MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
Name:
This test has 8 pages.
Work out everything as far as you can before making decimal approximations.
1.
Suppose that a string in a still fluid is tied down at x = 0 and x = 1. The
force of resistance to motion of the string coming from the fluid is proportional to
the velocity of the string. Which of the following is the equation of motion of this
string?
(a)
∂u
∂2u
∂u
= −2k
+ c2 2
∂t
∂x
∂x
(b)
∂u
∂2u
∂2u
= −2k
+ c2 2
2
∂t
∂x
∂x
(c)
∂2u
∂u
∂2u
= −2k
+ c2 2
2
∂t
∂t
∂x
(d)
∂u
∂2u
= c2 2
∂t
∂x
(e)
∂2u
∂2u
= sinh u +
2
∂t
∂x2
Which two of the following are modes of the string?
(a)
cos (πnct) sin (πnx)
(b)
(cos (πnct) − kt) sin (πnx)
(c)
cos (πnct) cos (πnx)
(d)
2
exp − (πnc) t sin (πnx)
(e)
p
e−kt sin t π 2 c2 n2 − k 2 sin (πnx)
(f)
sin (πnct) sin (πnx)
Date: March 22, 2002.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
(g)
2
exp − (πnc) t cos (πnx)
(h)
p
e−kt cos t π 2 c2 n2 − k 2 sin (πnx)
(i)
sinh(πnt) sin(πnx)
MATH 3150: PDE FOR ENGINEERS
2.
MIDTERM TEST #2 VERSION E
3
The total energy of a vibrating string of length L is
2
2 !
Z
∂u
∂u
1 L
2
H=
+c
dx.
2 0
∂t
∂x
(Warning: this is not what we called energy before, in talking about Fourier series.)
Calculate the total energy at time t of every mode
πnx sin
cos (λn t)
L
and
πnx sin
sin (λn t)
L
of the wave equation with ends of the string tied down (where λn = πnc/L). How
does the total energy of each mode vary in time?
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
3.
Suppose that a wire of length L = π and diffusivity c = 1 with initial
temperature 100o is placed in an insulating tube. One end is kept at 100o with a
thermostat, while the other is kept at 0o .
(a) Find the temperature u(x, t). Your answer should be a sum of a steady state
(in this case a linear function of x) and a solution of a similar problem with
0o at both ends.
(b) Draw a picture of what the temperature looks like at time t = 0 and at
large time t when it has not quite reached the steady state.
MATH 3150: PDE FOR ENGINEERS
4.
MIDTERM TEST #2 VERSION E
5
Consider the wave equation with gravitational force
∂2u
∂2u
= c2 2 − g.
2
∂t
∂x
(a) Find the steady state (i.e. u = u(x) independent of time) which satisfies
u = u0 fixed (u0 some constant) at x = 0 and u = u1 (some other constant)
at x = L.
(b) Draw a picture of what it looks like for small positive g (for example, on
the moon) and for large positive g (for example, on Jupiter).
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5.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
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 the string has length L = π, and at time t = 0 has 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? (By flat, I mean that
at that time t the height is u(x, t) = 0, for every point x.)
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E
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6.
A wire of length L = 1 with insulated ends and diffusivity c = 1 has
temperature f (x) = x at time t = 0. Find the temperature u(x, t) at time t and
position x.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION E