Question

Solve the wave equation for the vibration of a string stretched between the end-points x = 0 and x = L. Subject to the boundary conditions: 1) u(0, t) = 0 for t ? 0 (fixed at the end x = 0). 2) u(L, t) = 0 for t ? 0 (fixed at the end x = L). And the initial conditions: 3) u(x, 0) = F(x) (given initial displacement) Where F(x) = { x, 0 ? x ? L/2; L - x, L/2 ? x ? L 4) ?u(x,0)/?t = 0 for 0 ? x ? L (zero initial velocity)

          Solve the wave equation for the vibration of a string stretched between the end-points x = 0 and x = L.

Subject to the boundary conditions:
1) u(0, t) = 0 for t ? 0 (fixed at the end x = 0).
2) u(L, t) = 0 for t ? 0 (fixed at the end x = L).
And the initial conditions:
3) u(x, 0) = F(x) (given initial displacement)
Where F(x) = { x, 0 ? x ? L/2; L - x, L/2 ? x ? L
4) ?u(x,0)/?t = 0 for 0 ? x ? L (zero initial velocity)

Solve the wave equation for the vibration of a string stretched between the end-points x = 0 and x = L.

Subject to the boundary conditions:
1) u(0, t) = 0 for t ? 0 (fixed at the end x = 0).
2) u(L, t) = 0 for t ? 0 (fixed at the end x = L).
And the initial conditions:
3) u(x, 0) = F(x) (given initial displacement)
Where F(x) = x, 0 ? x ? L/2; L - x, L/2 ? x ? L
4) ?u(x,0)/?t = 0 for 0 ? x ? L (zero initial velocity)

Added by Randy H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/10/2023

Step 1

The wave equation in one dimension is given by \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\), where \(u(x, t)\) is the displacement of the string at position \(x\) and time \(t\), and \(c\) is the speed of wave propagation in the Show more…

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Solve the wave equation for the vibration of a string stretched between the end-points x = 0 and x = L. Subject to the boundary conditions: 1) u(0, t) = 0 for t ≥ 0 (fixed at the end x = 0). 2) u(L, t) = 0 for t ≥ 0 (fixed at the end x = L). And the initial conditions: 3) u(x, 0) = F(x) (given initial displacement) 4) ∂u(x,0)/∂t = 0 for 0 ≤ x ≤ L (zero initial velocity)