Solution to Problems for the 1-D Wave Equation 18.303 Linear Partial Differential Equations
نویسنده
چکیده
(i) Suppose that an " infinite string " has an initial displacement x + 1, −1 ≤ x ≤ 0 u (x, 0) = f (x) = 1 − 2x, 0 ≤ x ≤ 1/2 0, x < −1 and x > 1/2 and zero initial velocity u t (x, 0) = 0. Write down the solution of the wave equation u tt = u xx with ICs u (x, 0) = f (x) and u t (x, 0) = 0 using D'Alembert's formula. Illustrate the nature of the solution by sketching the ux-profiles y = u (x, t) of the string displacement for t = 0, 1/2, 1, 3/2. 1 � � x+t � u (x, t) = f (x − t) + f (x + t) + g (s) ds 2 x−t In this case g (s) = 0 so that 1 u (x, t) = (f (x − t) + f (x + t)) (1) 2 The problem reduces to adding shifted copies of f (x) and then plotting the associated u (x, t). To determine where the functions overlap or where u (x, t) is zero, we plot the characteristics x ± t = −1 and x ± t = 1/2 in the space time plane (xt) in Figure 1. For t = 0, (1) becomes 1 u (x, 0) = (f (x) + f (x)) = f (x) 2 1 Fall 2006
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تاریخ انتشار 2005