SD-Lecture07-Semi-Infinite-Media

SD-Lecture07-Semi-Infinite-Media - Soil Dynamics Lecture 07...

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Unformatted text preview: Soil Dynamics Lecture 07 Stress Waves in a Semi-Infinite Media © Luis A. Prieto-Portar, August 2006. Rayleigh waves. The stress waves we studied in Lecture #6 dealt with waves within bodies that are infinite, elastic and isotropic. There is another type of wave, called the Rayleigh wave , that can only exist near the boundary of an elastic half-space. This wave was first discovered by Lord Rayleigh in the year 1885 for an elastic half-space as shown below. A plane wave along the surface boundary of an elastic media. 1 The x y plane is the surface of the half space. Let u and w be the displacements in t he x and z directions, independent of y.Therefore, u and w ( ) x z z x where and are two potential functions. The dilation can be defined as φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ ε − − ∂ ∂ ∂ ∂ = + = − ∂ ∂ ∂ ∂ ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 x y z y , u v w x y z ( ) x x z z x z x z and the rotation in t he x z plane is given by, u w z x x z Re place ( ) and ( ) int o the compression wave di ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε φ ψ φ ψ φ φ φ ψ φ ψ φ φ φ ψ φ ψ φ φ φ ψ φ ψ φ φ ε φ ε φ ε φ ε φ ψ ψ ψ ψ ψ ψ ψ ψ ω ψ ω ψ ω ψ ω ψ ∂ ∂ ∂ = + + = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + − = + = ∇ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ = − = + = ∇ ∂ ∂ ∂ ∂ ( ) 2 2 2 fferential equation, u G G u t x ε ρ λ ρ λ ρ λ ρ λ ∂ ∂ = + + ∇ ∂ ∂ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 4 3 4 G G t x z x x z or G G ( ) x t z t x z and G G ( ) z t x t z x Equations ( ) and ( ) will be satisfied if φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ φ ψ ρ λ φ ρ λ φ ρ λ φ ρ λ φ φ ψ φ ψ φ ψ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ φ ψ φ ψ φ ψ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ ρ ρ λ φ ψ ∂ ∂ ∂ ∂ ∂ ∂ + = + ∇ + ∇ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = + ∇ + ∇ ∂ ∂ ∂ ∂ ∂ ∂...
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SD-Lecture07-Semi-Infinite-Media - Soil Dynamics Lecture 07...

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