MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2
Name:
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
f (x) = sin πx
g(x) = 0
with L = 1 and c =
its original shape?
1
π.
What is the first positive time t that the string returns to
Date: October 26, 2000.
1
2
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2
2. Use separation of variables to find the general solution of the heat equation in
a rectangular plate with edges of lengths a and b, and with all edges insulated.
MATH 3150: PDE FOR ENGINEERS
3.
MIDTERM TEST #2
3
Under rescaling a region by a factor of ε in the x and y variables, by rescaling
x → εx
y → εy
the operators
∂
∂
and
∂x
∂y
get rescaled by
∂
1 ∂
→
∂x
ε ∂x
∂
1 ∂
→
∂y
ε ∂y
Calculate the rescaling that must occur to the time variable t in the heat equation
2
∂u
∂ u ∂2u
= c2
+
∂t
∂x2
∂y 2
in order that the constant c stay the same, and the heat equation still be satisfied
by the same function with the rescaled arguments. Recalling the basic idea that a
mode consists of an exponential decay in time multiplied by a function of x and y,
say
exp(−κt)f (x, y)
how does this rescaling affect κ, the rate of decay of the mode? Given a replica of a
plate of 1/20 the scale of the original, made of the same material, how many times
faster will it cool or heat than the original?
4
4.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2
Show that the function
un (r, θ) =
r n
R
(an cos(nθ) + bn sin(nθ))
satisfies the Laplace equation
∇2 u = 0
on the disk of radius R, where in polar coordinates
∂ 2 u 1 ∂u
1 ∂2u
+
+
.
∂r2
r ∂r
r2 ∂θ2
Now adding these together, why does the function
∞ n
X
r
u(r, θ) = a0 +
(an cos (nθ) + bn sin (nθ))
R
n=1
∇2 u =
satisfy the Laplace equation?
Moreover show that if we pick these a0 , am and bm according to
Z 2π
1
a0 =
f (θ) dθ
2π 0
Z
1 2π
am =
f (θ) cos(nθ) dθ
π 0
Z 2π
1
bm =
f (θ) sin(nθ) dθ
π 0
(as Fourier amplitudes of f (θ)) then
u(R, θ) = f (θ)
so that this is the steady state of the heat equation in the disk with boundary held
fixed at temperature u(R, θ) = f (θ).
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2
5
1
0.8
0.6
0.4
0.2
–1
–0.5
y0
0.5
1
1
0.5
0
x
–0.5
–1
Figure 1. A round peak
5.
Looking at the wave equation
2
∂2u
∂ u ∂2u
2
=
c
+
∂t2
∂x2
∂y 2
and recalling second derivatives from calculus, explain why a round peak in the
graph of u (as in figure 1) will be accelerated downward.