the wave patterns do not change much. W e also did some numerical computations for the case of to = 1.57 and K = 0.1 and the wave pattern still remains the same. As to moves towards the second natural frequency 022 = 2.367644, the four points A, B, C, and D, where the maximum amplitude of the surface waves takes place, move towards the center o f free surface. The wave patterns are shown in Figs. 4 and 5. As to is in the neighborhood of 023 = 2.89227, two separatrices appear and four more maximum points emerge. The numerical results for this case are presented in Fig. 6. In some other numerical experiments, where different values of T and a are used, the wave patterns are basically the same but the solution behavior is very sensitive to the change of to. As to crosses each natural frequency, the solution becomes unbounded unless a solvability condition discussed in Section 4 is satisfied. The evolution of the amplitude of the surface waves with to near one of Omn will be studied by a nonlinear approach in a subsequent study. In all the figures except Fig. 2 where K - - 0.02, we choose m = 2, c~= 0.05, T = 0.0001, K = 0.1, and a = 1.2. W e note that the choice of the values of the parameters is based upon the estimates o f their orders of magnitude and the sensitivity of the numerical output to their changes. For example, h is in the range of 15 cm to 25 cm, and the surface tension of water ---70 d y n e s / c m . The unit for T is gh z and T is of order 10 -4. The radius of the basin is larger than its depth. Therefore we choose a = 1.2. At present we have no definite information about c~. On the other hand, K
A
,o1972
.0197/
-.01972
-.0018
/
D
B
C Fig. 2. The contour plot of the free surface, to= 1.6, K = 0.002.
408
M.C. Shen et al. /Capillary-gravity waves
A .09856
D
-
C Fig. 3. The contour plot of the free surface, to = 1.6.
..00152~ Fig. 4. The contour plot of the free surface, to = 2. should be small. H o w e v e r , for a g i v e n m the values o f a, K and T do not affect m u c h the qualitative properties o f the w a v e mot!on, and the d o m i n a n t p a r a m e t e r apparently is ~o.
4. U n i q u e n e s s result
In this section we attempt to reformulate the t i m e - r e d u c e d equations ( 1 ) to (5) in terms o f an integral equation
M.C. Shen et al. / Capillary-gravity waves
•
•
409
J .2
..04 .00
0 "
.1 .19
.!
Fig. 5. The contour plot of the free surface, to = 2.367.
Fig. 6. The contour plot o f the free surface, to = 2.88.
on S by means of Green's functions satisfying the Neumann or the Robin boundary condition. In order to incorporate the edge condition in the formulation of an integral equation to be derived later, we first construct the Green's function (Gs(~, ~), where :~(x, y, z), ~= (x', y' z' ) ~ S are Cartesian coordinates, satisfying
- V~2Gs+T-'Gs=~(.f-~), Gs, = 0
on F .
.f, ~ S ,
(36) (37)
Here S may be considered as a simply connected open domain with a simple closed curve F as its boundary in the x, y-plane. It is easy to show that the homogeneous problem of (36) and (37) possesses a trivial solution only. Let us define
M.C. Shen et al. / Capillary-gravitywaves
410
A=-V~z+T
II,
on D ( A ) = (u(x-)lu ~ C 2 ( S ) 0 C I (S), u. = 0 at F), where C " ( ~ ' ) denotes n-times continuously differentiable functions defined on a domain ~g/, and S is the closure of S. Then
(Au, U) = ~ f U(-- ~ + T - l l ) u dA >~T - ' ( u , U). s ,t
Furthermore,
(Au, v ) = ( u , Av),
u, ~,~D(A) .
Hence A is positive definite and its eigenvalues are greater than or equal to T nonnegative eigenvalues only. But it may not be positive definite. Next we construct another Green's function Gv satisfying
- V~3Gv=8(d-~),
d, ~ V ,
Gv: + ( o~2cr2/T)Gv=O Gv, = 0
~. Its inverse A - l exists and has
(38)
on S ,
(39)
on 3V,
(40)
where o- is a positive constant, V denotes a bounded open domain in E 3 with the flat surface S as the equilibrium liquid surface, 3V is the remaining rigid smooth boundary of the liquid body and F i s the contact line (Fig. 7). It is also easy to show that the homogeneous equations corresponding to (38) to (40) possess a trivial solution only. Now we are ready to deal with the following problem, which is more general than the time-reduced problem of (1) to (5), W3=q~=0
inV,
-TV~z~o.+q~z=w2~0+f(x-)
(41) onS,
(42)
q~. =g(x-)
on aV,
(43)
qL-, = h ( x ~
on F ,
(44)
where we assume that f, g and h are sufficiently smooth functions. By means of Gs, we can transform (42) to an integral form
ff
(Gs V~z~z,-~z, V~2Gs) dA
s
F
aV Fig. 7. The fluid domain.
