Stony Brook University - MAT 341 Midterm II

November 15, 2005

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

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

Total score = 100.

1. 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

   
   

   $$\phi''(x) + \lambda^2\phi(x) = 0         0\leq x\leq 2$$
   
   
   
   
   $$\phi(0) = 0         \phi'(2) = 0.$$
   
   
   1. (20 points) Calculate the eigenvalues and eigenfunctions for this problem.

   2. (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.

2. 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

   
   

   $$\phi''(x) - \phi(x) + \lambda^2\phi(x) = 0         0\leq x\leq 2$$
   
   
   
   
   $$\phi'(0) = 0         \phi'(2) - \phi(2)= 0.$$
   
   
   1. (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.

   2. (20 points) Calculate the first (the smallest positive) eigenvalue for this problem.

3. 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 $\wedge\wedge$.
   
   

   1. (25 points) Sketch the d'Alembert solution of this problem for $t = 3/(2c)$.
   2. (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?