Project 8, MATH 481, Spring 2008
Detecting land mines

2008–05–01: Project description

We compute ground surface temperatures in the presence or absence of buried land mines. We find that the amplitude of daily variations of temperature is significantly higher near a buried land mine.

Things to do

• The soil temperature T(x,t) in the absence of buried land mines is given in equation (14), page 362. (The textbook uses U for the temperature. I think T is a better notation. You choose.) What is the amplitude of the surface temperature? Call it A.
• Let Θ₁(t) = T(0,t) / A, where T(0,t) is the solution of the head equation in the absence of land mines. (Note: The Greek letter Θ is produced with \Theta in LaTeX).
• The ground surface temperature T(-d,t) in the presence of buried land mines is given by the real part of the expression in equation (7), page 367. Give the details of the derivation of equation (7).
• Let Θ₂(t)=T(-d,t) / A, where T(0,t) is the solution of the head equation in the presence of land mines, and where A was computed above.
• Use the data given on page 367 for heat condition properties of soil and a land mine:

  α₁ = 0.6 × 10⁻⁶ m²/s,   α₂ = 0.1 × 10⁻⁶ m²/s,   k₁ = 1.2 W/(m °K),   k₂ = 0.26 W/(m °K),

  where the subscript 1 refers to the soil properties and the subscript 2 refers to the land mine properties, and where m, s, W, °K mean meters, seconds, Watts, degrees Kelvin, respectively.

  Plot &Theta₁ and &Theta₂ against time for a period of one day. Note that the time is measured in seconds, thus in Maple we may do:

      plot([Theta1, Theta2], t=0..24*3600);
  

  The result is not ideal because the range on the horizontal axis goes from 0 to 86,400, which is ugly. It would be much better if the horizontal axis went from 0 to 1, indicating a period of one day. To fix this, we replace t by 24*3600*t in the expressions that are being plotted, then let the "new" t go from 0 to 1:

      plot(subs(t=24*3600*t, [Theta1, Theta2]), t=0..1);
  

  The resulting graphs should be identical to those shown on page 368.

• Suppose that the ground surface temperature varies between 20°C and 50°C where there is no buried land mine. What is the range of temperature variations at a location with a buried land mine? Is there an appreciable difference?

Project due date

Please turn in your writeup by Friday May 9th.