M.C. Shen et al. / Capillary-gravity waves
S
s
= T-to2,~s~o+ F(x-),
411
F
.fES
(45)
where we have used (37), (42) and (44). Then we make use of Gv to reduce (41) to (44) to an integral equation
(xO=f f f
dV
v
(46) s
S
ov
where (38) to (40) and (45) have been used. Since q~ is completely determined by its values in S, we may just consider Y~ S and denote the restriction of q~on S by q~sand the restriction of Gv on S by Gvs. (46) may be expressed as
q~s(X--)=T-%2ffGvs(Gs+o'2I)~osdA+Hs(xD S
= T - l co2,~vs(~ s + tr21) qOs+ Hs(x-),
~
S.
(47)
Here
s
av
and we define, for 2, ~ S, ~vs:~o~L2(S)--* ~vs~P =
ff
Gvs(X, ~)~P(~) dA,
S
ff s
where L2(S) denotes the space of square-integrable functions on S. Note that ffs is an extension ofA - 1 and Gs has a logarithmic singularity in S. Gv has a singularity like I~?- ~1 - ~ at .f= ~, and its restriction Gvs on S also has the same singularity. It is not difficult to show that both operators ~'vs and ffs are compact operators [9]. Furthermore, they are symmetric and ffs + tr2l is positive definite. Hence ~ v s ( ~ s + 0"21) is also compact, but not necessarily symmetric. However, we may define a new inner product (U, V)o-=(,~S
u, V )
U, v E L 2 ( S )
,
where # s = ~'s + Ov2I. Since # s is positive definite, (u, v),, is well defined. Then ~ v s # s is symmetric with respect to the new inner product, since
( ~vs#sU, v) ~ = ( #s~Cvs#sU, v) = ( #sU, ~vs#sV ) = ( u, ~vs#sV),,. Now we need only solve the integral equation q~s=T-ltoz,~ffgs+H,
~os~L2(S) ,
(48)
412
M.C. Shen et al. / Capilla~-gravity waves
where ,~ = . ~ v s ~ s and L~(S) is L2(S) with the new inner product. Since
(~sU, U)
u~L~(S),
(,fsU, U)>~T lw2o-2(u,u),
where lilts Ill is the operator norm O f 3 s , IJull = (u, u)J/2 and Ilull,, = (u, u)~/z are equivalent. Therefore, ff is also compact on L~(S). It follows that the Fredholm alternatives for a symmetric compact operator can be invoked to study a solution of (48). If T - ~w2 is not a characteristic value of if, (48) has a unique solution q~sand consequently q~ is also uniquely determined by (46). If T ~o92happens to be a characteristic value of if, then (48) possesses a solution provided that H is orthogonal to the eigenfunctions corresponding to T ~w2. It can be shown that ¢ in (46) does possess the necessary regularity and satisfy (41) to (44), and we also note that the boundary conditions together with the edge condition in particular are sufficient to ensure a unique solution of (41) to (44) if o92 is not an eigenvalue of the homogeneous equations. In principle, once Gvs and Gs are found, q~s can be determined by the Ritz-Rayleigh method or other variational methods. Needless to say, more general boundary conditions
qJr =f(z, O) e i~
on M ,
q~rz = a ( 0 ) e i~"
at F ,
than (4), (5) can be dealt with either by the Green's function method presented in Section 3 or by the integral equation method discussed here.
Acknowledgements The authors wish to thank Professor Dajun Wang of Peking University, China for several helpful discussions. The research of the first two authors was partly supported by the National Science Foundation under Grant CMS8903083.
