B 1 B 2 + α k 2 a = 0 γ + g / C 1 2 ρ 01 ω A ( ω k V 0 ) a = 0 δ 1 + g / C 2 2 ρ 02 ω B 1 + δ 2 + g / C 2 2 ρ 02 ω B 2 ω a = 0 δ 1 + g / C 2 2 ρ 02 ω exp ( δ 1 h ) B 1 + δ 2 + g / C 2 2 ρ 02 ω exp ( δ 2 h ) B 2 = 0 (22)

Equating the determinant of the system (22) to zero, we will get dispersion relation for the wave on the liquid surface taking into account surface tension force in the form of:

$\begin{array}{l}\frac{{\delta }_{1}{\delta }_{2}}{{\rho }_{02}\omega }\left(\omega -k{V}_{0}-\frac{\gamma }{{\rho }_{01}\omega }\alpha {k}^{2}\right)\left[\mathrm{exp}\left(-{\delta }_{1}h\right)-\mathrm{exp}\left(-{\delta }_{2}h\right)\right]\\ -\frac{\gamma }{{\rho }_{01}}\left[{\delta }_{2}\mathrm{exp}\left(-{\delta }_{1}h\right)+{\delta }_{1}\mathrm{exp}\left(-{\delta }_{2}h\right)\right]=0\end{array}$ (23)

Taking into account that on the sea level ${C}_{1}\cong 340\text{\hspace{0.17em}}\text{m}/\text{sec}$ , let’s consider the inequality

${\theta }_{1}=\frac{2k{C}_{1}^{2}}{g}>1⇒k>\frac{g}{2{C}_{1}^{2}}⇒\lambda <\frac{4\text{π}{C}_{1}^{2}}{g}=1.45×{10}^{5}\text{m}$

We can see, that to this inequality satisfies with the entire range of lengths of surface waves on the water, from capillary to tsunami. It is apparent that for the capillary waves length of which does not exceed a few centimeters, we have: ${\theta }_{2}\gg {\theta }_{1}\gg 1$ . Considering also that ${U}_{p}^{2}/{C}_{2}^{2}\ll {U}_{p}^{2}/{C}_{1}^{2}\ll 1$ , from (12), (15) and (16) we find $\gamma ={\delta }_{1}=-k$ , ${\delta }_{2}=k$ and then neglecting ${\rho }_{01}$ with respect to ${\rho }_{02}$ , the dispersion Equation (23) takes the form:

${\rho }_{02}{U}_{p}^{2}+{\rho }_{01}{V}_{0}{U}_{p}-\alpha k=0$ (24)

the solution of which is

${U}_{p}=\frac{-{\rho }_{01}{V}_{0}±\sqrt{{\rho }_{01}^{2}{V}_{0}^{2}+4\alpha k{\rho }_{02}}}{2{\rho }_{02}}$ (25)

4. Discussion of Results

In order to show the truthfulness of our results, let’s consider earlier results and show their drawbacks. As it was said in the introduction, in the monography  dispersion relation for the capillary-gravitational wave on the surface of incompressible liquid is presented in the form of:

${U}_{p}^{2}={\left(\frac{\omega }{k}\right)}^{2}=\left(\frac{g}{k}+\frac{\alpha k}{{\rho }_{02}}\right)th\left(kh\right)$ (26)

from which it follows that when

$\frac{g}{k}>\frac{\alpha k}{{\rho }_{02}}⇒k<\sqrt{\frac{{\rho }_{02}g}{\alpha }}⇒\lambda >2\text{π}\sqrt{\frac{\alpha }{{\rho }_{02}g}}=1.72\text{sm}$ (27)

the influence of the surface tension force is negligible, and the wave becomes purely gravitational. This conclusion contradicts to the classical experiment, in which a steel needle does not sink in a glass filled with water to the brim. This is because although the diameter of glass greatly exceeds the above specified length, the force of surface tension acts which balances the pressure produced by the needle. Thus, the dependence of phase speed on gravitational acceleration is excluded and consequently there is no existing condition that limits the length of capillary wave. Such a conclusion is quite understandable from the point of view of physics, because surface tension arises due to the interaction forces between molecules on the surface of a liquid that significantly exceed the gravitational force.

