Phase Structure of fluid fluctuations with a floating elastic ice plate under nonlinear interaction of progressive surface waves Текст научной статьи по специальности «Физика»

Phase Structure of Fluid Fluctuations with a Floating Elastic Ice Plate under Nonlinear Interaction of Progressive Surface Waves

A. E. Bukatov, A. A. Bukatov*

Marine Hydrophysical Institute, Russian Academy of Sciences, Sevastopol, Russian Federation

*e-mail: newisland@list.ru

Asymptotic expansions up to the third-order terms are constructed based on the multiple scales method for the fluid velocity potential of the finite-depth homogeneous fluid and elevation of the plate-fluid surface (ice-water surface). The obtained expansions constitute a foundation for analyzing dispersive properties of the fluctuations formed under interaction of progressive harmonics of the finite-amplitude surface waves. Changes of the fluctuation frequency taking place due to the contribution of the values of the first and second approximations conditioned by the taken into account non-linearity are considered. The effect of non-linearity of the ice plate vertical displacement acceleration on the amplitude, frequency and phase velocity of the wave disturbances is studied. It is shown that at the wave numbers exceeding the maximum resonance value, the oscillation frequency increases in case non-linearity of the plate vertical displacement acceleration is taken into account. It also grows with the plate thickness, and change of a sign (plus into minus) of the second interacting harmonic amplitude reduces the frequency value if the wave number is fixed. The phase velocity increase with allowance for the acceleration nonlinearity is more significant than without considering it. Under a negative amplitude of the second interacting harmonic, if the acceleration nonlinearity is taken into account, the phase velocity is less than when it is not.

Keywords: ice plate fluctuations, waves of finite amplitude, nonlinear interaction of waves, flexural deformation of plate, phase characteristics.

Acknowledgements. The present study is carried out within the framework of the State Order No. 0827-2018-0003.

For citation: Bukatov, A.E. and Bukatov, A.A., 2018. Phase Structure of Fluid Fluctuations with a Floating Elastic Ice Plate under Nonlinear Interaction of Progressive Surface Waves. Physical Oceanography, [e-journal] 25(1), pp. 3-17. doi:10.22449/1573-160X-2018-1-3-17

DOI: 10.22449/1573-160X-2018-1-3-17

© 2018, A. E. Bukatov, A. A. Bukatov © Physical Oceanography

Introduction. In a linear formulation, the study of the fluid fluctuations with a floating elastic plate was carried out in [1-8]. Analysis of the influence of the mass and elastic forces of the plate on the characteristics of forced unsteady flexural-gravitational waves of small amplitude is performed in [9]. In [10] the nonlinear fluctuations of an absolutely flexible plate (broken ice) floating on the surface of homogeneous ideal fluid of finite depth are studied. Vibrations of finite amplitude in homogeneous fluid with a floating elastic plate without taking into account the nonlinearity of the acceleration of its vertical displacements are considered in [11-13]. An estimate of the influence of the nonlinearity of the vertical displacement acceleration of a plate due to its deformation on the propagation of periodic surface waves is given in [14]. Formation of nonlinear oscillations of an absolutely flexible floating plate with a nonlinear interaction of traveling periodic wave harmonics is analyzed in [15].

The fluctuations formed in the system of an ice plate - liquid under the interaction of harmonics of traveling periodic surface waves of finite amplitude are studied in the present paper.

1. Problem statement. Suppose a thin elastic plate of constant thickness h is floating on the surface of the homogeneous ideal incompressible fluid of constant depth H. In the horizontal directions, the plate and liquid are unlimited. Let us consider the nonlinear fluctuations of a plate in the interaction of the first and second harmonics of progressive waves of finite amplitude, assuming that the fluid movement is potential, and the plate fluctuations are uninterrupted. In dimensionless

variables x = kxl, z = kzl, t = yfkgt1, Z = kZ*, and l=(k2/yfkg, where k is

the wave number; g is the gravity acceleration; t is the time; 9(x, z, t) is the fluid movement velocity potential; Z(x, t) is the plate or the plate-fluid surface elevation, the problem involves the Laplace equation solution