References [ 1 ] T.H. Havelock, "Forced surface waves on water", Philos. Mag. 8, 569 (1926). [2] D.V. Evans, "The effect of surface tension on the waves produced by a heaving circular cylinder", Proc. Cambridge Philos. Soc. 64, 833 (1968). [3 ] P.F. Rhodes-Robinson, " O n the forced surface waves due to a vertical wave maker in the presence of surface tension", Proc. Cambridge Philos. Soc. 70, 323 ( 1971 ). [4] L.M. Hocking, "Capillary-gravity waves produced by a heaving body", J. Fluid Mech. 186, 337 (1986). [5] N. Madal & S. Bandyopadhyay, " A note on the plane vertical wavemaker in the presence of surface tension", Quar. Appl. Math. 49, 627 (1991). [6] J. Miles & D. Henderson, "Parametrically forced waves", Ann. Rev. Fluid Mech. 22, 143 (1990). [7] Dajun Wang, " A research on the mechanical properties of cultural relics of ancient China" (in Chinese), Proceedings of Scientific and Technological Archeology, Chinese University of Science & Technology Publishing House, 51-56 ( 1991 ). [ 8 ] Dajun Wang, "Mysterious Chinese antiquities - The application of vibration theory to ancient Chinese antiquities and modem research on them", manuscript. [9] S.G. Mildalin, "'Mathematical Physics, An AdL,anced Course, North-Holland, Amsterdam (1977).
401
Forced capillary-gravity waves in a circular basin M.C. Shen and S.M. Sun Department of Mathematics, University of Wisconsin, Madison, W153706, USA
D.Y. Hsieh Department of Mathematics, The Hong Kong University of Science & Technology, Clear Water Bay, Hong Kong Received 5 January 1993, Revised 14 July 1993
A linear theory is developed to study the forced capillary-gravity waves generated on water in a circular basin by the horizontal oscillations of its side wall. A Green's function method is used to construct a solution of the linearized equations subject to an edge condition at the contact line. The problem is reformulated in terms of an integral equation on the equilibrium free surface, and a uniqueness result is obtained by invoking the Fredholm's alternatives.
I. Introduction
The problem of forced capillary-gravity waves generated by the harmonic motion of part of the boundary of a liquid body has been extensively studied in the past. On the basis of linearized equations governing the liquid motion, the classical plane wave-maker problem in the absence of surface tension was first studied by Havelock [ 1 ]. Evans [2] included the effect of surface tension in his investigation of surface waves produced by a heaving circular cylinder. He pointed out that the main difficulty is not the complexity of the free surface condition, but rather the uncertainty over the condition at a contact line, the intersection of a free surface and rigid boundary of the liquid body. An edge condition was prescribed by him at the contact line so that the wave amplitude at infinity is uniquely determined. The Havelock' s problem including the capillary effect was investigated by Rhodes-Robinson [ 3 ]. Hocking [4] formulated an edge condition dealing with the dynamical variation of the contact angle at a contact line as well as its hysteresis. More recently Mandel & Bandyopadhyay [ 5 ] have employed Fourier transform to restudy the problem considered by Rhodes-Robinson [ 3 ]. Up to now the uniqueness of a solution in relation to the edge condition has not been completely resolved. The nonlinear theory of forced surface waves was essentially developed for the investigation of the subharmonic excitation of the so-called Faraday waves in a basin. We refer readers to the review article by Henderson & Miles [6] for more details. The work reported here was stimulated by the observation of surface waves excited by the horizontal oscillation of the side wall of a basin of antiquity. The apparatus, invented in ancient China for entertainment, has two handles symmetrically fixed at the rim of the basin half-filled with water. When the handles are rubbed with both hands, a standing wave first appears with a number of nodes around the circular wall. After the pace of rubbing the handles becomes faster, the free surface starts to break up and eventually the water in the basin spurts high into the air. The basin is called a "Fish Wash", because the unknown inventor engraved four fish on the bottom of the basin and it looks as if the water spurted out of each fish's mouth. The mechanical properties of the Fish Wash have been studied both experimentally and theoretically by Wang [7,8]. To simplify our mathematical model, we shall assume that 0165 -2125 /93 / $06.00 © 1993 - Elsevier Science Publishers B.V. All rights re served
