By taking influence of pre-stress into account, the governing equation of wave propagation in elastic continuum material is derived. The results indicated that the nonhomogeneous pre-stress changed the governing equation of wave propagation. The definite problem of wave propagation in material with non-homogeneous pre-stress was verified by using mathematical physics method. The displacement-time relationship of wave propagation was explored, which indicated that the non-homogeneous pre-stress changed waveform and relationship of displacement-time histories.
The propagation process of elastic wave propagation in a material and its related seismic exploration has been widely studied. In civil and mining engineering, the problems involve tectonics and earth masses that are initially under stress state. The theory of elastic wave propagation in pre-stressed solids has a long history (Tolstoy 1982). A definite theory explaining the wave propagation of a material under initial stress was developed by Biot & Drucker (1965), which clearly showed that the effect of initial stress on wave propagation cannot be represented by a change in elastic coefficients.
A large number of laboratory measurements of P-wave and S-wave velocities and their characteristics of anisotropy, hysteresis, and splitting have been conducted on various types of rocks by many researchers, such as Birch (1960, 1961), Christensen & Ramananantoandro (1971), Christensen (1974, 1979), Manghnani et al. (1974), and others (Li & Tao 2015). The results indicated that velocity of seismic waves and elastic coefficients are much influenced by initial stress, and thus it is imperative to deal with the problem of wave propagation in a material with initial stress.