On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces Pham Chi Vinh ⁎, Pham Thi Ha Giang Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam article info abstract Article history: Received 30 December 2010 Received in revised form 1 May 2011 Accepted 5 May 2011 Available online 13 May 2011 Formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces are derived using the complex function method. The derivation also shows that if a Stoneley wave exists, then it is unique. By using the obtained formulas, we can easily reproduce the numerical results previously obtained by Murty [G. S. Murty, Phys. Earth Planet. Interiors 11 (1975), 65–79.] by directly solving the secular equation. © 2011 Elsevier B.V. All rights reserved. Keywords: Stoneley waves Stoneley wave velocity Loosely bonded interface Holomorphic function 1. Introduction Rayleigh surface waves [1] and Stoneley interfacial waves [2] in isotropic elastic solids discovered many years ago, in 1885 by Rayleigh and 1924 by Stoneley, respectively, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example. The velocities of Rayleigh waves and Stoneley waves are of great interest to researchers in various fields of science. The formulas for them provide powerful tools for solving the direct (forward) problems: studying effects of material parameters on the wave velocity, and especially for the inverse problems: determining material parameters from the measured values of the wave speed. The formulas for the velocity of Rayleigh waves and Stoneley waves are thus of theoretical as well as practical interest. For Rayleigh waves, some formulas for the velocity have been obtained recently, see for instance [3–13], while no formulas have been derived for Stoneley waves so far, to the best of the authors' knowledge. The main aim of this paper is to establish formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces by using the complex function method. It is shown from the derivation of these formulas that, if a Stoneley wave exists, then it is unique, as proved by Barnett et al. [14] by another technique. Using the obtained formulas, it is easy to recover the numerical results obtained previously by Murty [15] by directly solving the secular equation. 2. Secular equation In this section we present briefly the derivation of the secular equation of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces. For details, the reader is referred to the papers [15,16] by Murty. Let us consider two isotropic elastic solids Ω and Ω ⁎ occupying the half-spaces x 2 ≥ 0 and x 2 ≤ 0, respectively. Suppose that these two elastic half-spaces are not in welded contact with each other at the plane x 2 = 0 (see [15,16]). In particular, the normal Wave Motion 48 (2011) 647–657 ⁎ Corresponding author. Tel.: + 84 4 5532164; fax: + 84 4 8588817. E-mail address: pcvinh@vnu.edu.vn (P.C. Vinh). 0165-2125/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2011.05.002 Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wavemoti