﻿ MAT 461 Homework

# MAT 461 Homework

### Fall, 2013

Extra credit opportunity: Report to the class on Friday, August 30 (Week 1, Day 2)
Predict the sound frequencies of the two notes I made on the 22 cm flute mouthpiece.
Here are power spectra for the the flute mouthpiece with open and closed ends: flute 1 and flute 2 obtained with the Audacity program. Which is which?

In class on Friday, Nov. 1, I made predictions based on my assumption that the flute has a closed end near the mouth hole (embourchure). My predictuons did not make sense, because in reality the embourchure itself acts like an open end. So the mouthpiece with an open end has two open ends, and the mouthpiece with a closed ends has an open and a closed end. The length of the mouthpiece from the embourchure to the end is about 15.5cm. See open vs. closed pipes

Due Friday, Sept. 6 (Week 2, day 2)
First Assignment

Assignment 2
Due Friday, Sept. 13 (Week 3, day 3)

1 (a). Suppose that a coffee cup filled with water is placed on the desk in our classroom at t = 0. The water is originally at 95 degrees C, and the room is at 20 degrees C. Considering only the heat capacity of the water and the air in the room, and the conservation of energy, estimate the temperature in the room after waiting a long time. Assume that the windows and doors of the room are closed, and that the walls are insulating. There is a range of ``right'' answers. You will need to research the specific heat and density of air and water. You will also need to come up with reasonable estimates for the volume of the room and of the coffee cup.

1(b). Comment on your prediction for the temperature rise in part (a). How could you control the experiment to make the change in temperature measurable? (Think about windows, doors, and people.)

Do problems 1.2.1, 1.2.5, 1.3.1, 1.3.2, 1.4.1, and 1.4.6 from the textbook (Haberman 4e or 5e).

Assignment 3
Due Friday., Sept. 20 (Week 4, day 3)
1.5.1, 1.5.3, 1.5.11
2.2.2, 2.2.3

Assignment 4
Due Friday, Sept 27 (Week 5, day 3)
2.3.1, 2.3.2, 2.3.3(a-d), 2.3.8, 2.3.9

Hint: in problem 2.3.2(g) Just get an equation for the eigenvalue lambda. You cannot find the eigenvalues in closed form, so don't even try. (You would need to fix a value of L and then find the eigenvalues numerically, so don't even try. If you are interested, here is a Mathematica notebook that can be used to approximate the eigenvalues. This notebook also has a description of a possible class project.)

In class I was asked about the reason for equation (2.3.32) in the book, which computes the integral of sin(n pi x/L) sin(m pi x/L) from 0 to L. This scanned page, orthogonalityOfSines.pdf, proves it. The u substitution u = pi x/L is not necessary, but I find that it makes this integral, and many of the integrals in the homework, nicer. (Don't forget that you need to change the limits in the definite integral!)

Assignment 5
Due Oct 4 (Week 6, day 3)
2.4: 1(b-d), 2, 3, 6
2.5: 1(a-d), 2.
NOTE: 2.5: 3, 7, 8, 10. These are delayed until next week.
Here is a contour map of the solutions to 2.5.1g, with L/W =1 and L/W =3. This is a solution to Laplace's equation on the rectangle with three insulating boundaries, and a specified temperature (a step function) on the bottom boundary.

Ground-rules for the exam on Monday, Oct. 7.
The exam is open book and you may bring one page of notes with handwriting on both sides. You may bring any calculator, but a calculator is not needed. You may use any formula in the book without justifying it.
There will be two textbooks available for students to use.

Assignment 6
Due Oct. 11 (Week 7, day 3)
2.5: 3, 7, 8, 10
Non-book problem: Write the vector v = (1, 0, -1) as a linear combination of the orthogonal vectors v1 = (1, 2, 3), v2 = (3, 0, -1), and v3 = (1, -5, 3). See the Appendix to 2.3:

Assignment 7
Due Friday, Oct. 18 (Week 8, day 3)
3.2: 1(a, c, e, g), 2(d, e),
3.3: 1, 3(c), 8, 15
Use a computer to plot the series in 3.3.3(c) with different values of the cut off.

Assignment 8
Due Friday, Oct. 25 (Week 9, day 3)
3.4: 1, 2, 6
3.5: 2(a), 4
3.6: 1, 2

Due Mon, Oct. 28
Proposals for project. This can be as short as a paragraph, or as long as a few pages. I plan to talk about a few possible projects that involve numerical solutions to eigenvalue problems on Friday, Oct. 25.

Assignment 9
Due Monday, November 4 (Week 11, Day 1)
(a) Use trig. identities, or sin(x) = (ei x - e-i x)/(2i), or sin(x) = Im(ei x), to show that sin(x-h) -2 sin(x) + sin(x + h) is proportional to sin(x), and find the constant of proportionality.
(b) Use the computation from part (a) plus scaling, or a similar computation to the one in part (a), to find the eigenvalues and eigenvectors of the L matrix derived in class that approximates the eigenfunction problem φ'' = λ φ, φ(0) = φ(π) = 0. Label the eigenvectors with a positive integer n, so L vn = λnvn. The components of the eigenvector vn can be called vn, i. The eigenvalues will depend on n and N, the number of subdivisions of [0, π].
4.2: 1
4.3: 1
4.4: 1, 2, 3, 7

Assignment 10 (This part is delayed.)
Due Friday, November 8 (Week 11, Day 3)
12.2: 3, 4
12.3: 1, 6

Exam 2, on what we covered in chapters 3, 4, and 12, is scheduled for Wednesday, Nov. 13 (week 12, day 1). Here is the sample midterm I handed out in class.

Wednesday, Nov. 20 (Week 13, Day 2)
First draft of projects due by midnight. Email me a power point file or pdf of the presentation, or a pdf of the write-up if you plan to use the chalk board. I will grade these and get these back to you for corrections before the presentations to the class.

Assignment 11
Due Monday, November 25 (Week 14, Day 1)
5.3: 2, 3, 5
5.4: 3, 6
5.5: 1

Note: If this listing is incomplete or disagrees with what I assigned in class, please send me e-mail at Jim.Swift@nau.edu.