Applied Mathematics
Vol.4 No.3(2013), Article ID:28858,5 pages DOI:10.4236/am.2013.43072

The Conservation Laws and Stability of Fluid Waves of Permanent Form

Troyan A. Bodnar’

Department of Mathematics Technological Institute, Altai State Technical University, Biisk, Russia


Received December 13, 2012; revised January 15, 2013; accepted January 23, 2013

Keywords: Nekrasov Integral Equation; Kellogg Method; Successive Approximations Method; Wave Mechanical Energy; Wave Stability


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 obey 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˚ and the value 45˚ is supposed attainable, 2) the wave kinetic energy vanishes. The stability of a steady wave considered as a compound pendulum is analyzed.

1. Introduction

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 [3] as mathematical theorem follows that the kinetic energy vanishes on the boundaries of the segment and has the maximum at a point Thus we observe symmetry: at the points the full mechanical energy consists only of potential energy. The wave of constant shape may be considered as a compound pendulum with a suspension center in the origin of coordinates arranged on unperturbed surface of fluid.Then the wave stability is spotted as a compound pendulum stability.

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.

2. Solution of Nekrasov’s Integral Equation

Nekrasov’s integral equation describing steady state waves of unchangeable shape on the surface of a fluid with finite depth is written as [4]


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 [5] in [1,2]. Let us remark that the Kellogg method was applied to a non-linear integral equation and the discovered solution is not spectral. In accordance with [2], Equation (1.5) has a unique solution (the motionless point)

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


(in [2] is designated as).This means that the solution of the system (1.3) at exists and is unique if the wavelength


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


derived from (1.3) by substitution If requirements (1.4) are satisfied, then the function is defined on the segment (see [1]), or on the equivalent segment


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 [6]. We suppose that

3. The Conservation Law of Full Mechanical Energy and Stability of a Wave

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


and the function


Coefficients are determined as algebraic expressions of [2].

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 [4]


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



The solution of system (1.3) obtained in [2] allow us to calculate the function on the segment On the boundaries of this segment we have

The functions calculated on the segment are shown on Figure 1. The outcomes of these calculations allow to assert that the wave potential energy as function of parameter

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]. This theorem asserts that: 1) for any constant depth of fluid the wavelength 2) at the boundaries of the segment the wave velocity

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 [7]). Now it is possible to express the coefficients from the system (1.6) in terms of the integrals We have


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

Figure 1. The functions CT(hλ), CII(hλ), CE(hλ) for 0.08 ≤ hλ ≤ 0.4.

Let as remark that the convergence of series has been proved at a limited in [4] and for a presented as a converging series in [8].

3) Since the function vanishes at the boundaries of the interval it follows from the Rolles theorem [7] that exists at least a point

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 [2].

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 Figure 1). The full mechanical energy of a wave is a monotonically decreasing function on the segment, but vanishes only at the point We note also that the profiles of the waves calculated on the segment obey the law of mass conservation

The stability of Nekrasov’s waves has been considered in [9]. We proceed from the fact that a liquid maintaining the invariable shape without a vessel, does not suspect that is a fluid.We suppose that a steady state wave can be presented as a compound pendulum with a suspension center in the origin of coordinates. This wave is stable if

and unstable if The function

is presented on Figure 2. This function has the greatest value, is equal to zero

has the minimum

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

Figure 2. The dependence ych1(hλ) for 0.08 ≤ hλ ≤ 0.4.

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 [2]) and the coefficients b1, b2, …, b7 are given below: μ7(0.1) = 7.449619; a1 = 0.047452; a2 = 0.0195554; a3 = 0.00606262; a4 = 0.00183319; a5 = 0.000565461; a6 = 0.000178475; a7 = 0.0000568810; b1 = 0.047452; b2 = 0.0206812; b3 = 0.00700837; b4 = 0.00233431; b5 = 0.000787255; b6 = 0.000268797; b7 = 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 [2]) the wave parameters will be: This wave is unstable as

In summary it is necessary to note that in the accessible publications we have not discovered materials for comparison, except [6].

4. Conclusion

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 .


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  2. T. A. Bodnar’, “On Steady Periodic Waves on the Surface of a Fluid of Finite Depth,” Journal of Applied Mechanics and Technical Physics, Vol. 52, No. 3, 2011, pp. 378-384. doi:10.1134/S0021894411030072
  3. T. A. Bodnar’, “On Steady Waves on the Surface of a Finite-Depth Fluid,” Free Boundary Problems: Theory, Experiment, and Applications, 3rd All-Russian Conference with International Participation, Biisk, 28 June-3 July 2008, pp. 25-26.
  4. A. I. Nekrasov, “Exact Theory of Steady Waves on the Surface of a Heavy Fluid,” Izd.Akad.Nauk SSSR, Moskow, 1951. (In Russian)
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  7. R. Courant, “Differential and Integral Calculus,” Interscience, New York, 1936.
  8. L. N. Sretenskii, “Theory of Fluid Wave Motion,” Nauka, Moscow, 1977. (In Russian)
  9. T. A. Bodnar’, “Conservation Law of the Full Mechanical Energy and Stability of the Steady-State Waves on the Surface of a Fluid of Finite Depth,” In: IV All-Russian Conference with foreign participation on Free Boundary Problems: Theory, Experiment, and Applications, Biisk, 5-10 July 2011, pp. 18-19.