The contradiction associated with the influence of the gravitational field is eliminated by taking into account the compressibility of the fluid in the mass continuity equation. The solution of the problem for such a case is given in  , where the dispersion equation is obtained in the form:

${U}_{p}=\frac{\omega }{k}=\frac{{\rho }_{01}{V}_{0}th\left(kh\right)±{\left\{th\left(kh\right)\left[{\rho }_{02}\alpha k-{\rho }_{01}{\rho }_{02}{V}_{0}^{2}\right]\right\}}^{1/2}}{{\rho }_{02}}$ (28)

from which follows the condition of stability of capillary wave:

${V}_{0}\le \sqrt{\frac{\alpha k}{{\rho }_{01}}}$ (29)

From (29) it is easy to calculate that the wind with the speed of ${V}_{0}=5$ m/s, will blow off capillary waves whose length $\lambda >1.6$ cm. However, simple observations show that capillary waves exist at quite stronger winds. In addition, since the capillary wave is a purely surface phenomenon, its phase speed must not depend on the depth of the fluid.

As it is apparent in the Equation (25), this contradiction is eliminated if in Euler equation, we consider liquid as compressible. Capillary wave is stable in any wind, if only the wind force does not exceed intermolecular interaction force and in this case, setting of the problem becomes meaningless. We can also see that phase speed of capillary wave does not depend on the depth of fluid.

5. Conclusion

Contradictions that are present in the theory of surface waves are described in detail in the works  and  . In this work, through the example of capillary waves we have explicitly investigated the causes of these contradictions and showed how to overcome them. We can say with confidence that our recommendations will result in overcoming contradictions not only in the theory of capillary waves but also in the theory of gravity waves too.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

  Landau, L.D. and Lifshitz, E.N. (1988) Hydrodynamics, Vol. 6, Nauka, Moscow.  Stoker, J.J. (1957) Water Waves. Intersciense Publisher Inc., New York.  Whitham, G. (1974) Linear and Nonlinear Waves. John Wiley and Sons, Hoboken.  Kowalik, Z. (2012) Introduction to Numerical Modeling of Tsunami Waves. Institute of Marine Science University of Alaska, Fairbank.  Hassan, J.M., Mohamed, T.A., Mohammed, W.S. and Alawee, W.H. (2014) Journal of Fluids, 2014, Article ID: 325259.  Chand, R. and Rana, G.C. (2014) Journal of Fluids, 2014, Article ID: 479107.  Rabby, M.G., Shupti, S.P. and Molla, M.M. (2014) Journal of Fluids, 2014, Article ID: 757902. https://doi.org/10.1155/2014/757902  Eldabe, N.T.M., Agoor, B.M. and Alame, H. (2014) Journal of Fluids, 2014, Article ID: 525769.  Singh, M. and Gupta, R.K. (2014) Journal of Fluids, 2014, Article ID: 714150.  Troshkin, O. (2016) Mathematics and Statistics, 4, 47-57. https://doi.org/10.13189/ms.2016.040201  Campos Pinto, M. and Charles, F. (2016) Proceedings and Surveys, 53, 38-48. https://doi.org/10.1051/proc/201653003  Cotter, C.J., Eldering, J., Holm, D.D., Jacobs, H.O. and Meier, D.M. (2016) Journal of Nonlinear Science, 26, 723-765. https://doi.org/10.1007/s00332-016-9317-6  Cubos-Ramírez, J.M., Ramírez-Cruz, Salinas-Vázquez, J.M., Vicente-Rodríguez, W., Martinez-Espinosa, E. and Lagarza-Cortes, C. (2016) Computers & Fluids, 136, 212-227.  Rehman, K. and Cho, Y. (2016) Water, 8, 212. https://doi.org/10.3390/w8050212  Ogilvie, G.I. (2016) Journal of Plasma Physics, 82, 212. https://doi.org/10.1017/S0022377816000489  Kirtskhalia, V. (2016) Journal of Fluids, 2016, Article ID: 4519201. https://doi.org/10.1155/2016/4519201  Kirtskhalia, V. (2012) Open Journal of Acoustics, 2, 80-85. https://doi.org/10.4236/oja.2012.22009  Kirtskhalia, V. (2013) Journal of Modern Physics, 4, 1075-1079. https://doi.org/10.4236/jmp.2013.48144  Kirtskhalia, V. (2015) Applied Problems of Gas and Hydrodynamics. Academic Publishing “LAMBERT”, Saarbrucken. 