402
M.C. Shen et al. / C a p i l l a ~ - g r a v i ~ waves
the harmonic motion of the circular wall of the basin is prescribed, and our objective of this work is to develop a linear theory to study the capillary-gravity waves excited by the prescribed motion. We shall construct a solution to the linearized equations by means of the Green's function method, and a prescribed edge condition at the contact line originally formulated by Evans [2] is seen to be necessary to obtain a complete solution. Later the problem will be reformulated in terms of an integral equation on the equilibrium free surface and it is shown that the edge condition together with other conditions is also sufficient to ensure a unique solution of the linearized equations. Therefore the uniqueness question for the present problem is completely resolved. Needless to say, the linear theory fails when the amplitude of the excited surface wave is too large. We shall defer the development of a nonlinear theory to a subsequent study. This paper is organized as follows. In Section 2 the problem in terms of the linearized equations is formulated, and the corresponding eigenvalue problem is studied. In Section 3 we construct a solution of a related problem and use it to make the boundary condition on the side wall homogeneous. The derivative of a Green's function is found by constructing a solution for the case of zero edge condition, and in terms of it we finally obtain a solution of our problem. Some numerical results are presented to describe the surface wave patterns based upon the linear theory. In Section 4 we construct two Green's functions subject to standard boundary conditions and use them to reduce the linearized equations to an integral equation on the equilibrium free surface. The Fredholm's alternatives for an integral equation with a symmetric weakly singular kernel are invoked to determine the uniqueness of a solution of the problem.
2. Formulation
We consider a circular basin V with a side wall M and base B (Fig. 1 ) filled with an incompressible, inviscid liquid of constant density and assume that the liquid motion is irrotational. Let q~ be the potential function and z = Z(r, ~9, t) be the free surface. Then the linearized equations governing the liquid motion are the following [ 2 ] : V~3qb=0
inV,
(1)
V
M
T h
,Y
Fig. 1. The circular basin.
M.C. Shen et al. /Capillary-gravity waves qOt+gZ=TV~2Z)
onS,
(2)
z , = q~z
q0z=0
403
(3)
onB,
tier = f ( z ) (eimOq- e -imO) e iw¢ on M ,
(4)
C1)rz = ot( eima--}-e -imO) ei°,t
(5)
/",
where V~3 and V~2 denote respectively the three-dimensional and the two-dimensional Laplacian, g is the constant gravitational acceleration, pT is the constant surface tension coefficient where p is the water density, m is a positive integer, f ( z ) is a prescribed smooth function, a is a constant, S is the equilibrium free surface z = h , a n d / " i s the constant line, a circle with radius a. The condition on M is obtained by differentiation of the prescribed oscillatory displacement of the circular wall. The commonly observed value o f m is 2 [7]. The condition at I ' i s called the edge condition, the form of which is motivated by the one used in [2] and [3], and plays an important role in the uniqueness of a solution to the problem. It is obtained by prescribing Zr at/"in the same form as ~rz except a phase difference of xr/2 and then by differentiation. Since ( 1 ) to (5) are linear, we may just consider qbr = f ( z ) e i(+mO÷tot)
on M,
qbrz = ae i( + m a+ ~)
at F ,
to replace (4) and (5). Let
t~= ei( +m°+ °x)t]~(r, Z)
Z = e i( +ma+°x)rl( r ) .
We now measure r, z, Z and ~/in units of h, t in units of ( h / g ) 1/2, qb and ~b in units of gh 3/2, to in units of ( g / h ) l / a , T in units of gh 2, f ( z ) in units of (gh)l/2 and a in units of g l/2h-i/2. In terms of ~b and ~?, (1) to (5) become Y24~=
~r 2 + ---rOr - ( r e ~ r ) 2 +
-to~4,+4,~;r~,4,~=
4~=0
in V,
(6)
~ + rOr -(m/r)~ 4'z onS,
(7)
n= (ito)-~4~ z
~6z = 0 on B ,
(8)
~br = f ( z ) on M ,
(9)
~brz= a at F .
(10)
We first study the natural frequencies of the liquid in the basin and consider the eigenvalue problem .~2~b=0
inV,
-122qb+qbz=T.ZPl~bz
(11) on S,
(12)
~bn=0
on B U M ,
(13)
~)rz=O
at/'.
