**Question 7** (Unit 14) $\,$ - $\, 17$ marks

The vibration of a column of air in an organ pipe can be modelled by the partial differential equation $$\frac{\partial^2u}{\partial x^2} = \frac{1}{V^2}\frac{\partial^2u}{\partial t^2} \quad(0 < x < L, t > 0),\hspace{2.2in}(2)$$ where $V > 0$ is a constant.

One end of the pipe is closed, which corresponds to the boundary condition

$u(0,t) = 0$, for $t > 0$. The other end of the pipe is open, which corresponds

to the boundary condition $u_x(L,t) = 0$, for $t > 0$.

Applying the method of separation of variables, with $u(x,t) = X(x)T(t)$,

gives the boundary value problem $$X'' = \mu X,\quad X(0) = 0, \quad X'(L) = 0,\hspace{2.25in}(3)$$ and the differential equation $$\ddot{T} = \mu V^2T.\hspace{4.25in}(4)$$ (a) Suppose that $\mu < 0$, so $\mu = -k^2$ for some $k > 0$. Find the non-trivial

$\hspace{18px}$ solution $X(x)$ that satisfies equations (3), stating clearly what values $k$

$\hspace{18px}$ is allowed to take. $\hspace{4.5in}$ [8]

(b) Write down the general solution of equation (4) for the case $\mu = -k^2$. $\hspace{0.86in}$ [2]

(c) You may assume that if $\mu \geq 0$, then only the trivial solution satisfies

$\hspace{18px}$ equation (3). Use this assumption to write down the general solution of

$\hspace{18px}$ the partial differential equation (2) that satisfies the boundary

$\hspace{18px}$ conditions, by combining your solutions to parts (a) and (b). $\hspace{1.64in}$ [2]

(d) Write down the solution that corresponds to the initial conditions

$\hspace{23px} u_t(x,0) = 0$ and $u_t(x,0) = 2\sin\left(\frac52\frac{\pi x}{L}\right). \hspace{2.75in}$ [5]