PHYS 374 Homework 4---Due March 5
1) Express the function )(cos)( 4tAtf
ω
=in the form of a Fourier series
∑
∞
−∞=
−
=
n
tni
nectf
ω
)( . Explain why the series truncates, i.e. why n
c=0 for |n| >4.
2) This second problem is to get a feel for Fourier series. In class we argued that they
were useful for smooth periodic functions. Here I want you to show that they are
even useful to describe discontinuous functions where the approximation can do well
except at the point of discontinuity. Consider the square wave:
<≤−
<≤+
=1)(1
)(01
)(
1
2
1
2
1
1
tModfor
tModfor
tf where )(
1tMod is the fractional part of t, i.e.
ttMod =)(
1 for 10 <≤t, 1)(
1−=ttMod for 21 <≤t,2)(
1−=ttMod for 32 <≤t,
and so forth. Thus f(t) alternates between positive and negative one in a periodic
manner with a period of unity.
a) Assume f(t) can be written as a Fourier series of the form ∑
∞
−∞=
−
=
n
tni
nectf
π
2
)( and
show that the n
ccoefficients
=−
evennfor
oddnfor
cni
n0
2
π
.
b) Since f(t) is real we should have *
nn cc =
−verify that this true. Explain the
significance of the fact the n
c coefficients are pure imaginary.
c) Plot the exact expression and 4 approximations based on the Fourier series a
truncation for n<2, n<4, n<6 and n<8. Comment on the apparent converngence.
Where does the expansion do poorly and why?
3) Consider a particle of mass m moving in the following anharmonic potential:
4
24
1
2
2
1
)( xxkxU
α
+= with α>0. The particle is released from rest at x=A.
a) Use energy conservation to determine the speed of the particle when it reaches
x=0.
b) Use the approximate expression based on Fourier analysis to determine the
velocity at x=0. You should work at lowest nontrivial order (including the 3ω
term in the expansion)
c) Specify the condition on A for which you expect your answer in b) to be valid.
4) Show that the two expressions derived in problem 3a) and 3b) are consistent, i.e. they
differ only beyond the order calculated in the approximation