(14)
Condition (12), by using ( 11 ), can be rewritten as - 0 2 4'+ 4~z = - r4~=z .
(15)
404
M . C . S h e n et al. / C a p i l l a ~ - g r a v i O, w a v e s
We remark here that although ( 15 ) is much more convenient to use than (12), it masks the importance of the edge condition (14) in relation to the operator ~t~b. in (12). Let
q~=R(z)Z(z) , and (11), (13) to (15) imply
R"+r-lR'-r-2m2R+~r2R=O R'(a) ---0
O<~r
Z " - cr2Z=0
0
-,(22Z+Z'=-TZ"
atz=l,
Z'=0atz=0.
It is easy to find that the eigenfunctions are given by
d~m. =Jm(crm. r) cosh Crm.Z, n = 1, 2 . . . . .
(16)
where Jm(x) is the mth order Bessel function of the first kind and O'm,,aare the zeros of J~,,(~ra) = 0. The eigenvalues are given by O~,. = O'm,, tanh o-.,.( 1 + T~r2,..) .
(17)
Next we construct a function ff satisfying ~2~=0
inV,
on S U B ,
3.-=0
ffr=f(z)
onM.
By separation of variables again, it is not difficult to obtain
¢=Aorm+
y" A,,lm(n~r) cos n'rrz,
(18)
n--I
where I
Ao=(mam-i)
i I f ( z ) dz, *¢
0 I
A,, = 2 I f ( z )
cos nTrz d z / ( n T r l , ' ( n w a ) ) ,
n= 1, 2 ....
0
and Ira(X) is the ruth order modified Bessel function of the first kind. Now we let
4,= q,+,~. Then (6) to (10) become 5LY2~0=0 i n V ,
(19)
- oJ2q,+ q,~ - T_~, q~ = o~2,~} r/= (ion) -~(qJ~ + ~.-) ~0n = 0
onMUB,
qJr~ = c~ a t F ,
_
on S ,
(20) (21) (22)
M.C. Shen et al. / Capillary-gravity waves
405
where we assume f(z) is sufficiently smooth so that termwise differentiation of the series in (18) is permissible.
3. G r e e n ' s function and solution of the p r o b l e m
We first construct a Green's function satisfying .~2G = - r - 1 6 ( r - r ' ) 6 ( z - z
-oo2G+Gz- T.~,G~=O
Assume
')
inV,
(23)
onS,
(24)
Gn=0
onMUB,
(25)
Grz = 0
at F .
(26)
G(r, z; r', z') exists. We interchange the roles of (r, z) and (r', z')
and form the integral
f I f (GS~;q'-q'S~'~G)r' dr' dz' dO=2~rqJ(r,z)= f l(GqJz,-qtGz,)r' dr' dO V
S
=t°-2T f I jr
(Gz'S~]qlz'-OeZ~]Gz')r' dr' dO+
s
(oGz,r'dr' dO
((
l.t
s
=to-2T2~raGz,(r,z,r'=a,z'=l)a+ ff(oGz,r'dr'dO
(27)
S
where we have used (19) to (26) and 02
2~= ~
1
0
+ r'Or'
02
( m / r ' ) 2q- 0 z , 2 ,
As seen from (27), we need only to know problem with a = 0. .ZF2t~=0
¢ --~1-
02
1
0
0r,2 + r'Or'
(m/r')2
Gz, on S in order to construct ~0. Hence we look for a solution t~ of the
in V,
-o)2(O+~z-T(Ozz~=~o2~
(28)
(29) onS,
(30)
~.=0
on M U B ,
(31)
~z = 0
at F .
(32)
~=
b. .]m(O'mnr) c o s h ( O ' m n Z ) ,
(33)
Assume E tl=l
where B. are constants to be determined. Apparently ~ satisfies (29), ( 31 ) and (32). We substitute (33) for ~ in (30) to obtain
~B.J,.(tr.~r)( - t o 2 cosh 0-.,. +trm~ sinh 0-,.. + To-3n sinh Ormn) -,~-t02~.
M.C. Shen et al. / Capillary-gravity waves
406
Assume
amn = ~'~2,n --0)2:=/:0.
