2019-04-27T09:26:14+00:00 sample questions

Assuming that u(x,t) can be separated into a function of space x multiplied by a function of time t, show that the system is represented by two Ordinary Diﬀerential Equations

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Assuming that u(x,t) can be separated into a function of space x multiplied by a function of time t, show that the system is represented by two Ordinary Diﬀerential Equations (ODEs). State these equations, the two boundary conditions, and the two initial conditions that are necessary to solve for a unique solution. (8%)
2. Apply the boundary conditions to solve the spatial ODE. State the general solution to the time ODE. Then determine an expression for the displacement u(x,t) in terms of wave-speed, c, and the Fourier coeﬃcients. (5%)
3. After some lab testing, you discover that an initial velocity proﬁle shown in Equation 2 causes the web to vibrate continuously at a frequency of f3 = 540 Hz and that the wave-shape is exactly a 3rd mode sine wave with amplitude A = 1.0 × 10−3m. Use this information to determine the wave-speed, c, and state clearly the eigenvalues/eigenfunctions of the system. Hence present the complete analytic solution for this problem. (7%)

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