Phase Velocities of Three-Dimensional and Axis-Symmetrical Elastic Waves in Isotropic Cylindrical Shell

Based on use of Debye’s potentials one can find the direct solution of the problem of definit ion of the characteristic equations for wave numbers of three-dimensional and axis-symmetrical (flexural and longitudinal) elastic waves in steel and alumin ium cylindrical shells of various thickness.


Introduction
The Debye's potentials are used for the first time for a study of the three-dimensional flexural waves. The paper shows the existing of the three-d imensional and a xis-symmet rical flexural waves in cylindrical shell (contrary to the cylindrical bar). The paper demonstrates the calculated values of the phase velocities of axis-symmetrical and three-dimensional flexu ral waves, wh ile the phase velocities of the non-zero forms of three-symmetrical flexu ral waves are calculated for the first time.

The Three-Dimensional Flexural Waves in Cylindrical Shell
The first part of the article substantiates the effectiveness of usage of Debye's potentials [1 -3] for studying of three-dimensional flexural waves in cylindrical shell.
In contrast to the bar the flexural wave in cylindrical shell can be three-dimensional and two -dimensional (axis-symme trical). The deformation of cylindrical shell in the propagation of axis-symmet rical flexu ral (а) and longitudinal (b) waves in it is schematically shown in fig. 1.
Let's start with examination of three-d imensional fle xu ral wave in isotropic cylindrical shell. In this case the same mathematical apparatus is used as in the study of flexu ral waves in bar [4], but the number of unknown quantities and the number of boundary conditions increase with the account of the second (internal) boundary surface. Now the expansions of the potentials Ф, V, U [5 -15]   We desire the analytical form of boundary conditions as following: If we substitute (1) in the boundary conditions (2) -(4), we'll get the determinant of six order [5 -15]   Let's turn to axis-symmetrical longitudinal and fle xu ral waves. In accordance with [5 -15] in axis-symmet rical case the boundary conditions (2) -(4) become simp ler: the condition (3) disappears and the condition (2) takes the following form (in this case the index m=0 or 1): The determinant of the fourth order derived from boundary conditions takes the form [ In the present work the derivatives of cylindrical functions at radial coordinate r are marked the following way: [5] differently: Equating the determinant (7) zero and opening it, we get the characteristic equation for wave nu mbers of flexural and longitudinal axis-symmetrical waves.
The determinant for axis-symmetrical torsion waves are represented in [5]: Equating the determinant (8) zero and opening it, we get the characteristic equation for wave numbers of a xis-symmet rical torsion waves.

The Results of Numerical Experiment for Determination of Phase Velocities of Elastic Waves
The second part of the article investigates the results of numerical experiment for determination of phase velocities of elastic waves (axis-symmetrical and three-dimensional) in cylindrical shell and analyses the calculated dependences.
The In this case the boundary conditions (2) for normal tensions on the both surfaces of the shell transforms: where: 1 ρ -the solid ity of the external environ ment.
where: In axis-symmetrical case the condition (6) transforms: The added boundary conditions in axis-symmetrical case:  (axis-symmetrical problem) will add two columns. In this case only two lines in each of these two columns will be different fro m zero.
If the shell borders the liquid only from one side (and from the other is still the vacuum) the determinant for determination of wave numbers will have the seventh order in three-dimensional problem and the fifth order (in a xis-symmet rical case), that is to determinants (5) и (7) one line and one colu mn are added correspondingly.
The results presented in the article are received in the conducting of scientific research in the framework of State contract P 242 fro m 23 April 2010 FPP (Scientific and scientific -pedagogical personnel of innovative Russia for the years 2009-2013).