Then by (17) and the orthogonality property of Jm(°',,,.r), we find
B. = 0)2 i JffJ.,(~r.,.r)r dr/ ( Am,,IIJm(O'm,,r)II2c°sh
(34)
o'.,,,)
0
where a
II&.( cr.,.r) II2= f j2m( O%nr)r d r = ( 6r],,,a2- mZ)j2 ( O'm.a ) ( 2~r2,,,) -' 0 By using (34) for B. in (33) and assuming that ~b on S is sufficiently smooth, we can interchange integration and summation to obtain
(0= f f G:,(r,z; r',z'=l)¢r' dr' dO, S
where
G.,( r, z; r', z' = 1) = (2ax)
1o)2
Jm( cr..nr)J,,,( O-m,,r') cosh O-m,,Z/( Am,, IIJ,,,(~r..,r) II2c°sh O-ran) •
(35)
n=l Finally we gather all the results and obtain a solution of ( 1 ) to (5) : @P= ( ~ + ~ ) ( e i m ° ± e -im°) e i°~,
~=Aor'"+
~_~ A.lm(n'rrr) cos nrrz, tt= l
I
Ao=(ma
m I)ff(z)dz, o
I
An=2 f f(z) cosn~rzdz/(nTrl',(nrra)),
n=l, 2.....
o
~t
~=oe0) 2 T a G z , ( r , z , r ' = a , z ' = l ) +
f ~(r',z'=l)G.,(r,z;r',z'=l)r'dr
',
0
G:,(r, z; r', z '= 1) is given by (35) and r/is determined by (7). For the case o f f ( z ) = K = c o n s t a n t , an approximate expression for ~b may be obtained if I,a.,~l << l, but ( 0)21Am~[ ) - I remains finite. In other words, 0) is near the natural frequency J2mk. NOW (o=(ma m l)-lKrm
and
M.C. Shen et al. /Capillary-gravitywaves
407
a
f Jm(Ormkr)~(r)r dr=KaZ(mo',~k) -IJm + l ( Ormka ) 0
dp~ ( Totto2Jm( trmka) + Ka2( mtr,~k) - IJm+ l(O'mka) ) X
Jm(Ormkr) cosh(trmkZ) / (toz Amk(Cosh Ormk) IIJm(trmkr) II)"
Some numerical computations for ~7cos 20 in the case o f f ( z ) = K = constant are presented in Fig. 2 to Fig. 6, which indicate rather interesting results. Figs. 2 and 3 show a wave pattern similar to the initial stage o f the capillarygravity waves generated by rubbing the handles of the basin. In the numerical experiments as shown in these two figures, the forced frequency of the side wall is near the lowest natural frequency O2~ = 1.5861. In Fig. 3, the amplitude K of the forced oscillations is five times larger than that in Fig. 2. However, the wave patterns do not change much. W e also did some numerical computations for the case of to = 1.57 and K = 0.1 and the wave pattern still remains the same. As to moves towards the second natural frequency 022 = 2.367644, the four points A, B, C, and D, where the maximum amplitude of the surface waves takes place, move towards the center o f free surface. The wave patterns are shown in Figs. 4 and 5. As to is in the neighborhood of 023 = 2.89227, two separatrices appear and four more maximum points emerge. The numerical results for this case are presented in Fig. 6. In some other numerical experiments, where different values of T and a are used, the wave patterns are basically the same but the solution behavior is very sensitive to the change of to. As to crosses each natural frequency, the solution becomes unbounded unless a solvability condition discussed in Section 4 is satisfied. The evolution of the amplitude of the surface waves with to near one of Omn will be studied by a nonlinear approach in a subsequent study. In all the figures except Fig. 2 where K - - 0.02, we choose m = 2, c~= 0.05, T = 0.0001, K = 0.1, and a = 1.2. W e note that the choice of the values of the parameters is based upon the estimates o f their orders of magnitude and the sensitivity of the numerical output to their changes. For example, h is in the range of 15 cm to 25 cm, and the surface tension of water ---70 d y n e s / c m . The unit for T is gh z and T is of order 10 -4. The radius of the basin is larger than its depth. Therefore we choose a = 1.2. At present we have no definite information about c~. On the other hand, K
A
,o1972
.0197/
-.01972
-.0018
/
D
B
C Fig. 2. The contour plot of the free surface, to= 1.6, K = 0.002.