9xx +9zz = 0, -<»< x <<», -H < z < Z

(1.1)

for the velocity potential with boundary conditions on the plate-liquid surface

(z = Z)

ax4

Dk4 + Kk—

d_ dz

1 f dq]

dt

21 dx

= P =

(1.2)

dq 1

P = z — dt 2

dl] +fdq dx ) \ dz

and at the bottom (z = -H) of the basin

ck dz

In the initial time (t = 0)

8* = o.

dZ

Z = f(x), -jr = 0. dt

(1.3)

(1.4)

Here

D

Eh3

D =—, D = -1 Pg 12(1 - v2)

k = h Pi; P

E, h, p1 and v are the normal elasticity module, thickness, density and Poisson's ratio of the plate; p is the fluid density. The velocity potential and deflection of the plate at z = Z are bounded by the following kinematic condition

dt dx dx dz

(1.5)

In the dynamic condition (1.2), the expression with the factor k represents the inertia of the vertical displacements of the plate. The first term in the brackets of this expression characterizes the nonlinearity of the vertical acceleration of the plate.

2

2. Equations for nonlinear approximations. The problem solution (1.1)-(1.5) is obtained by the method of multiple scales [16]. Two new variables T = st, T2 = s2t are introduced. They are slowly varying in comparison with t = T0, where s is small but finite, and the validity of the expansions is assumed

Z = sZ0, 9 = s^Q, f = s/c, Z0 = + sZ2 + s2Z3 + O(s3), (2.1)

9c =91 + s^2 + s 2% + O(s3), /0 = f + sf2 + s2 /3 + O(s3).

Substituting 9 from the expressions (2.1) into (1.1) and (1.3), within the accuracy up to the the third-order values, the following expression is obtained

A 2* 3 A n d91 2 d92 3 d93 d2 d2

sAm, + s2Am9 + s3Am3 = Q, s—— + s—— + s3 —— = Q, A =—- +—- . 1 2 3 dz dz dz dx2 dz

Below the dynamic (1.2), kinematic (1.5), and initial (1.4) conditions are considered. The surface velocity potential of a plate - liquid z = sZQ is represented in the following form

9( x, sZ Q, t) = m( x,Q, t) + sZ q9 z (x,Q, t) +1 s 2Z 2m zz (x,Q, t) +.... (2.2)

Z = sZQ, f = sfQ, m(x, sZQ, t) and 9z (x, sZQ, t) are substituted in the corresponding conditions (1.2) and (1.5), bearing in mind that, according to the complex function differentiation rule, the partial derivative with respect to time is defined by the following expression

d d d 2 d

— =-+ s-+ s2-,

dt dTQ dT dT2

taking into account the dependence of Zc on x and t in the expression (2.2). Then, collecting the coefficients at equal powers of s and equating them to zero, the following equations are found

d 2 d 2

—9n + —% = Q, -<»< x <<», - H < z < Q, (2.3)

3r2 dz

Dlk4-Kk-d9n + Zn = F;, z = Q, (2.4)

1 dx4 dzdTQ dTQ n "

^ + = Zn, z = Q, (2.5)

dTQ dzn

d9n = Q, z = -H , (2.6)

dz

Zn = fn (x), ^ = Gn, t = Q (2.7)

dTQ

for the determination of nonlinear approximations. Here

F* = Fn + Fn0, F = F1q = L = G = Q, n = 1,2,3; PHYSICAL OCEANOGRAPHY VOL. 25, ISS. 1 (2018) 5

dT0dz

dTL

ôT0 dz

dx

+

dz

+ KkN,

N =

dT0 dz2

+Zi

dT0dz2

+

d 29l dTidz:

2 âx dx 1 dz2 5Tl

F3 =çinl +Z_+ ^

dT0dz

dT dT dx dx dz dz

_ + N_ + KkN3,

N = d_9i , -д_^Lд^LfdZi ^ ^

i ^ ....... dT dz

dTidz dT0dz

dzdx dx dz2 f

+1

_ dTndz2

N =-

ÔT0 dz

- + -

i _

N3 = ZN +i

dT0dz3

dz

3

+ -

dT„ 5T dx dx

ô39i , d\ f dZ_ , dZ

+ Z 2

dT0dz2

+

dz2

-+ ^

vÔT0 dT ,

+ ^ + N5 dzdTn

N = d +_дZIд_^L 1 d ^ 4 dT0dz2 ôT0 dz3 dz2ôTi '

