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{\large\bf Stony Brook University - MAT 341 Midterm II}

November 15, 2005\vs

This test is open book: Powers ``Boundary Value Problems'' may
be consulted. No other references or notes may be used. \vs

Students may use graphing calculators like TI-83, 85, 86; but
they may NOT use calculators with Computer Algebra Systems,
like TI-89.\vs

Total score = 100.
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\begin{enumerate}
 \item

 The heat equation in a laterally insulated bar of length 2
with fixed
temperature at one end ($x=0$) and insulated at the other ($x=2$) leads to the eigenvalue problem

$$\phi''(x) + \lambda^2\phi(x) = 0 ~~~~~~~~~0\leq x\leq 2$$
$$\phi(0) = 0 ~~~~~~~~~\phi'(2) = 0.$$

  \begin{enumerate}
   \item (20 points)
Calculate the eigenvalues and eigenfunctions for this problem.

   \item (10 points)
Are these eigenfunctions orthogonal, i.e. does
$$\int_0^2 \phi_m(x) \phi_n(x)~dx = 0$$
where $\phi_m(x)$ and  $\phi_n(x)$ are eigenfunctions corresponding
to two eigenvalues $\lambda_m^2 \neq \lambda_m^2$? Explain your answer
carefully.
  \end{enumerate}

\newpage


 \item The heat equation in a bar of length 2
 immersed in a medium of constant
temperature, insulated at its left end ($x=0$)
 and exchanging heat with the medium
along its length and at its right end ($x=2$),  leads (simplifying the constants)
to the eigenvalue problem

$$\phi''(x) - \phi(x) + \lambda^2\phi(x) = 0 ~~~~~~~~~0\leq x\leq 2$$
$$\phi'(0) = 0 ~~~~~~~~~\phi'(2) - \phi(2)= 0.$$

  \begin{enumerate}
   \item (15 points)
Are the eigenfunctions for this problem orthogonal, i.e. does
$$\int_0^2 \phi_m(x) \phi_n(x)~dx = 0$$
where $\phi_m(x)$ and  $\phi_n(x)$ are eigenfunctions corresponding
to two eigenvalues $\lambda_m^2 \neq \lambda_n^2$? Explain your answer
carefully.

 \item (20 points)
Calculate the first (the smallest positive) eigenvalue for this problem. 
  
  \end{enumerate}

 \item

The vibrations of a cord of length 2 fixed at both
ends are governed by the wave equation

$$ \frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 u} {\partial t^2}~~~~~~~~~0\leq x\leq 2$$
$$u(0,t) = 0 ~~~~~~~~~u(2,t) = 0.$$

Suppose that at $t=0$ the cord is not moving, i.e. $\partial u/\partial t(x,0) = 0$, and that the graph of the initial position
function $f(x) = u(x,0)$ shows two adjacent equilateral triangles of 
base 1,
like $\wedge\wedge$.\vs
  \begin{enumerate}
   \item (25 points)
Sketch the d'Alembert solution of this problem for $t = 3/(2c)$. 
   \item (10 points)
The sound spectrum of the resulting vibration will have
a component at frequency ${\displaystyle \frac{c\lambda_m}{2\pi}}$
if the eigenfunction corresponding to $\lambda_m$ occurs with
non-zero coefficient in the eigenfunction expansion of $f(x)$.
Which frequencies will be heard in the vibration of the cord with
this initial condition?

  \end{enumerate}
\end{enumerate}

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