ass2 - AMATH 351 Due Friday Assignment No 2 Fall 2005 1...

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AMATH 351 Assignment No. 2 Fall 2005 Due: Friday, September 30, 2005 1. Recall that in class, we studied the clamped string problem to arrive at the PDE, 2 y ∂t 2 = T ρ 2 y ∂x 2 . (1) Here y ( x, t ) denotes the displacement of the string from equilibrium. Using separation of variables and the boundary conditions y (0 , t ) = y ( L, t ) = 0, we arrived at the following solutions y n ( x, t ) = u n ( x )cos( ω n t ) , n = 1 , 2 , 3 , . . . , (2) where u n ( x ) = sin p nπx L P , ω n = L r T ρ . (3) Show that a linear combination of the above solutions, i.e., N s n =1 c n y n ( x, t ) , (4) is also a solution to the PDE in Eq. (1). 2. Now consider the PDE model in Eq. (1), but for a string that is clamped only at one end, i.e. y (0 , t ) = 0. (Once again, assume that the tension T and lineal density ρ are constant throughout the string.) Assume that the other boundary condition is given by ∂y ∂x ( L, t ) = 0. (This would also
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