TECHNISCHE UNIVESITElT Laboratorium voor ScheeohydromechanIca Archef Mekelweg 2, 2628 CD Deift Th.:O15-78a73-FaO15-781338
Observation of a standing kinK cross wave
arametricaily excited
D T. YAO-TSU Wu
1. Introduction
The phenomenon of parametrically forced water waves was first investigated by Faraday (18.31), who ôbserved subharmonic excitation of a liquid contained in a basin that is forced to oscillaté vertically with frequency twice the natural frequency of the excited waves in the basin. Faraday waves have been analysed by Rayleigh (1833a, b), Benjamin & Ursell (1954), Ockendon & Ockendon (1973), Miles (1984a, b), and others (for a review, see Miles & Henderson (1990)). Using small motion approximations, Rayleigh explained why wave motions so excited are subharmonic. Benjamin & Ursell studied the stability of parametrically forced water waves using linear theory and found that the resulting waves can be described by Mathieu 's equation. This analysis was extended by Ockendon & Ockendon to obtain some. Of the main features of nonlinear self-interaction. Miles applied the averaged .lagrangian method to calculate the nonlinear interaction coefficients Agreement between theory and experiment has been examined by Henderson & Miles (1990) with the effects of viscous dissipation and surface tension taken into account for circular and rectangular cylinders, and discussed iii the review, f this general subject by Miles &
Henderson...
One remarkable manifestation of the parametrically excited waves is the family called standing cross waves (or standing edge waves) generated on a layer of liquid contained in a rectangular basin of large aspect-ratio which is forced by a vertical harmonic oscillation. Within this family, Wu et al. (1984) first showed experimentally the existence of a localized standing soliton'-like wave on a layer of modest depth containea in a rectangular tank being subjected to vertical oscillation. At a forcing frequency shifted slightly below twice the natural frequency w0 of the (0, 1)-mode of the water waves within the tank, the fOrced oscillation generates standing cross waves sloshing to and fro across the width of the tank with their profile modulated by a hyperbolic-secant envelope along the tank's length. Subsequent theoretical work by Larraza & Putterman (1984) and Miles (1984b) showed that the standing soliton does not exist for values of kh 1.022 where h is the uniform quiescent water depth and k = it/b, b being the width of the tank.
Proc. R. Soc. Lond. A (1991) 434, 435-440
Printed in Great Britain 435
Califo . .: . o ogy, Pasadena, California 91125, U.S.A.
A layer of water in a cylindrical tank is known to be capable of sustaiñing standing. solitary waves within a certain parametric domain when the tank is excited uñdr vertical oscillation A new mode of forced waves is discovered to exist in a different parametric domain for rectangular tanks with the wave sloshing across the short sidé of the tank and with its profile modulated by one or more hyperbolic-tangent, or kink-wave-like envelopes. A theoretical explanation for the kink wave properties is provided. Experiments were performed to confirm their existence.
436 G. S. Guthart ind T. Y. -T. Wu
However, we observe that for kh 1.022 and for a forcing frequency slightly above twice the natural frequency ù0 for the (01 )-mode, a new family of surface wavesis
found to exist that sloshes across the width of the tank and is modulated
by a
hyperbolic-tangent, or kink-wave envelope that is stationary in nature along the length of the tank.In the existing literature on nonlinear 'nalysis of Faraday waves, we have noted
that no simultaneous cons 4rtion Lhas been given to both the longitudinal
modulation transverse to the cross waves aiìd an explicit calculation involving an
interaction parameter that measures the amplitude of excitation against
the nonlinear inertia effects, both of which are third-order terms in the equation of motion. In developing the theory, it is of importance to include the effects of the external forcing function rather than just the natural modes of a freely oscillatingsystem, since these effects are needed to provide a new interaction parameter and thus extend the parametric domain for determining the existence as well asthe stability of the modes of interest.
