Lamb's Problem with Internal Dissipation. I

A theoretical investigation is made of the response of an elastic half-space to a vertical impulse at the surface. Dissipative properties are incorporated in the medium and dissipation coefficients λ′, μ′ varying as the nth power of the frequency are assumed. These lead to dispersion and absorption. The response may be described by components of the three distinct types, P-, S-, and R-. Each is observed at the surface as a train of waves culminating in a characteristic phase about the instants r/α, r/β, and r/γ, respectively, where α, β, and γ are the velocities of propagation of P-, S-, and R- waves in nondissipative media and r is the focal range. These phases are discussed in detail; their “amplitude” and “span” are defined and the variations of each of these quantities and their ratios with r, density ρ and the material constants are derived. A schematic determination of λ and μ (the elastic constants), n, λ′, and μ′ from experimental data is outlined.