1. Consider the following partial differential equation (called the “diffusion equation”):
2
2
(,)
f xt
D
tx
∂∂
=
a.
Assuming
t
is time and
x
is distance, what are the units of
D
? (5 pts)
b.
Show that this one partial differential equation of two variables can be written as two
ordinary differential equations of one variable each: (10 pts)
2
2
()
dux
ux
dx
λ
=−
and
dv t
Dvt
dt
c.
What are the units of
? (5 pts)
d.
Solve these ordinary differential equations for the MOST GENERAL solutions
u(x)
and
v(t)
. (10 pts)
e.
If
1
( ,
0)
sin(
)
n
n
n
A
x
L
π
∞
=
==
∑
, then what are the coefficients A
n
? [HINT:
0
2
sin(
)sin(
)
L
nm
nm
xx
d
x
LL
L
ππ
δ
=
∫
] (10 pts)
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View Full Document2. If a point source emits
“monochromatic” light (of
only
one
wavelength)
isotropically incident on a
FabryPerot etalon of fixed
thickness, draw the pattern on
the screen. (5 pts)
3. If a string held fixed at
x=0
and
x=L
is driven with mechanical force at angular frequency
ω=3
s
π
/L (s is wave speed) and location
x
0
=L/3
, sketch a bar graph representing the magnitude of
coefficients for the response of each of the normal modes. (15 pts)
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 Spring '08
 buma
 pts, Partial differential equation

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