N =_dZi1d 5 dT dz2 dzdT '

L3 = ZiN-Z

ddZ2 dZi dZ2 f dZ

2 dz2

■ +

- +

c)x dx dx dx

+ -

ÔT dT_ dz l c)x

N =

dZid_9i d-1 z =-Kk d^i

c)x dzdx dz

_ dz

dxdz dx

F30 = -Kk

ZiN7 +-

- + -

drdz l c)x dx dz

d_9i f dZi d^ ^ ô^i ( ddZi d_9i

+

dx

- + -

dxdz dx dz2

N7 =

drdz

+

d39i dx dxdz2

G_ =-

dZi

dT

G3 =-

dZi dZ _

dT dT

Note that F_0and F30 terms appearing on the right-hand sides of the dynamic conditions (2.4) for the second (n = 2) and third (n = 3) approximations are due to the nonlinearity of the vertical displacements of the plate.

2

2

2

3. Expressions for the plate deflection and the fluid movement velocity potential. The equations (2.3)-(2.7) are obtained for the general case of unsteady fluctuations of finite amplitude. Solution of these equations in the case of interaction of traveling periodic wave harmonics Z11 = cos 0 and Z12 = «j cos 29, 9 = x + tT0 +P(Tj,T2) is to be obtained. The first approximation (n = 1) of the surface elevation of the plate-liquid is defined in the form as

Z1 = Z11 +Cl2.

(3.1)

where 01 is the constant of the order of unity, and p = 0 under t = 0. Satisfying the condition at the bottom and taking into account the relationship of the wave characteristics through the boundary conditions (2.4), (2.5), the following can be written

91 = t

ch(z + H) . Q , ch2(z + H) . "

—--^sm 0 + a---- sin20

shH

sh2H

(3.2)

t2 =(1 + D1k4)(1 + KkthH)-1thH .

Amplitude ai and phase shift P (Ti, T2) are determined from subsequent approximations.

Substituting Zi and 91 in the right sides of the dynamic (2.4) and kinematic (2.5) equations for the second approximation and solving the problem (2.3)-(2.7) for n = 2, taking into account the requirement of absence of the first and second harmonics in the particular solution, Z2 and 92 are obtained. In turn, Zi and 91 and Z2 and 92 determine the right sides of the dynamic and kinematic conditions when n = 3. Eliminating the terms generating secularity from them, Z3 and 93 are obtained.

As a result, the basin surface elevation Z and fluid movement velocity potential 9 in dimensionless variables up to the third-order quantities are determined from following expressions

3 3 .»----^ +

Z = s cos 0 +Vsnan cos 20+ n V anJ cos j0+s3 V a3n cos n0, (3.3)

n=1

n=2 j=3 3

n=5

9 = s- -ch(z + H )sin 0 + V snbn 2ch2(z + H )sin 20

shH ^

+

n=1

+

3 4

V sn V bj chj(z + H )sin j0 +s3 V b3nchn(z + H )sin n0 + V snbnot

n=2 j=3

(3.4)

0 = x + ct, c = t + sc1 +s c2, s = ak,

where a is the amplitude of the initial harmonic. Here

f

b12 = a1

sh2H

, a1 = ±

^ 2r1

\V2

f

r =

f

2cthH + th2H

v 4r2 (2T2cth2H + 4t2 Kk + ^ 2 )(1 + 2Kk th2H )

5xX

/

v

- + cth2HcthH - Kk I cth2H--cthH

2 I 2

A

+ - cthH + cth2H

(3.5)

cthH11cthH + 3Kk J-5 I |(t2(cthH + Kk) + ^1 ),

x

2

'2~

f

f

2cthH + th2H

cthH| -cthH + 3Kk I--

a, = -

//

4a1 (2x2cth2 H + 4t2 Kk + ^ 2)(1 + 2Kk th2H) /3 = -3aji(2cth2H + cthH), l4 = -4a2cth2H :

