PHYSICS 312. Midterm Examination, March 2 2001

Answer all three questions. Allowed aids two sheets of formulas. Calculator.

Problem 1 :
Consider the functions

$$f(x)=\sin(x)+\cos(x)$$

defined only in the restricted range $0<x<\pi$ and

$$g(x)=\left\{\begin{array}{lcl}\sin(x)+\cos(x)& for&2n\pi<x<(2... ...-\infty,\cdots -2,-1,0,1,2,\cdots\infty\\ \end{array}\right\}$$

defined for all real x.
a: Which of the following statements are true

1.

g(x) is the periodic extension of f(x).

2.

g(x) is the odd periodic extension of f(x).

3.

g(x) is the even periodic extension of f(x).

4.

none of the above.

Justify your answer. We make a Fourier expansion of the function g(x)

$$g(x)=\alpha_0+\sum_{n=1}^\infty(\alpha_n\cos(nx)+\beta_n\sin(nx)))$$

$$\alpha_0=\frac{1}{2\pi}\int_0^{2\pi}dxg(x)$$

$$\alpha_n=\frac{1}{\pi}\int_0^{2\pi}dxg(x)\cos(nx)$$

$$\beta_n=\frac{1}{\pi}\int_0^{2\pi}dxg(x)\sin(nx)$$

b: Which of the following are true

1.

all the coefficients $\beta_n=0$.

2.

the coefficients $\alpha_n=0$ for n>1.

3.

none of the above.

c: Calculate the coefficient $\alpha_0$.

Problem 2:
Consider the following eigenvalue problem

$$\frac{d^2\phi}{dx^2}+2\frac{d\phi}{dx}+\lambda\phi=0$$

$$\phi(0)=0$$

$$\phi(a)=0$$

a: Find the eigenvalues and eigenfuntions for $\lambda>1$.
b: Are there any eigenvalues for $\lambda<0$? for $\lambda <1$? Is $\lambda=1$ an eigenvalue?


Problem 3:
Consider the heat conduction problem

$$\frac{\partial}{\partial x}(\kappa(x)\frac{\partial T(x,t)}{\partial x})=\frac{1}{k}\frac{\partial T}{\partial t}$$

The ends x=0 and x=L are kept at the temperatures T=0 and T=T₀, respectively, and the thermal conductivy depends on the position

$$\kappa(x)=\frac{\kappa_0}{1+\alpha x},\;\; \alpha >0,\;\;\kappa_0>0$$

Find the steady state temperature T_S.
END OF EXAMINATION

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