2 11 33 13 0, 1,3,5 and 0 ii c i cc c > = − > { } 2 11 55 0 min , c c c < < World Applied Sciences Journal 16 (1): 73-75, 2012 ISSN 1818-4952 © IDOSI Publications, 2012 Corresponding Author: A. Khan, Mathematics, COMSATS Institute of Information technology, Islamabad, Pakistan 73 Rayleigh Waves in a Rotating Orthotropic Medium A. Khan, S. Islam and M. Khan Mathematics, COMSATS Institute of Information technology, Islamabad, Pakistan Abstract: The conditions that strain energy tensor and necessary condition on the roots of a constitutive equation must be positive definite produces some inequalities. These inequalities reduce the number of Rayleigh waves in orthotropic and transversely isotropic materials. Some earlier published work is found to be erroneous. Key words: Rayleigh wave Orthotropic and transversely isotropic waves INTRODUCTION orthotropic material. Rehman et al. [14-16] and Khan et al. Problem in hand is about the surface wave in rotating orthotropic material but ignored the necessary condition orthotropic medium. The problem can be considered as a for the existence of the Rayleigh waves. This condition model of earth which is rotating about an axis and plays a significant role on the wave propagation and behavior of Rayleigh waves are like earth quake waves reduces the number of waves. Similar condition for the which travels on the surface of earth. In this article it is rotating material was ignored in a recent published work interesting to note that in fast rotating medium, surface [14] and found that three Rayleigh waves can propagate waves does not exist but, slow rotating medium like earth in rotating and non-rotating mediums. Chadwick [1] and increases the speed of the surface wave. Pham & Ogden [13] also considered the condition of Chadwick [1] presented very comprehensive theory strain energy tensor to be positive definite for for the elastic wave propagation in a transversely transversely isotropic and orthotropic materials isotropic material and proved that three waves can respectively. propagate in the material. He also discussed the surface In this note, it is found that only one Rayleigh can waves in that material [2]. Achenbach [3], Bhatia and propagate in the medium. Speed of the waves in few Singh [4], Dieulesaint [5] and Fung [6] discussed, in orthotropic materials is also calculated. It is also detail, the wave propagation in elastic solids. Barnett interesting to note that Rayleigh wave cannot survive in and Lothe [7, 8] studied the surface waves in anisotropic a fast rotating medium. Results for transversely isotropic elastic half spaces by using impedance method. Carcione materials can be deduced from orthotropic materials by and Kashoff [9] also presented the theory of wave replacing elastic constants c by c . propagation in transversely isotropic material. Synge and Griffth [10] discussed the rate of change of displacement Speed of Rayleigh Waves in Orthotropic Materials: vector in a rotating medium where as Schoenberg and For orthotropic material it is shown by Pham and Ogden Censor [11] derived the secular equation for a rotating [13] that necessary and sufficient condition for strain isotropic elastic medium and proved that in a rotating energy tensor to be positive definite is: medium an additional wave appear and rotating isotropic (1) material behaves like anisotropic medium. Ahmed and Khan [12] checked the effect of rotation on transversely By considering these inequalities Pham and Ogden isotropic materials and proved that rotational effect [13] proved that necessary condition for existence of increases the speed of elastic waves. Pham and Ogden surface wave is as follows: [13] found secular equation for orthotropic material and also imposed a condition on the speed of Rayleigh waves. (2) This condition allows only one wave to propagate in [17, 18] introduced rotational effect on Rayleigh waves in 55 44