RESEARCH ARTICLE Transient Vibrations of a Half-Space Under a Massive Line Loading S. Asadollahi 1 • M. Dehestani 1 Received: 20 May 2015 / Revised: 2 June 2017 / Accepted: 6 December 2017 Ó The National Academy of Sciences, India 2018 Abstract Dynamic response of a half-space subjected to loads is of considerable practical interest for engineers and scientists. Recent investigations demonstrated that the inertial influences of moving loads are of importance in some cases. Although there are several works on moving inertial loads (masses), very few works were performed on the interaction of a half-space and a stationary inertial foundation on its surface. In this study, a new analytical– numerical method has been used to investigate the vertical interaction between a massive strip foundation and a homogenous, isotropic elastic half-space. Navier’s equa- tions of motion were transformed to a system of wave-type partial differential equations using the Helmholtz resolu- tion. The interactive tractions between the strip foundation and the half-space were imposed to the problem as boundary conditions. A concurrent two-sided and one- sided Laplace integral transform was used for the wave type partial differential equations subjected to the specific boundary conditions and eventually the solution in trans- formed form was obtained. In order to inverse the trans- formed solution, the Cagniard–De Hoop method accompanying a numerical procedure was implemented. Final results revealed the influence of the inertia of the massive strip foundation on the dynamic response of the half-space. Keywords Wave propagation  Inertia  Integral transform  Steady-state response  Parametric analysis 1 Introduction Propagation of waves in a half-space is subject of intensive research for engineers and scientists. This is mainly because of its applications in design of machine founda- tions, foundation of tracking radars and design of earth- quake resistant foundations for buildings. In recent years, significant progress in technologies induced wide range of speeds, dynamic loads and operating conditions to the foundations. To avoid any damage, mal-function and unacceptable motions due to dynamic loads, rigorous analysis and design are needed. Therefore, obtaining the exact influences of these loads is of great importance. Over the years, numerous studies have been conducted to investigate wave propagation in ground. In a pioneering work, Lamb [1] studied horizontal and vertical concen- trated loads on an elastic isotropic half-space in 2D and 3D cases. He used Fourier Integral transform in combination with Bessel functions and contour integration in his ana- lytical treatment. Vertical and torsional dynamic loads at the surface of an elastic half-space with a circular contact patch have been investigated by Reissner [2]. Between the 1950s and the 1960s, many researchers extended Reiss- ner’s solution to study elastic half-space with various modes of vibration and shaped loading areas [3–7]. Bycroft [3] derived the surface displacement for an elastic half-space under static stress distribution for low frequency range. Awojobi and Grootenhuis [4] obtained the dynamic stress distributions under the rigid circular and strip footing. The solution was based on the dual integral & M. Dehestani dehestani@nit.ac.ir S. Asadollahi sina.asdllhi@gmail.com 1 Faculty of Civil Engineering, Babol Noshirvani University of Technology, Postal Box 484, Babol 47148-71167, Islamic Republic of Iran 123 Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. https://doi.org/10.1007/s40010-017-0479-x