AMME2000_Wk_6_Solutions - 1 Tacoma Narrows Bridge The equation of the wave equation is ytt = c2 yxx To discretise the motion use the central difference

AMME2000_Wk_6_Solutions - 1 Tacoma Narrows Bridge The...

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1 Tacoma Narrows Bridge The equation of the wave equation is y tt = c 2 y xx . To discretise the motion, use the central difference stencils, which means it is second-order accurate in both time and space: y tt | t = n x = i = y n +1 i - 2 y n i + y n - 1 i Δ t 2 (1) y xx | t = n x = i = y n i +1 - 2 y n i + y n i - 1 Δ x 2 (2) The overall equation of motion can be written as: y n +1 i - 2 y n i + y n - 1 i Δ t 2 = c 2 y n i +1 - 2 y n i + y n i - 1 Δ x 2 (3) Simplifying gives: y n +1 i = 2 y n i - y n - 1 i + Δ t 2 c 2 Δ x 2 ( y n i +1 - 2 y n i + y n i - 1 ) (4) The eigenvalues of the wave equation can be solved through the separation of variables: y = F ( x ) G ( t ) F ( x ) G tt ( t ) = c 2 F xx ( x ) G ( t ) (5) The equation can be split into two ODEs: F 00 ( x ) + p 2 F ( x ) = 0 (6) G 00 ( t ) + c 2 p 2 G ( t ) = 0 (7) Due to the boundary conditions, F ( x ) has the form of F n ( x ) = B n sin( px ), where p = nπ/L . The initial tem- poral BC of y ( x, 0) = 0 . 125 sin(3 πx/L ), with spatial boundary conditions on the ends y (0 , t ) = y ( L, t ) = 0. Equating coefficients, gives the only non-zero F n ( x ) at n equals to 3.
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