By Helmohltz’s theorem [22] the displacement vector can be written as
(9)
where the two functions and are known in the theory of elasticity, by Lame’s potentials irrotational and rotational parts of the displacement vector , respectively. The cylinder being bounded by the curved surface, the stress distribution includes the effect of both and . It is possible to take only one component of the vector to be nonzero as follows: . From (9) we obtain
(10a)
(10b)
Substituting equations (10a) and (10b) into equations (8a) and (8b), we get two independent equations for and as follows:
(11a)
(11b)
To study the propagation of harmonic waves in the z-direction, we assume a solution in the form
(12a)
(12b)
where γ is the wave number, ω is the angular frequency. Substituting equations (12a) and (12b) into equations (11a) and (11b) and omitting the factor exponential throughout, we have
(13a)
(13b)
Similar results were obtained by Elnagar and Abd-Alla [23], the former deriving the constitutive equation for Rayleigh waves in an elastic medium under initial stress, and the latter deriving the constitutive equation for thermoelastic problems in an infinite cylinder under initial stress. From equations (13a) and (13b), we get equations (14a) and (14b) as follows:
(14a)
(14b)
Equation (14a) represents the shear wave, and equation (14b) represents the longitudinal wave. The solution of equations (14a) and (14b) can be written in the following form:
(15a)
(15b)
where
, , and are arbitrary constants, is the Bessel function of the first kind and of order zero, is the Bessel function of the second kind and of order zero. is the Bessel function of the first kind and of order n, is the Bessel function of the second kind and of order n. From equations (12a), (12b) and (15a), (15b) we get
(16a)
(16b)
Substituting equations (16a) and (16b) into equations (10a) and (10b), we obtain the final solution of displacement components in the following form:
(17a)
(17b)
Substituting equations (17a) and (17b) into equations (3a)-(3d), we obtain the final solution of the stress components of solid in the following form:
(18a)
(18b)
(18c)
(18d)
where and are the Bessel functions of the first order. In the following section, solutions of hollow circular cylinders with three different boundary conditions are performed.