Existence and uniqueness of Rayleigh waves with normal impedance boundary conditions and formula for the wave velocity

The existence and uniqueness of Rayleigh waves propagating in compressible isotropic elastic half-spaces subject to tangential impedance boundary conditions (at the surface of half-spaces, the tangential stress is proportional to the horizontal displacement and the normal stress vanishes) were investigated by Godoy et al. (Wave Motion 49:585–594, 2012) and Vinh and Xuan (Eur J Mech A 61:180–185, 2017). In this paper, the half-spaces are assumed to be subject to normal impedance boundary conditions (at the surface of half-spaces, the normal stress is proportional to the vertical displacement and the tangential stress is zero). Our main aim is to establish the existence and uniqueness of Rayleigh waves and to derive formula for their velocity. These problems have been settled by using the complex function method which is based on Cauchy-type integrals. The sets of material and impedance parameters on which a Rayleigh wave is possible and is impossible have been found along with explicit formula for the wave velocity. It has been proved that if a Rayleigh wave exists, then it is unique. Interestingly, unlike the case of tangential impedance boundary conditions where a Rayleigh wave is always possible, for the case of normal impedance boundary conditions, there not always exists a Rayleigh wave. This fact suggests a way to prevent the propagation of surface Rayleigh waves which are main destructor when earthquakes occur.