408
M.C. Shen et al. /Capillary-gravity waves
A .09856
D
-
C Fig. 3. The contour plot of the free surface, to = 1.6.
..00152~ Fig. 4. The contour plot of the free surface, to = 2. should be small. H o w e v e r , for a g i v e n m the values o f a, K and T do not affect m u c h the qualitative properties o f the w a v e mot!on, and the d o m i n a n t p a r a m e t e r apparently is ~o.
4. U n i q u e n e s s result
In this section we attempt to reformulate the t i m e - r e d u c e d equations ( 1 ) to (5) in terms o f an integral equation
M.C. Shen et al. / Capillary-gravity waves
•
•
409
J .2
..04 .00
0 "
.1 .19
.!
Fig. 5. The contour plot of the free surface, to = 2.367.
Fig. 6. The contour plot o f the free surface, to = 2.88.
on S by means of Green's functions satisfying the Neumann or the Robin boundary condition. In order to incorporate the edge condition in the formulation of an integral equation to be derived later, we first construct the Green's function (Gs(~, ~), where :~(x, y, z), ~= (x', y' z' ) ~ S are Cartesian coordinates, satisfying
- V~2Gs+T-'Gs=~(.f-~), Gs, = 0
on F .
.f, ~ S ,
(36) (37)
Here S may be considered as a simply connected open domain with a simple closed curve F as its boundary in the x, y-plane. It is easy to show that the homogeneous problem of (36) and (37) possesses a trivial solution only. Let us define
M.C. Shen et al. / Capillary-gravitywaves
410
A=-V~z+T
II,
on D ( A ) = (u(x-)lu ~ C 2 ( S ) 0 C I (S), u. = 0 at F), where C " ( ~ ' ) denotes n-times continuously differentiable functions defined on a domain ~g/, and S is the closure of S. Then
(Au, U) = ~ f U(-- ~ + T - l l ) u dA >~T - ' ( u , U). s ,t
Furthermore,
(Au, v ) = ( u , Av),
u, ~,~D(A) .
Hence A is positive definite and its eigenvalues are greater than or equal to T nonnegative eigenvalues only. But it may not be positive definite. Next we construct another Green's function Gv satisfying
- V~3Gv=8(d-~),
d, ~ V ,
Gv: + ( o~2cr2/T)Gv=O Gv, = 0
~. Its inverse A - l exists and has
(38)
on S ,
(39)
on 3V,
(40)
where o- is a positive constant, V denotes a bounded open domain in E 3 with the flat surface S as the equilibrium liquid surface, 3V is the remaining rigid smooth boundary of the liquid body and F i s the contact line (Fig. 7). It is also easy to show that the homogeneous equations corresponding to (38) to (40) possess a trivial solution only. Now we are ready to deal with the following problem, which is more general than the time-reduced problem of (1) to (5), W3=q~=0
inV,
-TV~z~o.+q~z=w2~0+f(x-)
(41) onS,
(42)
q~. =g(x-)
on aV,
(43)
qL-, = h ( x ~
on F ,
(44)
where we assume that f, g and h are sufficiently smooth functions. By means of Gs, we can transform (42) to an integral form
ff
(Gs V~z~z,-~z, V~2Gs) dA
s
F
aV Fig. 7. The fluid domain.
M.C. Shen et al. / Capillary-gravity waves
S
s
= T-to2,~s~o+ F(x-),
411
F
.fES
(45)
where we have used (37), (42) and (44). Then we make use of Gv to reduce (41) to (44) to an integral equation
(xO=f f f
dV
v
(46) s
S
ov
where (38) to (40) and (45) have been used. Since q~ is completely determined by its values in S, we may just consider Y~ S and denote the restriction of q~on S by q~sand the restriction of Gv on S by Gvs. (46) may be expressed as
q~s(X--)=T-%2ffGvs(Gs+o'2I)~osdA+Hs(xD S
= T - l co2,~vs(~ s + tr21) qOs+ Hs(x-),
~
S.
