Question

5. Vibration of a string of length $\pi$ that is subject to tension and damping is governed by $\frac{\partial^2 u}{\partial t^2} + 2\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$, $0 < x < \pi$, $t > 0$, (5) where $u(x, t)$ is the transverse displacement of the string at a position $x$ along the string and time $t$. The ends of the string are fixed, hence the boundary conditions are $u(0, t) = u(\pi, t) = 0$, $t > 0$. (6) Suppose that the initial displacement and velocity are $u(x, 0) = 2\sin x$, $u_t(x, 0) = 0$, $0 < x < \pi$, (7) respectively. Use separation of variables to solve equation (5) subject to boundary conditions (6) and initial conditions (7). If applicable, you may use the result that the eigenvalues of the boundary-value problem $X'' - \lambda X = 0$, $X(0) = 0$, $X(\pi) = 0$, are $\lambda_n = -n^2$ for $n = 1, 2, \dots$, with corresponding eigenfunctions $X_n(x) = \sin nx$. [TOTAL 14 marks]

          5. Vibration of a string of length $\pi$ that is subject to tension and damping is governed by
$\frac{\partial^2 u}{\partial t^2} + 2\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$, $0 < x < \pi$, $t > 0$,
(5)
where $u(x, t)$ is the transverse displacement of the string at a position $x$ along the string
and time $t$. The ends of the string are fixed, hence the boundary conditions are
$u(0, t) = u(\pi, t) = 0$, $t > 0$.
(6)
Suppose that the initial displacement and velocity are
$u(x, 0) = 2\sin x$, $u_t(x, 0) = 0$, $0 < x < \pi$,
(7)
respectively.
Use separation of variables to solve equation (5) subject to boundary conditions (6) and
initial conditions (7).
If applicable, you may use the result that the eigenvalues of the boundary-value problem
$X'' - \lambda X = 0$, $X(0) = 0$, $X(\pi) = 0$,
are $\lambda_n = -n^2$ for $n = 1, 2, \dots$, with corresponding eigenfunctions
$X_n(x) = \sin nx$.
[TOTAL 14 marks]

5. Vibration of a string of length π that is subject to tension and damping is governed by
(∂^2 u)/(∂ t^2) + 2(∂ u)/(∂ t) = (∂^2 u)/(∂ x^2), 0 < x < π, t > 0,
(5)
where u(x, t) is the transverse displacement of the string at a position x along the string
and time t. The ends of the string are fixed, hence the boundary conditions are
u(0, t) = u(π, t) = 0, t > 0.
(6)
Suppose that the initial displacement and velocity are
u(x, 0) = 2sin x, ut(x, 0) = 0, 0 < x < π,
(7)
respectively.
Use separation of variables to solve equation (5) subject to boundary conditions (6) and
initial conditions (7).
If applicable, you may use the result that the eigenvalues of the boundary-value problem
X” - λ X = 0, X(0) = 0, X(π) = 0,
are = -n^2 for n = 1, 2, …, with corresponding eigenfunctions
Xn(x) = sin nx.
[TOTAL 14 marks]

Added by Josefa B.

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