2. Theory
We present here a theory for a dispersive, weakly, nonlinear and weaklyforced system sustaining Faraday waves. The analysis is considerablysimplified under the assumption that only one primary mode and its next higher harmonic are resonantly excited and that none of the natural frequencies of the secondary modes are nearly equal to the forcing frequency and no internal resonance occurs. A reference frame is taken fixed to the rectangular tank with the x-axis directed alongthe long side of the tank of length i, the y-axis across the tank of width b, b 4 i. The wave number k of the standing cross wave generated by the vertical oscillation is it/b and ki = 0(1/e) i is a measure of the aspect-ratio of the tank. The z-axis points vertically upward such that z = O 'at the quiescent water surface and z = - h at the tank bottom. With the fluid assumed inviscid and incompressible and the motion assumed irrotational and free of capillary effects,' the velocity potential çS(x, y, z, t) and free surface elevation (x,y,t) satisfy the field equation V2ç5 = O for h z
the wall condition for the normal velocity n Vçb = O on the tank walls and bottom, and the free surface conditions
w = çS = (1)
cb + (Vçb)2 + (t) = 0, ' (2)
on z = C(x, y, t), the subscripts denoting partial differentiation. Here the forced acceleration of the tank is absorbed in the gravity term so that (t) = g( 1 +fcos (2ut)), f being the amplitude of the vertical, oscillatory acceleration imposed on the tank and scaled with respect to the constant gravitational acceleration g.We assume that ç5 = O(e) and = O(e) in the motion weakly forced withf=e2f,f being of Q(i), where e a/h is the nonlinearity parameter for a typical wave amplitude a, with e 4 1. Expanding the conditions in
(1) and (2) about z = O in terms of e and
eliminating iii favour of ç5, we obtain (following Whitham (1976), now with additional t-dependence in (t))ç + gçS - (çb/g) (ç.S +gçb ± [( Vçi5)2 + VçS .V[(Vç5)2]
+ g'(ç5 + gç5) [g' (ç5),
(Vç5)2]+ g2(çS)2(ç?St + gç5)A parametrically excited standing kink cross wave 437
on z = O. In this equation, use has been made of the expansion of C up to O(e2),
gC= (z = O). (4)
The effects of the weak forcing excitation appear in the third-order terms of (3) and (4), however, they are needed oniy in (3).
For the phenomenon under investigation, we look for solutions of the above equation representing motions that slosh acrpss the tank like cos (ky) with frequency w very nearly equal to the priay natural frequency 00 and that are modulated along the x-direction by an envelope depending on a slow time, r; and a long space, . The appropriate scales of these variables are t = ot for the fast time, r c2wt for the slow time and = ex for the long space. Thus, with the expansions
e(y,z,t;,r),
(5)n-1
(0=w0+C(01+e2(02+..-., (6)
the original field equation V2çb = O becomes
Vç51 = O, Vç2 = O, Vç53=
ç,
(7)where V = E, + and , =
=
Proceeding to solve these equations with the wall conditiòns and the free surface condition (3) taken by orders of e, we obtain for çb1 the solutionçb1 = R1(t; ,r) cos (ky) cosh (k(z ± h))/cosh (kh),
R1(t;,r) = k(,r)exp(it)+c.c.
(c.c. denoting the complex conjugate of its preceding term) together with w =
gk tanh (kh) gkT for the dispersion relationship. Suppression of the secular terms at the second-order analysis requires that cJ1 = O and gives for c2 the particular solution
= R2(t; ,r)[C1 + C2 cos (2ky) cosh (2k(z+h))/cosh (2M)],
with R2(t; , r) =iLr(g, r)2 exp (2i1) + c.c.,
C1 =(k2/8w0)(1+3T2) (T tanh(kh)), C2 = (3k2/8w07Y)(lT4).
In the third-order analysis,, suppression of the secular terms yields for 1i- the equation
2iw 1r+c2fr+2w0w2 1r+fw fr* +AIfrI2fr = 0, (8)
where
A = k4(-9T2+ 16-5T2+6T4),
c2 = (g/2k)(T+kh(1T2)),
and * denotes the complex conjugate. In terms of the real and imaginary components
ofr=p+iq, (8) gives
= 0,
2wp7+c2q+(2w2 w0fw)q+Aq(p2 +q2) = O.
In the absence of forcing excitation, the reduced equation of (8) agrees with that given by Larraza & Putterman (1984). Furthermore, equation (8) is consistent with a nonlinear Mathieu analysis when there is no dependence. Notice that when the
438 0. S. Guthart and T. Y.-T., Wu
forcing is of order e, instead of the present order e2,, the foregoing analysis gives an equation similar to equation (8) but without the cubic term and the dependence and
With the slow time given by i = aut. Stich an equation is consistent with the linear Mathièu analysis of Benjamin & TJrsell (1954).
3. The soliton and standing cross waves
Without forcing excitation, (8) reduces to a cubic Schrodinger equation which possesses solitary wave solutibns (Whitham Ï974.; Miles 1984b). In fact, with f = O and
fr = exp [i(rsr)] !P(X), X
(8) becomes
d2/dX2+c2W+Ac2
= 0, (9)provided r - w U/c2 and = 2w0 w2 - c2r2 + .sw, where is introduced in place of s for algebraic convenience Solutions of (9) are noted to depend upon the signs of and A. With h = I as the length scale, A is plotted versus k in figure 1 which shows that A crosses zero from below at k i.O22. For <0 and k> 1.022 (A > 0), (9) has the solution
= (-2cz/A)isech[(cL/c)iX] (10)
1fr =0 and s = O (U = O and = 2w0 w2), this gives for w2 <0 (ci <0) the solution for the standing soliton which is stationary with respect to the slow time as reported
by Larraza & Putterman (1984) and Miles (1984 b). Since w2 <0, the wave frequency w is slightly lower than the pripary mode frequency, w0.
