PHYSICS 312
1999 Midterm eview problem set:

Problem 1: In a heat conduction problem the steady state temperature T_S satisfies the differential equation

$$\frac{d^2T_S}{dx^2}=h(T_S-T_0)$$

with boundary conditions

T_S(0)=T_S(a)=0

find T_S.

This problem was assigned again as problem 1 of Problem set 2, year 2000, for solution click here.

Problem 2: At Watson Lake in Yukon the average temperature through the year is -2^oC. The average daily temperature reaches a maximum of 14^oC in July 15. Assume that the average daily temperature varies sinusoidally through the year.

a:

Obtain an expression for the temperature as a function of time and distance below the ground surface.

b:

At what depth will the ground be permanently frozen. Assume one year is $3\;10^7\;s$, and that the constant k in the heat equation is $k=3\;10^{-7}m^2s^{-1}$ a

Problem 3:

a:

For which values of $\lambda$ will

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

have solutions satisfying

$$\phi(0)=\phi(a)=0$$

b:

Find the eigenfunctions $\phi_\lambda(x)$

This problem was assigned again as problem 1 of Problem set 3, year 2000, for solution click here.

Problem 4:

a:

Find the solution to the wave equation

$$\frac{\partial^2u}{\partial t^2}=\frac{1}{c^2}\frac{\partial^2u}{\partial t^2},\;-\infty<x<\infty;\; t>0$$

satisfying

$$u(x,0)=a\; for \;\vert x\vert<a; u(x,0)=0\; otherwise$$

$$\frac{\partial u(x,t)}{\partial t}\vert _{t=0}=0$$

Hint: try solutions on the form

f(x+ct)+g(x-ct)

b:

Sketch the solutions for times

$$t=\frac{a}{2c},\;\frac{a}{c},\;\frac{2a}{c}$$

This problem was assigned again as problem 2 of Problem set 3, year 2000, for solution click here.

Problem 5: Can you find functions $\phi(x-ct)$ that satisfy the one dimensional heat equation

$$\frac{\partial^2u(x,t)}{\partial x^2}=\frac{1}{k}\frac{\partial u(x,t)}{\partial t}$$

i.e. can you find traveling wave solutions to the heat equation? Can such a solution be realized physically?

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