Reference no: EM131077412
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
![644_Figure.png](https://secure.expertsmind.com/CMSImages/644_Figure.png)
(b) Calculate the Fourier transform g˜(α) of the function
![2208_Figure1.png](https://secure.expertsmind.com/CMSImages/2208_Figure1.png)
(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
![1698_Figure2.png](https://secure.expertsmind.com/CMSImages/1698_Figure2.png)
(e) Optional for the enthusiasts. Without doing any integration, calculate the Fourier transform of
![1769_Figure3.png](https://secure.expertsmind.com/CMSImages/1769_Figure3.png)
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 Tt = 95oC, the room temperature is Tr = 20oC, and the milk is kept in a refrigerator at Tm = 5 oC. Let the volume of the tea be Vt = 200 ml and the volume of the milk be Vm = 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 Tr. 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?