GATE 2026 MA – Question 43
$u_{tt}=c^2u_{xx}$ on ℝ, t > 0,c > 0, with initial displacement 1 for |x| < 1 and zero otherwise, and initial velocity zero. Find maximum u(0,t) for t > 0.
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Correct answer: (A) 1
Explanation
D’Alembert gives $u(0,t)=[f(-ct)+f(ct)]/2$. For 0 < t < 1/c both terms equal one, so u = 1; for t ≥ 1/c both are zero under the given boundary convention. Maximum is **1, A**.