By Kelchner, L M
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1) yields µ∇ 2 [∇ψ L + ∇ ∧ ψ T ] + (λ + µ)∇∇ · [∇ψ L + ∇ ∧ ψ T ] ∂2 = ρ 2 [∇ψ L + ∇ ∧ ψ T ]. ∂t Since ∇ · ∇ψ L = ∇ 2 ψ L and ∇ · ∇ ∧ ψ T = 0, we obtain upon rearranging terms ∇[(λ + 2µ) ∇ 2 ψ L − ρ ψ¨ L ] + ∇ ∧ [µ∇ 2 ψ T − ρ ψ¨ T ] = 0. 5), respectively. 4) are uncoupled wave equations. 9 Wave motion generated by a point load 45 substantial mathematical complications, the use of the displacement decomposition simplifies the analysis. To determine the solution to a boundary-initial value problem one may simply select appropriate particular solutions of Eqs.
To answer this question, the expression for a plane wave, Eq. 2), is substituted into the homogeneous form of the displacement equation of motion given by Eq. 20). By employing the relations ∇ · u = ( p · d) f (x · p − ct), ∇∇ · u = ( p · d) f (x · p − ct) p, ∇ 2 u = f (x · p − ct) d, u¨ = c2 f (x · p − ct) d, where the prime denotes differentiation with respect to the argument, we obtain [(λ + µ)( p · d) p + µd − ρc2 d] f (x · p − ct) = 0. 32 Wave motion in an unbounded elastic solid Hence (µ − ρc2 )d + (λ + µ)( p · d) p = 0.
The force is applied at the origin of the coordinate system, and it acts in the direction of the unit vector a. The stresses and displacements generated by a point force of time dependence g(t) = δ(t) comprise the basic singular solution of the field equations for elastodynamics. The displacements generated by a point force of arbitrary time dependence were first worked out by Stokes (1849); see also Love (1892). , Achenbach (1973). , g(t) = F0 e−iωt . 2) 46 Wave motion in an unbounded elastic solid The magnitude of F0 is generally taken as unity; this symbol is introduced to keep track of the dimensions.