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The solution of Nekrasov’s integral equation is described. By means of this solution the wave kinetic, potential, and full mechanical energies are defined as functions of fluid depth and wavelength. The wave obeys the laws of mass and energy conservation. It is found that for any constant depth of fluid the wavelength is bounded from above by a value denoted as maximal wavelength. At maximal wavelength 1) the maximum slope of the free surface of the wave exceeds 38^{o} and the value 45^{o} is supposed attainable,2) the wave kinetic energy vanishes. The stability of a steady wave considered as a compound pendulum is analyzed.

This paper is based on Nekrasov’s integral equation solution obtained in [1,2]. This solution allows us to find the profile and velocity of a gravitational wave, and the calculation of the wave kinetic and potential energy is possible. At fixed depth of fluid the solution of Nekrasov’s integral equation exists on a limited segment of wavelength The potential and full mechanical energies are monotonically increasing functions on the segment. From the law of the change of wave’s kinetic energy presented in [

The plan of the paper is as follows. In Section 2, we describe the method of Nekrasov’s integral equation solution. Here this method has been used for evaluation of the maximum wavelength boundaries and for estimation of the maximum slope of the wave free surface.

In Section 3, geometrical and energy properties of a wave are explored and the theorem about the change of the wave velocity on the segment is proved. As a result we have gained the laws of the change of wave’s kinetic and potential energy. Here we have defined the wave’s center of mass as a function of depth-towavelength ratio and have made some suppositions concerning the wave stability considering it as a compound pendulum.

Nekrasov’s integral equation describing steady state waves of unchangeable shape on the surface of a fluid with finite depth is written as [

where - is the polar angle, - is the angle that the wave surface makes with the horizontal, - is the wavelength, - is an arbitrary constant,

From Equation (1.1) follows that required function represents a trigonometric series

The evaluation of scalar product

taking account of Equations (1.1) and (1.2) gives a system of nonlinear integral equations

where the subscript denotes that in Equation (1.1) a truncated kernel with instead of is used. Below the subscript of any function will be omitted if

where - is a suitable from accuracy point of view small number The system (1.3) containing unknowns is underdetermined and has a set of solutions including the trivial.

We assume that the first coefficient is independent of and and can be calculated from the linearized on system (1.3) at Thus

satisfy the equation

This equation has been solved by Kellogg’s method [

Now we fix in (1.3) and consider as the initial approximation of After that, the system of Equations (1.3) contains n unknowns and has a solution that cannot be trivial.It has been shown [1,2] that coefficient satisfy system (1.3) at if

where

(in [

where

The lower boundary of the segment on which coefficient satisfy system (1.3) at can be obtained by solving the system

(1.6)

derived from (1.3) by substitution If requirements (1.4) are satisfied, then the function is defined on the segment (see [

The system (1.6) containing n unknowns

was solved by successive approximations method at limited by computer’s throughput. The results of calculations of are given below:

The function is of low accuracy at the point and allows only to write the inequalities From these inequalities follows that the maximal wavelength is bounded on both sides

The results of calculations of for are given below:

These numerical results assert that will remain the solution of system (1.3) at for any

From the solution of the system (1.6) we obtain the coefficients of the function. This function achieves the maximum at the points

The maximum slope of the free surface in degrees exceeds the value received in [

For evaluation of a wave square and full mechanical energy we need the parametric equations of wave’s surface coordinates

(2.1)

and the function

Coefficients are determined as algebraic expressions of [

The coordinate origin O is on vertical line through the wave crest at distance h from the bottom,the axis is directed upward, and the axis to the right. In a coordinate system attached to the wave the bottom moves from right to left at velocity [

Using (2.1)-(2.3) at fixed, we obtain the expressions for the properties of the wave: the surface area

the coordinates of the center of mass

the kinetic energy

where constant denotes the fluid density; the potential energy

the full mechanical energy

The full mechanical energy of an unperturbed fluid layer with depth h and wavelengths is considered as

Substituting for in (2.4)-(2.9), we get

and where

are integrals depending on parameter Using these relationships we can write the Equations (2.6)-(2.8) in the form

where

The solution of system (1.3) obtained in [

The functions calculated on the segment are shown on

monotonically diminishes on the segment

In order to calculate the greatest value of potential energy of a wave we need the solution of system (1.6) instead of (see above).

The law of the wave kinetic energy change is formulated in the form of a mathematical theorem in [

3) there is a value at which

and the wave velocity is maximum, i.e.

Let’s prove the points 1) - 3). 1) From (1.7) follows that and therefore

2) Suppose the solution of (1.6) is known and satisfy (1.4). Using this solution we write the integrals

The first integral is known the integral tends to zero as (see [

Taking into account as from (2.12) follows that series is converging independently of convergence or divergence of the series Now if we recall (1.7), (2.3), and (2.12), we get

Let as remark that the convergence of series has been proved at a limited in [

3) Since the function vanishes at the boundaries of the interval it follows from the Rolles theorem [

such that As the numerator and denominator of right side of Equation (2.3) are strictly increasing functions of the point is single and The values

have been calculated in [

The kinetic energy of the wave vanishes at boundaries of the segment and reaches the maximum at a point (This point is not presented on

The stability of Nekrasov’s waves has been considered in [

and unstable if The function

is presented on

has the minimum

and monotonically increases on the interval reaching the value only at the point If we continue to con-

sider the wave behaving as a compound pendulum then it is stable at and unstable when

In particular let’s consider a wave of length λ = 1000 m on the surface of a fluid with depth and density The solution of system (1.3) at the point (see [_{1}, b_{2}, …, b_{7} are given below: μ_{7}(0.1) = 7.449619; a_{1} = 0.047452; a_{2} = 0.0195554; a_{3} = 0.00606262; a_{4} = 0.00183319; a_{5} = 0.000565461; a_{6} = 0.000178475; a_{7} = 0.0000568810; b_{1} = 0.047452; b_{2} = 0.0206812; b_{3} = 0.00700837; b_{4} = 0.00233431; b_{5} = 0.000787255; b_{6} = 0.000268797; b_{7} = 0.0000921207. Using these outcomes in Formulas (2.1)-(2.10) we get the parameters of the wave: the crest coordinate the trough coordinate the amplitude the coordinates of mass center the kinetic energy the potential energy the velocity This wave similar to a tsunami is stable as

For and (The solution of the system (1.3) and coefficients for are given in [

In summary it is necessary to note that in the accessible publications we have not discovered materials for comparison, except [

Solving the Nekrasov’s integral equation we avoided the “Liapunov-Schmidt” method and other methods of searching the solution in the neighbourhood of eigenvalues of Nekrasov’s linearized equation. We sought the solution of this equation in the neighbourhood of the motionless point of the nonlinear integral Equation (1.5). This is the point .