a dy 4.) Ideal Wave Equation (20 pts.) Consider the 1-D ideal wave equation for the displacement y(t, x) on a vibrating

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A Dy 4 Ideal Wave Equation 20 Pts Consider The 1 D Ideal Wave Equation For The Displacement Y T X On A Vibrating 1 (80.45 KiB) Viewed 30 times

a dy 4.) Ideal Wave Equation (20 pts.) Consider the 1-D ideal wave equation for the displacement y(t, x) on a vibrating string as a function of position (2) and time (t): dy dt? (3) d.x2 where c denotes the propagation speed on the string. (a) Write down the general form of the solution of the wave equation (3). (b) Derive the symmetric finite-difference form of the discretized second-order derivative for an arbitrary smooth function f(1). (c) Apply your result from part (b) to show that the discretized space-time evolution of y can be obtained as Yn+1,1 = r²(yn,i+1 + yn,1-1) +2(1 – r?)yn i – Yn-1,i , where the notation yn i = y(tr, Ii) has been used with tn = nAt and Ii = iAt, and r=c/(Ac/At). Comment on a suitable choice for r in the numerical calculations. (d) If i = 1, ..., 20 and n= = 1, ..., 30, how many discrete initial values ynî are needed to solve the time evolution for the entire string? = = =