Loose Waves in Viscoelastic Cylindrical Wave Guide with Radial Crack

ABSTRACT

The main features are the length of the waveguide in one direction, as well as limitations and localization of the wave beam in other areas. There is described the technique of the solution of tasks on distribution of waves in an infinite cylindrical waveguide with a radial crack. Also numerical results are given in the article. Viscous properties of the material are taken into account by means of an integral operator Voltaire. Research is conducted in the framework of the spatial theory of visco elastic. The technique is based on the separation of spatial variables and formulates the boundary eigenvalue problem that can be solved by the method of orthogonal sweep Godunov. In the given paper we obtain numeric values of the phase velocity depending on of wave numbers. The obtained numerical results are compared with the known data. This work is continuation of article [1]. Statement of the problem and methodology of partial solutions are described in [1]. In this work, we present a complete statement of the problem, methods of solution and discuss the numerical results.

The main features are the length of the waveguide in one direction, as well as limitations and localization of the wave beam in other areas. There is described the technique of the solution of tasks on distribution of waves in an infinite cylindrical waveguide with a radial crack. Also numerical results are given in the article. Viscous properties of the material are taken into account by means of an integral operator Voltaire. Research is conducted in the framework of the spatial theory of visco elastic. The technique is based on the separation of spatial variables and formulates the boundary eigenvalue problem that can be solved by the method of orthogonal sweep Godunov. In the given paper we obtain numeric values of the phase velocity depending on of wave numbers. The obtained numerical results are compared with the known data. This work is continuation of article [1]. Statement of the problem and methodology of partial solutions are described in [1]. In this work, we present a complete statement of the problem, methods of solution and discuss the numerical results.

KEYWORDS

The Wave Guide, Wave, Cylinder, Crack, Integral Operator, Differential Equations, Relaxation Kernel Orthogonal Sweep, Approximation, Partial Derivatives, The Phase Velocity, Frequency, Damping Factor

The Wave Guide, Wave, Cylinder, Crack, Integral Operator, Differential Equations, Relaxation Kernel Orthogonal Sweep, Approximation, Partial Derivatives, The Phase Velocity, Frequency, Damping Factor

Cite this paper

Safarov, I. , Akhmedov, M. and Boltaev, Z. (2014) Loose Waves in Viscoelastic Cylindrical Wave Guide with Radial Crack.*Applied Mathematics*, **5**, 3518-3524. doi: 10.4236/am.2014.521329.

Safarov, I. , Akhmedov, M. and Boltaev, Z. (2014) Loose Waves in Viscoelastic Cylindrical Wave Guide with Radial Crack.

References

[1] Safarov, I.I., Akhmedov, M.Sh., Nuriddinov, B.Z. and Sharipov, D.Sh. (2014) Waves in Viscoelastic Cylinder with the Radial Crack. Magazine “Young scientist”, 4, 61-66.

[2] Pochhammer, L. (1876) &UUML;ber die Fortpflanzung segechwindigkeiten schwingungen in einem unbergrawzten isotropen kreiscylinder. J.reine und angew. Math, 81, 324-336.

[3] Cri, C. (1886) Longitudinal Vibrations of a Corcablar Bar. Quart. J. Pure and Appl. Math, 21, 287-298.

[4] Rayleigh, J.W. (1885|1886) On Waves Propagated along the Plane Surface of an Classic Solid. Proceedings of the London Mathematical Society, 17, 4-11.

[5] Lamb, H. (1917) On Waves in an Elastic Plate. Proceedings of the Royal Society A, 93, 114-128.

[6] Hrinchenko, V.T. and Meleshko, V.V. (1981) Harmonic Oscillations and Waves in Elastic Bodies. Naukova Dumka, Kiev, 283 p.

[7] Safarov, I.I., Teshaev, M.K.H. and Boltaev, Z.I. (2012) Wave Processes in a Mechanical Waveguide. LAP LAMBERT Academic Publishing (Germany), 217 p.

[8] Kravchuk, A.S., Majboroda, V.P. and Urzhumaev, Y.S. (1985) Mechanics of Polymeric and Composite Materials: Experimental and Numerical Methods. Publishing Science, Moscow, 304 p.

[9] Marchuk, G.I. (1977) Methods of Computational Mathematics. Publishing Science, Moscow, 456 p.

[10] Bozorov, M.B., Safarov, I.I. and Shokin, Y.I. (1996) Numerical Simulation of Oscillations Dissipative Homogeneous and Inhomogeneous Mechanical Systems. Publisher Siberian Branch of the Academy of Sciences, Novosibirsk, 188 p.

[1] Safarov, I.I., Akhmedov, M.Sh., Nuriddinov, B.Z. and Sharipov, D.Sh. (2014) Waves in Viscoelastic Cylinder with the Radial Crack. Magazine “Young scientist”, 4, 61-66.

[2] Pochhammer, L. (1876) &UUML;ber die Fortpflanzung segechwindigkeiten schwingungen in einem unbergrawzten isotropen kreiscylinder. J.reine und angew. Math, 81, 324-336.

[3] Cri, C. (1886) Longitudinal Vibrations of a Corcablar Bar. Quart. J. Pure and Appl. Math, 21, 287-298.

[4] Rayleigh, J.W. (1885|1886) On Waves Propagated along the Plane Surface of an Classic Solid. Proceedings of the London Mathematical Society, 17, 4-11.

[5] Lamb, H. (1917) On Waves in an Elastic Plate. Proceedings of the Royal Society A, 93, 114-128.

[6] Hrinchenko, V.T. and Meleshko, V.V. (1981) Harmonic Oscillations and Waves in Elastic Bodies. Naukova Dumka, Kiev, 283 p.

[7] Safarov, I.I., Teshaev, M.K.H. and Boltaev, Z.I. (2012) Wave Processes in a Mechanical Waveguide. LAP LAMBERT Academic Publishing (Germany), 217 p.

[8] Kravchuk, A.S., Majboroda, V.P. and Urzhumaev, Y.S. (1985) Mechanics of Polymeric and Composite Materials: Experimental and Numerical Methods. Publishing Science, Moscow, 304 p.

[9] Marchuk, G.I. (1977) Methods of Computational Mathematics. Publishing Science, Moscow, 456 p.

[10] Bozorov, M.B., Safarov, I.I. and Shokin, Y.I. (1996) Numerical Simulation of Oscillations Dissipative Homogeneous and Inhomogeneous Mechanical Systems. Publisher Siberian Branch of the Academy of Sciences, Novosibirsk, 188 p.