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i. Show by direct substitution that u(x, t) = 1⁄2H(x − at) + ²H(x + at), where H is an arbitrary function, is a solution to the wave equation - a²uxx, aЄR, t≥0, -∞ < x <∞. Utt ii. Explain concisely how the initial displacement, u(x, 0), will evolve over time? You may include a plot if it helps your explanation.

i. Show by direct substitution that u(x, t) = 1⁄2H(x − at) + ²H(x + at), where H is an arbitrary function, is a solution to the wave equation - a²uxx, aЄR, t≥0, -∞ < x <∞. Utt ii. Explain concisely how the initial displacement, u(x, 0), will evolve over time? You may include a plot if it helps your explanation.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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i. Show by direct substitution that = H (x+at), 1/H(x+at), 2 where H is an arbitrary function, is a solution to the wave equation a²uxx, a eR, t≥0, -∞ < x < ∞. Utt - u(x, t): = 1/2H(x- H(x − at) + ii. Explain concisely how the initial displacement, u(x, 0), will evolve over time? You may include a plot if it helps your explanation. b. An infinitely long string with zero initial displacement is subject to the initial velocity 0, if x < -1 10(x+1), if −1≤ x ≤ 0 10(1x), if 0<x<1 if 1 < x. (x) = 0, If the wave speed is c = = 1, find the displacement of the string, u(x, t), at subsequent times.
Transcribed Image Text:i. Show by direct substitution that = H (x+at), 1/H(x+at), 2 where H is an arbitrary function, is a solution to the wave equation a²uxx, a eR, t≥0, -∞ < x < ∞. Utt - u(x, t): = 1/2H(x- H(x − at) + ii. Explain concisely how the initial displacement, u(x, 0), will evolve over time? You may include a plot if it helps your explanation. b. An infinitely long string with zero initial displacement is subject to the initial velocity 0, if x < -1 10(x+1), if −1≤ x ≤ 0 10(1x), if 0<x<1 if 1 < x. (x) = 0, If the wave speed is c = = 1, find the displacement of the string, u(x, t), at subsequent times.
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