l7 = OjT2

11

1

--cth2 HcthH + Kk I 5cth2H--cthH

„ 2 I 2

l8 = a2x2 (5 - cth22 H + 4Kk cth2H), l3^3 + 3l7 t

b23 =

b24 =

3sh3H(^3 - 9KkT2 - 3T2cth3H) 4 + 418t

3sh4H(^4 - 16KkT2 - 4T2cth4H)

a23 = m*3

1 (/7 + 3t&23 (ch3H - kk3sh3H)),

a24 = m*4

(/8 + 4tb24 (ch4H - kk4sh4H)),

5 3 2 3

j3 = -—t - — a1 t - 6b24ch4H - —a24TcthH + 3a23a1,

j4 = -—a1T - 6b23ch3H - 2a23TcthH + 4a24a1,

69

5

j5 =--a1 t - 10b24ch4H--a24TcthH - 5a1I — b23ch3H - a23Tcth2H I,

8

3

m.

2 \ 2 j6 = -5aj3T - 6a1 (2b24ch4H - a24Tcth2H),

l+-

= Tf - a1a1 + 2b24ch4H (2th4H - cthH )| +1 t2 | ^cthH (1 - 23a2)+ 7a2cth2H -

2 I 4

—3a

24

+ 3b23a1ch3H + Kk (T(2b24sh4H (11cth4H - 4cthH)+ 3a1a1 +

+ I 2cth2H + 1cthH | + t2

21 7 1 13 a21--2cth22H — cthHcth2H I----a24cthH -

11 8 2 I 8 2 24

- 1cth2H ] + 9b23a1sh3H^

m,

= t| 4c1a12 + 3b23ch3H(5th3H - cthH)| + 2t2 I a1cth2H -1 a1cthH

a1cthH + a23 I +

+ + Kk[{3b23 Sh3H№ - 3chH) + ^^ | +

24

I

+ T

37 , „ 3 21

m,

a, I--4cth2HcthH—cth2H I + 2a23cthH

v \ 4 4 ) 23 ,

= t| 2b24ch4H(6th4H -cthH) + 3b23a1ch3H| -th3H -cth2H | | +

A A

+ 16b24c1sh4 H

+ t2 I - a12cth2H - 7 a12cthH + 5a23a1 + — a24 | + Kk It (2b24sh4H (19cth4H - 4cthH )

7

5

11

3

11

+ 3b23a1sh3HI — cth3H-3cth2H | | + t2I a/1 — 6cth2H--cth2HcthH | +

+ 10a23a1cth2H + — a24cthH

i6 = 4Tb24a1c h4H (4th4H - cth2H )+ T2a1 (afcth2H + 6a24)+ - 2Kka1 (4Tb24sh4H(5cth4H - 2cth2H)+ t2 (6a24cth2H - a2(1 + 4cth22H

f

ft =M1

3

2

3 15

—b23a1ch3H -tI---a1 + a23a1cth2H

8 4

+ t a1 - ^ c1 +

+ 3b„ch3HI -th3H + cth2H

2

f

2 1 I 5 2 9afcth2H + a1a23 + —cthH I--af

) v 4 v 2 ))

+ r

A

+

+ Kk

I f T2a1

vv

3b23sh3H| -cth3H + 3cth2H I - 2c1cth2H + -^cthH

1

A

+

+ t3 12a23a1cth2H +1 T2cth2 H + 3 + a? 18cth22H + cthHcth2H + 39 q2 = (3b23ch3H + 4b24a1ch4H + T(a23cthH + 2a24a1cth2H - 3af ))

+

3

2t2 I - b23ch3H (cthH - th3H )+ 4b24a1cth2Hch4H - c1

2

+

- 2t3 (a23 + a1 (3a1cthH + 2a24 + cth2H (5a2

- 2 +

KkI 2t2| -b23sh3H(3cthH - 5cth3H)+ 8b24a1sh4H(2cth2H - cth4H) -cxthH

+ 2t3

2a? (3 + 4cth2H)+ a1 ( 4cth2H(a24 + cthH)+icth2H - 3 I + a23cthH

1

+

Чу + 42

Ц 2

, h„ =

ЛЦ n + nmn T

n shnH(цn - n2x2кк - nx2cthnH)

n = 3...6,

43n

= (nxb3n(chnH + 3KkshnH)+mn)ц-1, n = 3...6 , цn = 1 + n44, n = 1...6,

b20 = T

b30 = а1Т

- (l + cth2 2H )+1 (l + cth2 H )+ кк | yCthH + 4a2cth2H

2cth2H + ^cthH + кк| - + 2cth2HcthH -^cth2H 2 l 4 4

Wherein b__ = ^ = a_ = 03 = /i = l_ = /5 = l6 = j = j_ = «i = = 0.

