Write with , and take plane-wave potentials travelling along , and…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Elastic waves in a solid and at its surface

Problem

Write u=∇ϕ+∇×ψ\mathbf u=\nabla\phi+\nabla\times\boldsymbol\psi with ∇⋅ψ=0\nabla\cdot\boldsymbol\psi=0, and take plane-wave potentials travelling along xx, ϕ=ϕ0ei(ωt−kx)\phi=\phi_0e^{i(\omega t-kx)} and ψ=ψ0z^ ei(ωt−kx)\boldsymbol\psi=\psi_0\hat{\mathbf z}\,e^{i(\omega t-kx)}. ((a)) Find the displacement produced by each potential separately. Show that the ϕ\phi-wave has displacement along xx (longitudinal) with non-zero dilatation ∇⋅u\nabla\cdot\mathbf u, while the ψ\boldsymbol\psi-wave has displacement along yy (transverse) with zero dilatation. ((b)) Substitute each potential into the elastodynamic Navier equation to show that ϕ\phi obeys a wave equation with speed cL=(λ+2G)/ρc_L=\sqrt{(\lambda+2G)/\rho} and ψ\psi obeys one with speed cT=G/ρc_T=\sqrt{G/\rho}, and hence derive cL2cT2=2(1−ν)1−2ν.\frac{c_L^2}{c_T^2}=\frac{2(1-\nu)}{1-2\nu}. ((c)) Evaluate this ratio, and cL/cTc_L/c_T, for a rock with ν=0.20\nu=0.20, and compare with the smallest value the ratio can take over the physically allowed range of ν\nu.

Browse the problem library