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pset_5b - ACM 95/100c Problem Set 5 27th April 2006 Due by...

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Unformatted text preview: ACM 95/100c Problem Set 5 27th April 2006 Due by 5:00PM on 5/5/2006 Please deposit all problem sets in the slot in 303 Firestone Please remember to include your section number and section instructor Each problem is worth 10 points - each part of a multi-part problem is weighted equally Problem 1 (No collaboration) Consider the wave equation for a vibrating rectangular mem- brane ( < x < L , < y < H ): ∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 subject to the initial conditions u ( x, y, 0) = f ( x, y ) . ∂u ∂t ( x, y, 0) = g ( x, y ) Solve the initial value problem if (a) u (0 , y, t ) = 0 , u ( L, y, t ) = 0 , u y ( x, , t ) = 0 u y ( x, H, t ) = 0 (b) u x (0 , y, t ) = 0 , u x ( L, y, t ) = 0 , u y ( x, , t ) = 0 u y ( x, H, t ) = 0 Problem 2 (No collaboration) Consider the eigenvalue problem ∇ 2 φ + λφ = 0 with boundary conditions φ x (0 , y ) = 0 φ ( x, 0) = 0 φ x ( L, y ) = 0 φ ( x, H ) = 0 (a) Show that there is a doubly in nite set of eigenvalues 1 (b) If L...
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