A semi-infinite string is located initially (at time t = 0) along the positive X-axis, 0 < x <∞. The string is looped around a vertical support at the 'far end', corresponding to x → ∞o, which exerts no vertical force on the tape: limu(x, t) = 0. At the near end, corresponding to x = 0, the tape is tethered to a support, which fixes its position. The initial velocity is zero throughout the tape, i.e., at ди t = 0, = 0. at ²u The displacement u(x, t) of the string obeys the wave equation - A t, where c is a at² given positive constant with units m s¹, and A is a given positive constant with units m s ³. Solve the PDE analytically for the displacement of the tape at any location at any time t > 0. = c². ² əx²
A semi-infinite string is located initially (at time t = 0) along the positive X-axis, 0 < x <∞. The string is looped around a vertical support at the 'far end', corresponding to x → ∞o, which exerts no vertical force on the tape: limu(x, t) = 0. At the near end, corresponding to x = 0, the tape is tethered to a support, which fixes its position. The initial velocity is zero throughout the tape, i.e., at ди t = 0, = 0. at ²u The displacement u(x, t) of the string obeys the wave equation - A t, where c is a at² given positive constant with units m s¹, and A is a given positive constant with units m s ³. Solve the PDE analytically for the displacement of the tape at any location at any time t > 0. = c². ² əx²
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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Transcribed Image Text:A semi-infinite string is located initially (at time t = 0) along the positive X-axis, 0 < x < ∞ . The
string is looped around a vertical support at the 'far end', corresponding to x→ ∞, which exerts no
0, the tape is
Ә
vertical force on the tape: limu(x, t) = 0. At the near end, corresponding to x =
х→00 дх
tethered to a support, which fixes its position. The initial velocity is zero throughout the tape, i.e., at
ди
t = 0, = 0.
at
อใน
əx²
The displacement u(x, t) of the string obeys the wave equation
A t, where c is a
given positive constant with units m s´¹, and A is a given positive constant with units m s ³. Solve the
PDE analytically for the displacement of the tape at any location at any time t > 0.
²u
at²
C²
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