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BATEMAN VARIATIONAL PRINCIPLE FOR A CLASS OF PROBLEMS OF DYNAMICS AND STABILITY OF SURFACE WAVES I. A. Lukovskii and A. N. Timokha

UDC 517.958;532.595

A generalization is made of the Bateman-Luke principle for the problem concerning acoustic interaction with the free surface of a bounded volume of fluid. Extremal criteria are presented for the stability of capillary-sound equilibrium forms.

In [i] a nonlinear evolutionary boundary value problem was considered involving an unknown surface separating two regions arising in the theory of bounded volumes of immiscible continuous media. A particular solution was constructed for the differential problem indicated in the framework of the general theory of perturbations, the solution having the sense of a dynamic position of equilibrium; in addition, the concept of a capillary-sound equilibrium form (CSEF), defined as the position of the surface of separation averaged over time, was introduced. The problem was described in variations relative to the CSEF. After application to the latter of a procedure for separation of motions and averaging over the rapidly changing variables, a problem was formulated concerned with characteristic oscillations of the system relative to the CSEF and theorems were established concerning properties of the spectrum of this problem as well as the stability of the aforementioned equilibrium forms. In [2], for the case in which the region is cylindrical, it was shown that the problem concerning the CSEF can be obtained from a stationary condition of some functional; in addition, two variational principles were formulated. Moreover, in [3] (also for a cylindrical region) it was shown that the initial differential evolutionary boundary value problem with an unknown surface separating two regions admits a variational formulation involving a Bateman-Luke type functional [4, 5] employed earlier for derivation of equations and solution of problems of the dynamics of bodies with cavities partially-filled with a fluid [6]. In the present paper the indicated variational principles are generalized to the case of an arbitrary convex piecewise-smooth domain and their interdependence is established. It will be shown that the variational principles formulated in [2] can be obtained, similar to what was done in [7, 8], by carrying-out an average procedure in the Bateman functional, and the CSEF stability problem can be solved through an analysis of extremal properties of the stationary functional introduced in [2]. It will also be shown that the condition for a strict maximum of the latter is equivalent to the spectral criterion for stability of the CSEF obtained in [I]. i. Differential Formulation of the Problem. We consider a convex piecewise-smooth bounded domain Q c R 3 ( W ~ , y,z)~O) and a simply connected smooth surface E dividing the domain into two subdomains Q1(t) and Q2(t); S i = 8Q i \ z; we select a surface S o c S I. The nonlinear problem in question assumes the form [i] Pi

\ Po~ ]

'

Pit -K div (PiV~i) = 0 in Qt,

PIV(~it+ 0,5(V~i) ~ + b~lgsX) = - - V p i i n Qi, @~$]@n= 0 in Si;

i = ], 2,

O~j/On= (-- l)l+l~/[V~ [ in E, j = I, 2,

(i)

P2 + ~188(K1 + K=) : ~lSPl in Z, (vW, vDII vW I = cos ~ I v~l in 0~,

p~O%/On = e--~- V (x, y, z) sin t in So,

Institute of Mathematics, Academy of Sciences of the Ukraine, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1181-1186, September, 1991. Original article submitted April 2, 1991.

1106

0041-5995/91/4309-1106512.50

9 1992 Plenum Publishing Corporation

where ~i, Pi, P i : ( O , ' - l - ~ 1 7 6 -+~; l O .

basis)

This completes the proof of the theorem. LITERATURE CITED

i.

2.

I. A. Lukovskii and A. N. Timokha, "On a class of boundary value problems in the theory of surface waves," Ukr. Mat. Zh., 43, No. 3, 365-370 (1991). I. A. Lukovskii and A. N. Timokha, "Nonlinear dynamics of the surface separating a liquid and a gas when a high-frequency acoustic field is present in the gas. Stationary regimes of motion," Preprint, Inst. of Math., Acad. of Sciences of the Ukr. SSR, Kiev

(1988). 3.

4.

5. 6. 7. 8.

I. A. Lukovskii and A. N. Timokha, "Variational formulation of a nonlinear problem with an unknown surface separating two regions," in: Stability of Motion of Solid Bodies and Deformable Systems [in Russian], Inst. of Math., Acad. of Sciences of the Ukr. SSR, Kiev (1989), pp. 4-9. H. Bateman, Partial Differential Equations of Mathematical Physics, Dover, New York (1944). J. C. Luke, "A variational principle for a fluid with a free surface," J. Fluid Mech., 27, 395-397 (1967). I. A. Lukovskii, Introduction to Nonlinear Dynamics of a Solid Body with Cavities Containing a Fluid [in Russian], Naukova Dumka, Kiev (1990). I. I. Blekhman and B. P. Lavrov, "On an integral criterion for stability of motion," Prikl. Mat. Mekh., 24, No. 5, 938-941 (1960). I. I. Blekhman, "Basis for an integral criterion for stability of motion in problems concerning auto-synchronization of vibrators," Prikl. Mat. Mekh., 24, No. 6, 1100-1103

(1960). 9.

A. D. Myshkis (ed.), Hydrodynamics of Weightlessness [in Russian], Nauka, Moscow (1978).

k DISTRIBUTION OF THE NUMBER OF s

IN A RANDOM BINARY SEQUENCE

SUBJECT TO SOME CONSTRAINTS UDC 519.21

V. I, Masol

There are investigated the joint distribution of random variables Pkn(s .... , Pkn(Zs), and distributions of some functionals of Dkn(Z), for n ~ . Here ~kn(s 1~l~n--I is the number of s in a binary sequence (b.s.), selected randomly and equiprobably from the totality of all n-dimensional b.s. that have a prescribed number of ones and k l-steps. By an s of a b.s. we understand a configuration of the form i...0, where the ellipsis stands for an (s - l)-dimensional b.s.

Let ~(k, m 0, m l) be the totality of binary sequences, containing m 0 zeros, m I ones, m 0 + m I = n, and k 1-steps, k/> l, m o ~ k , m1~>k. LEMMA I. A subset S(k, m0, m1~ P) of the set ~(k, m 0, ml), consisting of binary sequences, having ik i-steps, i = i, p, has the cardinality

t ~ (~' /nO' ml, ~) I

=

c~.-(._,).c~,_co_,)..

(1)

Kiev Institute. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1186-1193, September, 1991. Original article submitted May ii, 1989.

Iii0

0041-5995/91/4309-1110512.50

9 1992 Plenum Publishing Corporation