In addition; however,.we observe that for > O and A <0 (k < 1.022), there is a new solution of (9) of the form
(/_A)itanh[(z/2c2)kKJ. (11)
For r = O and s = 0, this solution exists for > O (w > O rendering the wave frequency w slightly greater than the primary mode frequency, w0) and is a wave envelope neither propagating nor varying with the slow time.
In the presence of forcing excitation, f 0, and with A <0, we seek solutions of (8), that are similar in form to the f = O solution (following Miles 1984 b), i.e.
fr = ae18tanh (Kg),
where O is a phase angle and a is a real positive constant. Substituting the above expression into (8) leads to two possible solutions.
O = 0, K2 = (w2w0+Uw)/c2, O = 7t, K2 =
(w2w0fw)/c2,
and a2 = 2K2c2/( A). Returning top and q, a solution corresponding to O = Os given by
q0, p=atanh(sc),
where K = [(w2 w0
+fw)/c2]i,
a = [2K2c2/ _A]i.The surface elevation corresponding to this modulating envelope is
C(x,.y,t) = (2ew/g)atanh(Kex)sin (wt)cos(ky). (12)
A parametrically excited stznding kink cross wave 439 0.2
AO
0.2
0.12834 -0.06417 -0.25668 -16 254Figure 2. Computer generated plot of the surface elevation for the sloshing kink wave with
b = 2.54 cm, h = 0.5 cm, w2 i and e = 0.5, based on equation (12).
f = 0, 0 = O and e = 0.5. These values are typical for observing the standing kink waves experimentally. It is of great significance to note that in determining which of the two families of standing waves will arise, the hyperbolic-tangent modulated wave or the hyperbolic-secant modulated wave, the criterion is no longer decided by the sense of o2 and A alone, but by w2, A and f as the three underlying parameters. It is interesting to note that the change in sign of A with respect to k may be regarded, qualitatively, as analogous to a change from a hard to a soft spring in nonlinear mechanics, as observed by Ockendon & Ockendon (1973). However, in the present case, the mechanism distinguishing the two stationary solutions of equation (8) is complicated, involving the interplay of three parameters.
We have done some preliminary experimental investigation on the two families of standing solitary and kink waves, with the results confirming qualitatively all the salient features of these waves and their existence within the respective parametric
domains as established by the foregoing
analysis. However, topresent a
comprehensive comparison between theory and experiment, considerable further efforts will be required which are underway.
o 0.4 0.8
k
Figure 1. The nonlinear coupling coefficient, A, against the wavenumber k, for cross waves in a
440 G. S. Gutliart and T. Y.-T. Wu
We are grateful to the referee and to Dr George T. Yates for valuable discussions on the general
subject. This work was done with ONR Grant N00014-89-J1971 and NSF Grant 4DMS-8190- 144M
jointly sponsored by the NSF Applied Mathematics, Computational Mathematics and Fluid
Dynamics/Hydraulics Programs. The numerical calculations were performed on the CRAY Y-MP
at the San Diègo Supercomputer Center (operated by the National Science Foundation).
References
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical
periodic motion. Proc. R. SOc. Lond. A 225, 505-515.
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by
groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299!34O. Henderson, D. & Miles, J. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Meh. 213,
95-109.
Larraza, A. & Putterman, S. 1984 Theory of nonpropagating surface-wave solitons. J. Fluid Mech. 148, 443-449.
Miles, J. 1984a Nonlinear Faraday resonance. J. Fluid Mech. .146, 285-302. Miles, J. 1984b Parametrically excited solitary waves. J. Fluid Mech. 148, 451-460.
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. A. Rev. Fluid Mech. 22,
143-165.
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397-413. Rayleigh, Lord 1883a On maintained vibrations. Phil. Mag. Ï5, 229-235. (Reprinted (1900) in
Scientific papers, vol. 2, pp. 188-193. Cambridge University Press.)
Rayleigh, Lord 1883 b On the crispations of fluid resting upon a vibrating support. Phil... Mag. 16, 50-58. (Reprinted (1900) in Scientific papers, vol. 2, pp. 212-219. Cambridge University Press.) Whitham,. G. B. 1974 Linear and nonlinear waves, pp. 602-603. Wiley-Interscience.
Whitham, G. B. 1976 Nonlinear effects in edge waves. J. Fluid Mech. 74, 353-368.
Wu, J., Keolian, R. & Rudnick, I. 1984 Observation of a nonpropagating hydrodynamic soliton.
Phys. Rev. Lett. 52, 1421-1424.