Formulas (3.3) - (3.4) for Z and 9 determine the wave disturbance being formed, and in case the nonlinearity of the vertical displacement acceleration of ice in the dynamic condition (2.4) is not taken into account. However, in this case it should be taken into account that

л = x

- + cth2HcthH -кк| 2cth2H +-cthH

2 l 2

1

+ цу| IcthH + cth2H I,

l7 = ayT2

— - cth2HcthH + ккI 6cth2H - - cthH

2 l 2

mQ

= t| 2b24ch4H(2th4H - cthH) + -а1ст1 | +1 т21 a21 7cth2H - ^cthH | + 1cthH -

- 3a24) + 3b23CT1ch3H + ккт| 3а1ст11 ^cthH + 2cth2H |+ — т| 15aj2 +1 - a24cthH

2 l 4 4

+ 24b24ch4H + 9b23CT1sh3H ),

= xf -b23ch3H(5th3H - cthH) + 4c1a12 j + 2x21 a11 cth2H - ^cthH

m

" ',2 23 4 ' 1 1 J l \ 4

+ 4b24c1ch4H + кkx(2(9b23ch3H + 4a12c1cth2H )+ x(5a1 + 2a23cthH ) +

+

+a

+

16b24c1sh4H ),

m.

= t| 2b24ch4H(6th4H -cthH)+ 3b23a1ch3H| 7th3H - cth2H | | +

+ T217a21cth2H - 1cthHj + 5^|a23a1 +1 a24j j + 5ккт|8b24ch4H + — b23a1ch3H +

17 1

+ t|— a12 + 2a23a1cth2H + — a24cthH

+

6 = 4b24xa1 ch4H (4th4H - cth2H )+x2a1 (a2cth2H + 6a24 ) ^x^b^^H + x(6a2 + 12a24cth2H )),

1

a 2 =T

4

a

+

f

q1 =^1

15 3

-—t-a, I — a, t + Tcth2H + — b„ ch3H

8 1V 4 1 23 2 23

+ a1T2

f

1

3b23ch3H th3H +

+ cth2H) -1 c11 + t3 1cthH15 - a2 j + 9a2cth2H + a1a23

9, 1

-KkT I a1 I — b23ch3H + c1I 2cth2H--cthH I + tI — a1 - 2a23a1cth2H + —

2 23 1V 2

j 'l 2 „ OA

48

q2 = (3b23ch3H - 3a3T + 4b24a1ch4H + a23TcthH + 2a24Ta1cth2H)+ + t2 (8b24a1cth2Hch4H - 3b23ch3H(th3H - cthH) - 2c1) + t3 (2a2 (5a1cth2H + + 3cthH) + 4a1 (a24 - cth2H)+a23)- 2KkT2 (9b23ch3H + 16a1b24ch4H + c1cthH + + T^a? + 4a1 (1 - a24cth2H) - a23cthH ))

b30 = a1T212cth2H + 1cthH j.

In dimension variables (Z*, 9*, X1, z1, ft, a) the following is obtained

( 2 3 2 \ / 2 32 \

a ka23 + a k a33 Jcos30 + a ka24 + a k a34 Jcos40 +

3 2 2 2

+ a k a35 cos 50 + a k a36 cos 60,

9 = a( g j ch(z + H )sin 0 + b12ch2(z + H )sin 20 j + a2 fig (b23ch3(z + H)+

+ b24ch4(z + H )sin 40 + b20t) + a3k*Jkg (b33ch3(z + H )sin 30 + b34ch4(z + H )sin 40 + + b35ch5(z + H )sin 50 + b36ch6(z + H)+ b30t),

0 = kx + -Jkg (t + akc1 + a 2k 2c2 )t.

Here and below, the index 1 for Latin symbols x, z, t and the asterisk sign for Greek Z h 9 are omitted.

