Fourier transform for an oscillating string. Transverse oscillations of an elastic string are described by the displacem

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Fourier Transform For An Oscillating String Transverse Oscillations Of An Elastic String Are Described By The Displacem 1 (170.81 KiB) Viewed 27 times

Fourier transform for an oscillating string. Transverse oscillations of an elastic string are described by the displacement amplitude u(x, t), which obeys the wave equation 32 1 32 2 t2 (a) Consider the spatial Fourier transform (FT) of the displacement amplitude, rਨੇ (z, |u(x, t) = 0. ülk,t) ſulx,t)e-ikt da. [5] Define the inverse FT, substitute it into the wave equation and thus derive the ordinary differential equation obeyed by ū(k, t). Show that the general solution of this equation can be written as ū(k, t) = Ã(k)eickt + B(k)e-ickt where Ã(k) and B(k) are arbitrary (differentiable) functions of k. (b) Consider the initially undistorted string that was plucked at t = 0 at a point x = 0 with the transverse velocity v as '(x,t=0) = vd(x/a), where a is a positive constant. Reduce the general solution found in part (a) to ūo(k, t) that satisfies the initial condition. (c) Find the particular solution, uo(r,t), of the wave equation obeying the initial condition by taking the inverse FT of the uo(k, t) found in part (b). ( Hint: You may find the relation . [cik(z+c) – (2-es)] = S £ dę usetul. [et– ( [5] [5] ct eikt Jr-