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1Tacoma Narrows BridgeThe equation of the wave equation isytt=c2yxx. To discretise the motion, use the central difference stencils,which means it is second-order accurate in both time and space:ytt|t=nx=i=yn+1i-2yni+yn-1iΔt2(1)yxx|t=nx=i=yni+1-2yni+yni-1Δx2(2)The overall equation of motion can be written as:yn+1i-2yni+yn-1iΔt2=c2yni+1-2yni+yni-1Δx2(3)Simplifying gives:yn+1i= 2yni-yn-1i+Δt2c2Δx2(yni+1-2yni+yni-1)(4)The eigenvalues of the wave equation can be solved through the separation of variables:y=F(x)G(t)F(x)Gtt(t) =c2Fxx(x)G(t)(5)The equation can be split into two ODEs:F00(x) +p2F(x) = 0(6)G00(t) +c2p2G(t) = 0(7)Due to the boundary conditions,F(x) has the form ofFn(x) =Bnsin(px), wherep=nπ/L. The initial tem-poral BC ofy(x,0) = 0.125 sin(3πx/L), with spatial boundary conditions on the endsy(0, t) =y(L, t) = 0.Equating coefficients, gives the only non-zeroFn(x) atnequals to 3.