4. Analysis of the results. The obtained solution (3.6) is valid outward small neighborhoods of the resonance values of the wave number k, which are positive

real roots k = k1, k = ki, k = k3, k = k4 of the equations

^ -n2T2Kk -nT2cthnH = 0, n = 3...6 (4.1)

respectively. Wherein k > k2 > k3 > k4. Note that one of these resonance values k =

= k1, obtained from the equation (4.1) when n = 3, coincides with the smaller of the two resonance values obtained in the case when the first approximation Z1 is given in the form Z1 = Z11 [14].

For quantitative estimate of the influence of the elastic and mass forces of the plate on the characteristics of the fluctuations formed, numerical calculations were carried out under the values of the modulus of elasticity E = 0.5-109 - 3-109 N/m2, Poisson's ratio v = 0.34 and density p1/p = 0.87, corresponding to the ice plate [17].

Fig. 1 and 2 show the ice-water surface elevation profiles (ice plate bending) along the direction of the wave propagation (negative direction of the x-axis) for E = 3-103 N/m2 and the values h = 1 m, H = 100 m when t = 17 min, k = 0.21 m-1, s = ak = 0.21 (Fig. 1) and t = 10 h, k = 7-10-3 m-1, s = 1.4-10-2 (Fig. 2). The profiles Z(x) in Fig. 1, a and 2, a correspond to a positive value of the amplitude of the second interacting harmonic (a1 > 0); in Fig. 1, b and 2b - to the negative one (a1 < 0). Dashed lines are the profiles obtained taking into account the nonlinearity of the vertical displacement acceleration of the ice plate (F20 ^ 0, F30 ^ 0), and solid ones - without taking into account (l'i' = 0, F3° = 0).

1,5

Fig. 1. The wave profiles of the flexural deformation of the ice plate with the amplitude of the initial harmonic a = 1 m and wave number k = 0.21 m-1, in case a1 > 0 (a) and a1 < 0 (b)

Fig. 2. The wave profiles of the flexural deformation of the ice plate with the amplitude of the initial harmonic a = 2 m and wave number 7-10-3 m-1, in case a1 > 0 (a) and a1 < 0 (b)

From the analysis of Z(x) graphs, it follows that under the interaction of wave harmonics, the nonlinearity of vertical displacement acceleration of the ice plate during its flexural deformation hastens the wave profile displacement in the negative direction of the x axis (the direction of wave motion) and slightly changes the amplitude of the flexural wave, depending on the wave number and the characteristics of the plate. When a1 > 0 the maximum displacements of the basin surface are reached at the tops of the wave elevations, and the minimum ones - in the soles of the troughs. Variation of the sign of the amplitude a1 of the second interacting harmonic causes a substantial deformation of the flexural profile. In the case of a1 < 0, the maximum profile displacements appear in the form of troughs. In this case, taking into account the nonlinearity of vertical displacement acceleration of the plate during the flexion causes the slowing down of the wave propagation velocity. As the wavelength of the fundamental harmonic increases, the contribution of the higher harmonics in the troughs and elevations on the profiles of Z(x) becomes more notable, and with its decrease - the wave profile approaches the harmonic form.

Distribution of the fluctuation frequency in the linear case and due to the frequency displacement nonlinearity in the approximation of the order of smallness of s and in the approximation of order s2 is shown in Fig. 3, 4 and 5, respectively, 12 PHYSICAL OCEANOGRAPHY VOL. 25, ISS. 1 (2018)

when H = 100 m. In Fig. 3 the curves 1, 3 and 4 are obtained when h = 0.5 m, and the curves 2, 5 and 6 - when h = 1 m. The lines 3, 5 correspond to the modulus of elasticity E = 109 N/m2, and the lines 4, 6 - to the value E = 3-109 N/m2. Curves 1, 2 correspond to the case of an absolutely flexible plate (E = 0) simulating broken ice. The graphs above show an increase in the linear component of the fluctuation frequency with increasing of the ice thickness and its modulus of elasticity, which agrees with [9, 17]. The shorter the wavelength (larger value of the wave number), the more pronounced the influence of h and E is. In the long wavelength range, it is practically absent.

