Problem Set 1 (1) Let U = 2 r z x xy 3 y3 z3 + 2 2 + 3 xz 4 and B = x̂ + 2 y 3 zyˆ + 2 2 ẑ y x x z (a) Calculate the gradient of U . r (b) Calculate the divergence of B . (c) Calculate the Laplacian of U . (d) Calculate the curl of the gradient of U . r r (e) Calculate B ⋅ (B × ∇U ) . (2) Solve following problems: (a) Using cijkl = λδ ij δ kl + µ (δ ik δ jl + δ il δ jk ) , ∆ = u k ,k , ε ij = 1 (ui , j + u j ,i ) , 2 and τ ij = cijkl ε kl , show that τ ij = λδ ij ∆ + 2 µε ij . (b) Show that u = ∇φ + ∇ ×ψ , ∇ ⋅ψ = 0 and u&& = ( ⎧ λ + 2µ lead to ∇ ⎨( ⎩ ρ λ + 2µ µ )∇(∇ ⋅ u ) − ( )∇ × ∇ × u ρ ρ ⎫ ⎧ µ ⎫ )∇ 2φ − φ&&⎬ + ∇ × ⎨( )∇ 2ψ − ψ&& ⎬ = 0 ⎭ ⎩ ρ ⎭ (3) Solve following problems: ⎧2 1 3 ⎫ ⎪ ⎪ (a) For the stress tensor σ = ⎨1 1 - 2 ⎬ , ⎪3 2 5⎪ ⎭ ⎩ find the traction on, i) the x-y plane, ii) the y-z plane, iii) the plane with normal (3, 2, -1). ⎧0 2 0⎫ ⎪ ⎪ (b) For the stress tensor σ = ⎨2 0 0⎬ , i) find the principal stresses and their ⎪0 0 0 ⎪ ⎭ ⎩ associated directions, ii) find the surfaces on which the maximum tangential traction occurs, and the value of this traction. ⎧3 0 0⎫ ⎪ ⎪ (c) For the strain tensor e = ⎨0 1 1 ⎬ , find the corresponding stress tensor, ⎪0 1 2 ⎪ ⎭ ⎩ assuming an isotropic solid with Lamé constants λ and µ. (4) For waves propagating in an arbitrary direction given by the wavenumber vector k, (a) Show that the P-wave displacement due to the scalar potential φ ( x, t ) = e i (ωt − k ⋅x ) is parallel to the propagation direction. (b) Show that the S-wave displacement due to vector potential γ ( x, t ) = Ae i (ωt − k ⋅ x ) , A=(Ax, Ay, Az), is perpendicular to the propagation direction. (5) The radii of the Earth, Moon, and Sun are 6,371 km, 1,738 km, and 695100 km, respectively. Make a rough estimate of how long it takes a P-wave to traverse the diameter of each body. (6) Estimate how many seconds earlier a vertically upgoing S-wave will arrive at a seismic station in the middle of Canada, compared to a station in the eastern Pacific. Ignore any topographic or crustal thickness differences between the sites; consider only the integrated travel time difference though the upper mantle.