EML 5060 Homework Set 4 Fall 2002

Page HW Class Topic

33 3.22ade intro

35 3.38 intro

35 3.39 intro (see note 1 below)

35 3.40 intro (see note 2 below)

35 3.41 intro (see note 3 below)

50 4.18 intro

50 4.19 intro

50 4.20 intro (see note 4 below)

17 2.19cfg 2.19e Classification

17 2.20 Classification

17 2.21ac 2.21b Classification: assume u=u(x,y,z[,t])

18 2.25 2.24 Canonical form

18 2.26 Canonical form

18 2.22bf 2.22d Characteristics

18 2.27b 2.27d 2D Canonical form

18 2.28egj 2.28nml 2D Canonical form

98 7.20 7.19 Unsteady heat conduction in a bar

98 7.21 7.22 Unsteady heat conduction in a bar

98 7.25 7.24 Unidirectional viscous flow

99 7.27 7.28 Acoustics in a pipe (use two methods)

99 7.35 7.36 Steady supersonic flow

99 7.37 7.37 Steady heat conduction in a plate

99 7.39 7.38 Potential flow inside a cylinder

00 Unsteady heat conduction in a disk

Note 1: Guess the solution for x²+y²<1

Note 2: Use 3.37 with f nonzero only in and see what part of the interior becomes nonzero.

Note 3: Assume instead of 1. Then make a physical argument based on the physical interpretation of steady heat conduction in a circle with a heat flux 2 entering through the perimeter.

Note 4: Solutions must be of the form .See when they satisfy the given conditions.

Also solve the following problem:

NO working together on the problem below! If you get stuck ask the instructor or TA.

Solve the 2D unsteady vibrations of a circular membrane of radius r₀ if the membrane is fixed to a nonmoving drum at its perimeter, i.e. . Use the separation of variables (eigenfunction expansion) method.

Initially, the membrane is at rest, but then at time t=0 it is hit by a drum stick a distance away from the center. You may assume that the initial displacement of the membrane is still zero, but that the initial velocity is a delta function positioned at and :.

To solve the problem, you are required to answer the following questions in the order asked (list question number with your answer):

1.

The governing P.D.E. is the two-dimensional wave equation where a is the given wave propagation speed. Write this equation out in polar coordinates.

2.

Identify the spatial domain of the problem.

3.

Identify the boundary conditions.

4.

Identify the initial conditions.

5.

You will need two eigenfunction expansions to reduce this P.D.E. in three variables, r, , and t, into ordinary differential equations with respect to t only. These expansions are similar to those of the heat conduction problem covered in class, but not the same.

6.

Write down the Sturm-Liouville problem for completely, including the boundary conditions by substituting a term of the form into the homogeneous PDE. Solve it, or find a place in the lecture notes where that problem has been solved before (same problem with same boundary conditions.)

7.

Find the transformation formulae that for an arbitrary function produce the Fourier coefficients Fⁱ_n(r,t) of that function in the eigenfunction expansion:

8.

Now that an eigenfunction expansion in the coordinate has been found, find a subexpansion in the r coordinate by substituting a single term of the form Tⁱ_n(t) Rⁱ_n(r) into the equation for the Tⁱ_n(r,t). Make Rⁱ_n satisfy the right boundary conditions. Where is the Sturm Liouville problem different from the problem solved in class? How does that affect the solution?

9.

Find the transformation formulae that for an arbitrary function with first Fourier coefficients Fⁱ_n(r,t) produces the second Fourier coefficients Fⁱ_nm(t) of the double eigenfunction expansion:

10.

Find the net transformation formulae that for an arbitrary function produce the Fourier coefficients Fⁱ_nm(t) of that function in the eigenfunction expansion above directly. (In terms of a multiple integral.) Where is it different from the heat conduction problem?

11.

Expand everything in the PDE in terms of the double eigenfunction expansions obtained and solve the resulting O.D.E. for the uⁱ_nm.

12.

Expand everything in the IC in terms of the double eigenfunction expansions obtained. While finding the Fourier coefficients of the given initial conditions involves double integrals, they can be found by noting that for Dirac delta functions the following is true:

13.

Work everything out as far as possible.

14.

What is the lowest frequency produced by the drum? What is its amplitude?

7.21 solution 7.37 solution 7.39 solution