Wj^C"1

Fig. 3. Distribution of the fluctuation frequency with respect to the wave number in the linear case with an ice thickness 0.5 m (the curves 1, 3, 4) and 1 m (the curves 2, 5, 6) in the case of the elastic modulus E = 109 N/m2 (the curves 3, 5), E = 3109 N/m2 (the curves 4, 6), E = 0 (the curves 1, 2)

Fig. 4. Distribution of the vibration frequency component due to the nonlinearity in the first approximation, without taking into account the factor s = ak in case ai > 0 when h = 0,5 m (the curves 1, 2) and h = 1 m (the curves 3, 4), if E = = 109 N/m2 (the curves 1, 3) and E = 3109 N/m2 (the curves 2, 4)

Fig. 5. Distribution of the fluctuation frequency displacement component of the second order according to the wave number within the accuracy of a factor when a1 > 0, in case k < k4 (a) and k > h (b)

Distribution of the nonlinearity dependent component of the fluctuation frequency with respect to the wave number in the approximation of order s, without taking into account the factor s = ak, is shown in Fig. 4 for ai > 0, where the curves 1, 2 correspond to h = 0.5 m, and the curves 3, 4 - to the value h = 1 m. Curves 1, 3 were obtained when E = 109 N/m2, and curves 2, 4 - when E = = 3-109 N/m2. The dashed lines are constructed when the nonlinearity of vertical displacement acceleration of the plate is taken into account, and the solid lines -when it is not. From the figure, it can be seen that the acceleration nonlinearity effect (F20 - 0, F30 - 0) is manifested in an increase in the frequency displacement component in the approximation of order s. Moreover, the contribution of accounting for the acceleration nonlinearity increases with increasing the wave number (wavelength decrease). In case when ai < 0, the given component value is preserved, but the sign changes to the opposite one. It can be seen from the expression ci, where ai is present in the denominator. Consequently, a change of ai sign leads to a change in ci sign.

Dispersion curves connecting the displacement component of the second-order frequency with the wave number are shown within the accuracy of a factor of s2 = = a2k2 in Fig. 5 for fixed s in the case ai > 0. At that, in Fig. 5, a they are given for the wave number values from the range k < k4, and in Fig. 5, b - for k from the range k > ki. All the indications in Fig. 5 are the same as in Fig. 4. Behavior of the graphs indicates that in the range k < k4 there is a wave number value, passing through which the sign of the component of the order s2 changes from plus to minus.

In the range k > ki this component is negative under the considered values of k. Taking into account the nonlinearity of vertical displacement acceleration of the plate reduces the component modulus value under a fixed value of k. The plate elasticity increase leads to an increase in the absolute value of the displacement frequency. A similar effect is observed with increasing thickness of the ice plate. The distribution character of the second-order displacement frequency with respect to the wave number under a fixed s for ai < 0 is qualitatively the same as under ai > 0 (Fig. 5).

Fig. 6. Distribution of the fluctuations frequency formed in the case of nonlinear interaction of wave harmonics, by wavenumber when ai > 0 (the lines 1, 3) and ai < 0 (the lines 2, 4), in case h = i m (the lines u 1,2) and h = 0,5 m (the lines 3, 4)

Fig. 7. The influence of taking into account the nonlinearity of vertical displacement acceleration of the plate on the phase velocity of the generated flexural-gravitational wave

Distribution of the fluctuations frequency (c = ^[kg (t + akc1 + a2k2c2)) formed in the nonlinear interaction of wave harmonics is shown in Fig. 6. Here, the lines 1, 3 are given for a1 > 0, and the lines 2, 4 - for a1 < 0. The lines 1, 2 correspond to a plate thickness of 1 m and the lines 3, 4 - to the thickness of 0.5 m. The dashed curves show distribution frequency of the fluctuations with allowance for the nonlinearity of vertical displacement acceleration of the plate during its flexural deformation, and the solid ones without taking it into account. As expected, taking into consideration the acceleration nonlinearity leads to the fluctuation frequency increase. c also grows with increasing of the elastic plate thickness. The change in the sign of the amplitude a1 from plus to minus decreases the fluctuation frequency value under a fixed wave number.

