MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
Name:
This test has 8 pages.
Work out everything as far as you can before making decimal approximations.
1. Consider d’Alembert’s solution
Z x+ct
1
1
u(x, t) = (f (x − ct) + f (x + ct)) +
g(s) ds
2
2c x−ct
of the wave equation for a vibrating string, where f (x) is the odd function with
period 2L which on the interval 0 ≤ x ≤ L gives the initial position of the string,
and similarly g(x) gives the initial velocity on 0 ≤ x ≤ L and is odd and 2L periodic.
Suppose that at time t = 0 the string has length L = π, initial position
f (x) = sin x
and initial velocity
g(x) = 0 .
What are all of the times t at which the string will be flat?
Date: July 16, 2001.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
Figure 1. A hollow cylinder
2. If we slit open a hollow cylinder, as in figure 1, it unfolds to a rectangle. Take
coordinates x, y on that rectangle. To glue back together the hollow cylinder, we
have only to ask that any functions we work with on the rectangle have equal values
on the left and right sides. You will find the general solution of the heat equation
in a cylinder with insulated top and bottom edges, thought of as a rectangle of with
left and right sides of length b and top and bottom sides of length a. In order to
get back the behaviour of heat on a cylinder with insulated ends, we pose the usual
heat equation on the rectangle, but ask that the temperature u(x, y, t) be equal at
corresponding points of the left and right sides:
u(x = 0, y) = u(x = a, y)
∂u
∂u
(x = 0, y) =
(x = a, y) .
∂x
∂x
Find the general solution of the heat equation satisfying these conditions.
MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION C
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION C
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
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3. Take a disk of unit radius and heat the top half of the edge of the disk to 1o
and the bottom half of the edge to 0o . Let the disk sit with these temperatures on
its edges for a long time, until it reaches a steady state. Center the disk at origin
of coordinates. What is the temperature at the point
1
(x, y) = − , 0 ?
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Hint: recall that if we write the temperature of the edge of the disk as f (θ), and
write the Fourier amplitudes of f (θ) as
Z 2π
1
a0 =
f (θ) dθ
2π 0
Z 2π
1
am =
f (θ) cos(mθ) dθ
π 0
Z
1 2π
bm =
f (θ) sin(mθ) dθ
π 0
then the steady state temperature inside the plate, in polar coordinates, is
∞
X
rm (am cos (mθ) + bm sin (mθ))
u(r, θ) = a0 +
m=0
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4.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
Suppose that u(x, t) satisfies the heat equation
∂2u
∂u
= c2 2
∂t
∂x
in a wire of length L. Assume that each of the ends of the wire is either insulated
or is kept at 0o (it doesn’t matter which of these conditions holds or at which end).
Show that
Z
L
u2 dx
0
decreases over time unless the heat is in steady state. Hint: differentiate in t, and
make use of the heat equation. Then notice that
∂
∂u
∂u ∂u
∂2u
u
=
+u 2
∂x
∂x
∂x ∂x
∂x
(one of these terms appears in the integral after you make use of the heat equation)
and use the fundamental theorem of calculus.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
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Figure 2. Temperature at some time
5. The two pictures in figures 2 and 3 are of the temperature of a wire with ends
kept at 0o . Which one is at the later time? Explain your answer.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION C
Figure 3. Temperature at some other time