(47)
Here
s
av
and we define, for 2, ~ S, ~vs:~o~L2(S)--* ~vs~P =
ff
Gvs(X, ~)~P(~) dA,
S
ff s
where L2(S) denotes the space of square-integrable functions on S. Note that ffs is an extension ofA - 1 and Gs has a logarithmic singularity in S. Gv has a singularity like I~?- ~1 - ~ at .f= ~, and its restriction Gvs on S also has the same singularity. It is not difficult to show that both operators ~'vs and ffs are compact operators [9]. Furthermore, they are symmetric and ffs + tr2l is positive definite. Hence ~ v s ( ~ s + 0"21) is also compact, but not necessarily symmetric. However, we may define a new inner product (U, V)o-=(,~S
u, V )
U, v E L 2 ( S )
,
where # s = ~'s + Ov2I. Since # s is positive definite, (u, v),, is well defined. Then ~ v s # s is symmetric with respect to the new inner product, since
( ~vs#sU, v) ~ = ( #s~Cvs#sU, v) = ( #sU, ~vs#sV ) = ( u, ~vs#sV),,. Now we need only solve the integral equation q~s=T-ltoz,~ffgs+H,
~os~L2(S) ,
(48)
412
M.C. Shen et al. / Capilla~-gravity waves
where ,~ = . ~ v s ~ s and L~(S) is L2(S) with the new inner product. Since
(~sU, U)
u~L~(S),
(,fsU, U)>~T lw2o-2(u,u),
where lilts Ill is the operator norm O f 3 s , IJull = (u, u)J/2 and Ilull,, = (u, u)~/z are equivalent. Therefore, ff is also compact on L~(S). It follows that the Fredholm alternatives for a symmetric compact operator can be invoked to study a solution of (48). If T - ~w2 is not a characteristic value of if, (48) has a unique solution q~sand consequently q~ is also uniquely determined by (46). If T ~o92happens to be a characteristic value of if, then (48) possesses a solution provided that H is orthogonal to the eigenfunctions corresponding to T ~w2. It can be shown that ¢ in (46) does possess the necessary regularity and satisfy (41) to (44), and we also note that the boundary conditions together with the edge condition in particular are sufficient to ensure a unique solution of (41) to (44) if o92 is not an eigenvalue of the homogeneous equations. In principle, once Gvs and Gs are found, q~s can be determined by the Ritz-Rayleigh method or other variational methods. Needless to say, more general boundary conditions
qJr =f(z, O) e i~
on M ,
q~rz = a ( 0 ) e i~"
at F ,
than (4), (5) can be dealt with either by the Green's function method presented in Section 3 or by the integral equation method discussed here.
Acknowledgements The authors wish to thank Professor Dajun Wang of Peking University, China for several helpful discussions. The research of the first two authors was partly supported by the National Science Foundation under Grant CMS8903083.
References [ 1 ] T.H. Havelock, "Forced surface waves on water", Philos. Mag. 8, 569 (1926). [2] D.V. Evans, "The effect of surface tension on the waves produced by a heaving circular cylinder", Proc. Cambridge Philos. Soc. 64, 833 (1968). [3 ] P.F. Rhodes-Robinson, " O n the forced surface waves due to a vertical wave maker in the presence of surface tension", Proc. Cambridge Philos. Soc. 70, 323 ( 1971 ). [4] L.M. Hocking, "Capillary-gravity waves produced by a heaving body", J. Fluid Mech. 186, 337 (1986). [5] N. Madal & S. Bandyopadhyay, " A note on the plane vertical wavemaker in the presence of surface tension", Quar. Appl. Math. 49, 627 (1991). [6] J. Miles & D. Henderson, "Parametrically forced waves", Ann. Rev. Fluid Mech. 22, 143 (1990). [7] Dajun Wang, " A research on the mechanical properties of cultural relics of ancient China" (in Chinese), Proceedings of Scientific and Technological Archeology, Chinese University of Science & Technology Publishing House, 51-56 ( 1991 ). [ 8 ] Dajun Wang, "Mysterious Chinese antiquities - The application of vibration theory to ancient Chinese antiquities and modem research on them", manuscript. [9] S.G. Mildalin, "'Mathematical Physics, An AdL,anced Course, North-Holland, Amsterdam (1977).