Original Article Journal of Intelligent Material Systems and Structures 1–16 Ó The Author(s) 2018 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1045389X18786464 journals.sagepub.com/home/jim Approximation of surface wave velocity in smart composite structure using Wentzel–Kramers–Brillouin method Manoj Kumar Singh 1 , Sanjeev A Sahu 1 , Abhinav Singhal 1 and Soniya Chaudhary 1,2 Abstract In mathematical physics, the Wentzel–Kramers–Brillouin approximation or Wentzel–Kramers–Brillouin method is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. An attempt has been made to approximate the velocity of surface seismic wave in a piezo-composite structure. In particular, this article studies the dispersion behaviour of Love-type seismic waves in functionally graded piezoelectric material layer bonded between initially stressed piezoelectric layer and pre-stressed piezoelectric half-space. In functionally graded piezoelectric material stratum, theoretical derivations are obtained by the Wentzel–Kramers–Brillouin method where variations in material gradient are taken exponentially. In the upper layer and lower half-space, the displacement compo- nents are obtained by employing separation of variables method. Dispersion equations are obtained for both electrically open and short cases. Numerical example and graphical manifestation have been provided to illustrate the effect of influ- encing parameters on the phase velocity of considered surface wave. Obtained relation has been deduced to some exist- ing results, as particular case of this study. Variation in cut-off frequency and group velocity against the wave number are shown graphically. This study provides a theoretical basis and practical utilization for the development and construction of surface acoustics wave devices. Keywords Love-type wave, Wentzel–Kramers–Brillouin, functionally graded piezoelectric material, piezoelectric, initial stress Introduction Piezoelectricity is an electric charge that accumulates in some solids or hard materials after applying a mechanical stress. The use of piezoelectricity has increased much faster in modern engineering processes. Functionally graded material (FGM) is of great use in several manufacturing process. To satisfy the require- ments of modern technologies and material science, functionally graded piezoelectric material (FGPM) was discovered in the line of FGMs. Transference of Love- type waves in different media and channels such as pre-stressed piezoelectric medium and in FGPM gives valuable information for designing and development of seismic devices. Transference behaviour of Love-type waves in piezo-composite structures has been studied by several authors (viz. Chaudhary et al., 2017; Du et al., 2007) medium. Recently, the surface wave pro- pagation in FGPM was explained by Chaudhary et al. (2018a). Wentzel–Kramers–Brillouin (WKB) approximation method is similar to the two-time scale method which is applied to solve the problem dependent on multiple time scale. In the early 1990, the WKB method was dis- covered by WKB and hence named after them. Remarkable works have been done to reveal the prop- erties of Love-type waves in FGPM coating layer set- tled on a layer or half-space by applying WKB method. Singhal et al. (2018a), Ezzin et al. (2017) and Liu and Tani (1991, 1992) have carried out some significant work on wave transference in FGPM and FGM. Singhal et al. (2018b) and Sahu et al. (2017) have suc- cessfully applied the WKB method for modelling of wave propagation in smart material structures. Wang and Pan (2010) discussed the two-dimensional 1 Indian Institute of Technology (Indian School of Mines), Dhanbad, Dhanbad, India 2 Madanapalle Institute of Technology & Science, Madanapalle, India Corresponding author: Sanjeev A Sahu, Indian Institute of Technology (Indian School of Mines), Dhanbad, Dhanbad 826004, Jharkhand, India. Email: ism.sanjeev@gmail.com