19 1. Wave propagation Table 1 shows the range of values of propagation velocities, VP and VS, and the densities of principal rock types. It also gives the expressions of the main mechanical modules (Poisson’s coefficient, Young’s modulus…). 1.1 Seismic wave equation In seismic prospecting, the energy generated by the seismic source is relatively weak and the medium can be considered as elastic, obeying Hooke’s laws. For small deformations, each stress tensor (σi,j) is a linear combination of the elements of the deformation tensor (εi,j). The constants of proportionality for a homogeneous and isotropic medium are Lamé’s constants λ and m. The parameter m is termed the shear modulus. The displacements Ui (components of the displacement vector) that are observable at all points within the medium and, particularly, on the surface are solutions to the wave equation. In a three-dimensional rectilinear frame of reference (xi, i=1 to 3) the wave propagation in the x-direction is written as: ρ σ ∂ ∂ = ∂ ∂ ( ) = ∑ 2 2 1 3 U t x i j i j j , with: σ λ ε δ µ ε i j k k k i j i j , , , , = ( ) + ( ) = ∑1 3 2 stress εi j i j j i U x U x , = ∂ ∂ + ∂ ∂ 1 2 strain where δi j , are the Kronecker symbols: δi j i j i j , = = ≠ 1 0 For each element of the stress tensor (σi,j) the first index i indicates the stress component in the reference system (xi, i=1 to 3); the second index j is the surface undergoing the stress, the surface being defined by its normal in the reference system. All waves (body waves and surface waves) are solutions to the wave equation. The compressional P-waves correspond to longitudinal vibrations which, at every point in the medium, have a particle motion parallel to the direction of propagation (Figure 1.1-a). The propagation velocity for compressional waves is equal to: VP 2 = (λ + 2m)/ρ The shear S-waves correspond to transverse vibrations which, at every point in the medium, have a particle motion perpendicular to the direction of propagation (Figure 1.1-b and c). The propagation velocity for shear waves is equal to: VS 2 = m/ρ
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