MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D
Name:
This test has 8 pages.
Work out everything as far as you can before making decimal approximations.
1.
Which of the following is the solution u(x, t) (via Fourier’s method, not
d’Alembert’s) to the wave equation describing a vibrating string of length L = π/2
and wave velocity c = 1 with initial velocity g(x) = x cos x and initial position
f (x) = 0?
(a)
∞
8 X (−1)n+1
u(x, t) =
sin (2nx) sin (2nt)
π n=1 (4n2 − 1)2
(b)
u(x, t) =
∞
8X
1
sin (2nx) sin (2nt)
π n=1 (4n2 − 1)2
u(x, t) =
∞
8X
1
sin (2nx) cos (2nt)
π n=1 (4n2 − 1)2
(c)
(d)
u(x, t) =
1
1
(x − t) cos(x − t) + (x + t) cos(x + t)
2
2
(e)
u(x, t) =
1
2
Z
x+t
s cos(s) ds
x−t
(f)
u(x, t) = 0
(g)
u(x, t) = x cos x cos t
Date: March 14, 2002.
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION D
MATH 3150: PDE FOR ENGINEERS
2.
MIDTERM TEST #2 VERSION D
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Consider again the same vibrating string as in the previous problem.
(a) Draw a rough graph of the function g(x).
(b) Examining the Fourier amplitudes in the solution, how many terms in a
partial sum do you think you would need to get a good picture of u(x, t)?
(c) Could you see this by just looking at the picture of g(x)?
(d) Draw what you expect the graph of u(x, t) looks like for time t = 0 and for
time t = 0.1.
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3.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D
Consider the heat equation
∂u
∂2u
= c2 2
∂t
∂x
on a wire of length L = 1 with c = 7 and with initial temperature u(x, 0) = x(1−x).
Suppose that the total heat at time t is given by
Z 1
Q=
u(x, t) dx
0
(a) What is the rate of change of the total heat at time t = 0? Hint: don’t
use any Fourier series: just differentiate Q with respect to t and bring the t
derivative under the integral sign. Then use the heat equation to turn time
derivatives into space derivatives.
(b) Is it cooling down or heating up?
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D
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4. Solve the heat equation for a wire with insulated ends, with initial temperature
u(x, 0) = 273o K.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D
5. Solve the heat equation for a wire of length L = 1 with ends held at u = 100
at x = 0 and u = 0 at x = L with diffusivity constant c = 1 and initial temperature
u = 100(1 − x) + 30 sin (πx) at time t = 0.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D
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6. Take a flat string of length L = 1 with constant c = 1. If you tap it at one end
(so you create an initial velocity g(x) in the string which is zero except very close
to x = 0) how long does it take for the tap to be noticed at the other end? Explain
using d’Alembert’s formula.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION D