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Assignment_8

# Assignment_8 - t x s that satisfies the PDE and the BCs(d...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 8 Question 1 . A wire of length L under tension slides on a vertical rod at 0 = x and is fixed at L x = , as shown on the sketch below. The wire displacement () t x s , is governed by the 1-D wave equation with accompanying BCs and ICs: PDE 2 2 2 2 2 x s a t s = where constant known 2 = = A F a T ρ BCs () 0 , 0 = t x s and ( ) 0 , = t L s ICs () ( ) x f x s shape given some 0 , = and () 0 0 , = x t s (zero initial velocity) (a) Assume () ( ) ( ) t x t x s τ ϕ = , , and separate the PDE and BCs. Show that the separation constant must be negative. Find the eigenvalues and eigenfunctions. (b) ( ) t τ
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Unformatted text preview: ( ) t x s , that satisfies the PDE and the BCs. (d) Suppose ( ) x f in the IC above is given as ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = L x H x s 2 3 cos , π , where L H , are known. Complete the solution for ( ) t x s , , sketch its behavior, and identify the oscillation frequency in Hertz. (e) Suppose ( ) x f in the IC above is given as ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 2 1 , x H x s , , where L H , are known. Outline how the solution for ( ) t x s , would be completed. (Do not take the time to actually evaluate the definite integrals which appear)....
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