Math 121A: Homework 7-

1. Consider solving the wave equation

∂²f/∂t² = c² ∂²f/∂x²

for f(x, t) in the interval -π ≤ x ≤ π, subject to the boundary conditions

∂f/∂x|_x=-π = ∂f/∂x|_x=π = 0

and initial conditions

f(x, 0) = g(x) = (|x| - π)²,              ∂f/∂t|_t=0 = 0.

(a) Find the Fourier series of g(x).

(b) Search for separable solutions of the form f(x, t) = X(x)T(t) where X(x) is even.

(c) By using parts (a) and (b), write down a general solution for f(x, t) in terms of an infinite series.

(d) The kinetic energy and potential energy for the system can be defined as

K(t) = ½ _-π∫^π(∂f/∂t)² dx,                 P(t) = ½ _-π∫^πc²(∂f/∂x)² dx.

respectively. Let the total energy be given by E(t) = K(t) + P(t). By considering the time derivative of E and making use of integration by parts, show that the total energy is constant.

(e) By using the initial conditions for f and ∂f/∂t calculate E(0).

(f) Calculate E(t) using the series solution from part (c), and show that it is constant.

(g) Optional for the enthusiasts. Use a computer to plot f(x, t) over the range from t = 0 to t = 2π/c.

2. (a) Calculate the Fourier transform f^˜(α) of the function

 (b) Calculate the Fourier transform g^˜(α) of the function

 (c) What is f^˜(α)/g^˜(α) and why should this be expected?

(d) By using the previous answers and using basic properties of Fourier transforms, without doing any integration, determine the Fourier transform of

(e) Optional for the enthusiasts. Without doing any integration, calculate the Fourier transform of

where n is a positive integer. Plot the imaginary component of q^˜_n(α) over the range -2 ≤ α ≤ 2 for the cases of n = 10, 20, 30 and interpret the shapes of the graphs.

3. In the United Kingdom people greatly enjoy drinking tea, particularly with a splash of milk. However, there is often some debate about the correct procedure for adding the milk.

Suppose that the tea is initially at T_t = 95^oC, the room temperature is T_r = 20^oC, and the milk is kept in a refrigerator at T_m = 5 ^oC. Let the volume of the tea be V_t = 200 ml and the volume of the milk be V_m = 50 ml. Assume that the rate of change in the tea's temperature is given by λ multiplied by the difference between the tea's temperature and T_r. Derive a differential equation for the temperature of the tea over time.

Suppose now that λ = (log2)/5 min^-1 = 0.1386 min^-1. Determine the temperature of the tea after the following two alternative procedures:

• The milk is added to the tea, and the tea is left to stand for 20 minutes.
• The tea is left to stand for 20 minutes, and then the milk is added.

After which procedure is the tea hotter?