Influence of the consideration of the nonlinearity of vertical displacement acceleration of the plate during its flexion to the phase velocity of the formed finite amplitude flexural-gravitational wave is demonstrated by the graphs u(k) shown in Fig. 7 for k > ku They are obtained when a1 > 0, H = 100 m, h = 1 m, E = = 3-109 N/m2, both taking into account (dashed line) and without taking into account (thin solid line) the vertical acceleration nonlinearity. A thick solid line represents the phase velocity in the linear approximation. Analysis of the results of numerical calculations in the range of considered wave numbers shows that taking into account nonlinearity causes an increase in the displacement velocity of the flexural wave. At the same time, the increase in velocity with allowance for the nonlinearity of vertical displacements is greater than without taking it into account. Note that the value of the phase velocity increases with increasing of plate thickness or modulus of elasticity [9, 17].

If a1 < 0, values of u(k) if the acceleration nonlinearity is taken into account are less than without taking it into account. In this case, u(k), both with and without allowance for acceleration nonlinearity, assumes smaller values than in the linear approximation.

In the range of wave numbers k < k4 (long waves), the influence of plate characteristics on the distribution of u over k is not practically manifested [9, 17]. The dependences of u u(k) with and without allowance for the terms F20 and F30 in the expression (2.4) coincide between each other both for a1 > 0 and a1 < 0. In the case a1 > 0, the graph of u(k) passes above the corresponding approximation, and in the case of a1 < 0 it is lower, although insignificantly.

Conclusion. Asymptotic expansions up to the third-order terms are constructed based on the multiple scales method for the fluid velocity potential of the finite-depth homogeneous fluid and elevation of the plate-fluid surface (ice-water surface). The estimation of the contribution of the values of the shift frequency of fluctuations due to nonlinearity in the first- and second-order smallness approximation is given. Dependence of these values on the sign of the second interacting harmonic amplitude and on the account of the nonlinearity of vertical acceleration of the ice plate is shown. Sign change of the second harmonic amplitude leads to a change in the sign of the frequency displacement component in the first approximation. The influence of the nonlinearity of vertical

displacements acceleration of the plate is manifested in the increase of this frequency displacement component.

Taking into account the nonlinearity of vertical displacement acceleration of the plate reduces the value of the modulus of the second-order smallness displacement component. Increase of the plate elasticity (or thickness) leads to an increase in the absolute value of this component.

Fluctuation frequency, taking into account the displacement components nonlinearity under the wave numbers larger than the maximum critical value, increases with allowance for the nonlinearity in the vertical displacement acceleration of the plate. The frequency also grows with the plate thickness increase. The second harmonic amplitude sign change from plus to minus decreases the frequency value under a fixed wave number. In this range of the wave numbers, the nonlinearity causes an increase in the velocity of flexural wave displacement under a positive second harmonic amplitude. The growth of velocity with allowance for the nonlinearity of vertical displacement acceleration is greater than without taking it into account. If the second harmonic amplitude is negative, and the acceleration nonlinearity is taken into account, the phase velocity is less than without it is not.

In the long-wave range under the wave numbers less than the minimum critical value, the influence of plate characteristics on the phase velocity distribution by the wave number is not practically manifested. The phase velocity distributions along the wavelength with and without allowance for the acceleration nonlinearity coincide with each other both for positive and for negative values of the second harmonic amplitude. Moreover, for a positive value of the amplitude, the phase velocity is greater, and for a negative value, it is less than in the linear approximation.

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About the authors:

Aleksei E. Bukatov, Head of Oceanography Department, FSBSI MHI (Sevastopol, Russia),

Dr.Sci. (Phys.-Math.), Professor, ORCID ID: 0000-000i-8666-7938.

Anton A. Bukatov, Senior Researcher, Department of Marine Information Systems and

Technologies, FSBSI MHI (Sevastopol, Russia), Ph.D. (Phys.-Math.), ORCID ID: 0000-0002-ii65-

8428, newisland@list.ru.

Contribution of the co-authors:

Aleksei E. Bukatov - scientific guidance, formulated and statement of the problem,

analysis and revision of the text.

Anton A. Bukatov - statement of the problem, derivation of an analytical solution,

analysis, reviewing the literature.

All authors have read and approved the final manuscript.

The authors declare that they have no conflict of interest.