2. In this problem you'll determine the motion of an infinite string which is initially flat and at rest, and is given a

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2 In This Problem You Ll Determine The Motion Of An Infinite String Which Is Initially Flat And At Rest And Is Given A 1 (135.85 KiB) Viewed 11 times

2. In this problem you'll determine the motion of an infinite string which is initially flat and at rest, and is given an impulse at time t = 0 by a moving block of width 1 cm, travelling at 1 cm/sec. The vertical displacement u(x, t) of the string is the solution of the initial-value problem on the line: where Utt = 4uxx u(x,0) = 0 u₁(x,0) = g(x) 0 g(x) = 1 0 a. Let G(x) = g(w) dw. Express G(x) as a piecewise-defined function of x. Sketch the graph of y = G(x) on the line. (Hint: G(z) will be given by different formulas for x ≤ 3, for 3 ≤ x ≤ 4, and for 4 ≤ x. One good way to find these formulas correctly is to start from the definition of G(x) as f 9(w) dw, and split this integral up into a sum of integrals appropriately. For x ≤ 3, finding the value of G(z) is easy (why?). For 3≤x≤ 4, you can write G(z) = ²*9(w) dw = = S² ⁹0 and, for x ≥ 4, you will want to write. for x < 3 for 3 ≤ x ≤ 4 for > 4. [²9 G(x) = f g(w) dw= (-∞0<x<∞, t > 0) (-∞0 < x <∞0) (-∞0 < x <∞0) S g(w) dw+ g(w) dw+ 1²9 g(w) dw; S g(w) dw+ g(w) dw. To understand the piecewise-defined formula for G(x), it helps to draw the graph of g and think about the area that the value of G(x) = f 9(w) dw represents.) b. The D'Alembert formula for the solution of the initial-value problem for the wave equation on the line can be written as 1 pz+at u(x, t) = ½{f(x + at) + f (x − at)] + 2 + 9(w) dw 2a Jx-at = [ƒ(x + at) + f(x − at)] + Use this formula and your answer to part a to sketch the graph of u(x, t) as a function of a at each of the values t = 1, t = 1.75, and t = 3. How would you describe the motion of the string in words? (Hint: this can be done by hand, but it is a little tricky to get the details right. It is easier to use Desmos: first define G(x) and then graph u(x, t)=G(x+at) - G(x-at)] using a slider.) [G(x+at) - G(x-at)].