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Fourier Analysis
Workshop 1: Fourier Series
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1424th & 27th September
2013
1. By writing sinA and cosB in terms of exponentials, prove
that
2 sinA cosB = sin(A+B) + sin(A� B).
2. If f(x) and g(x) are periodic with fundamental period X, show
that the following are alsoperiodic with the same period:(a) h(x) =
a f(x) + b g(x)(b) j(x) = c f(x) g(x)where a, b, c are
constants.
3. Find the fundamental periods for the following functions:(a)
cos 2x(b) 3 cos 3x+ 2 cos 2x(c) cos2 x(d) | cos x|(e) sin3 x.
4. Show thatZ
L
�Ldx sin
⇣m⇡xL
⌘sin
⇣n⇡xL
⌘=
⇢0 m 6= nL m = n
Printed: September 22, 2013 1
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5. (a) Sketch f(x) = (1 + sin x)2 and determine its fundamental
period.(b) Using a trigonometric identity for sin2 x in terms of
cos 2x, write down the Fourier Seriesfor f(x) (don’t do any
integrals to obtain the coe�cients).
6. Show that the Fourier Series expansion of the periodic
function
f(x) =
⇢�1 �⇡ < x < 0+1 0 < x < ⇡
is
f(x) =4
⇡
1X
k=0
sin[(2k + 1)x]
2k + 1.
7. (a) Show that the Fourier Series for f(x) = x in the range �⇡
< x < ⇡ is
f(x) = 21X
m=1
(�1)m+1
msinmx.
(b) Hence, by carefully choosing a value of x, show that
1� 13+
1
5� 1
7. . . =
⇡
4.
8. Using a trigonometric identity, or otherwise, compute the
Fourier Series for f(x) = x sin xfor �⇡ < x < ⇡, and hence
show that
⇡
4=
1
2+
1
1⇥ 3 �1
3⇥ 5 +1
5⇥ 7 � . . .
Printed: September 22, 2013 2
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Fourier Analysis
Workshop 2: More on Fourier Series
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/141st & 4th October
2013Handin Deadline: 4 p.m. Friday 11th October 2013
1. Consider the function f(x) = | cos x|.(a) What is its
fundamental period?(b) Sketch the function for �2⇡ < x <
2⇡(c) Show that the Fourier Series expansion for f(x) is
f(x) =2
⇡+
4
⇡
1X
m=1
(�1)m+1
4m2 � 1 cos(2mx).
2. Let f(x) = 1 + cos2(⇡x).(a) Sketch f(x) and determine its
fundamental period.(b) Using a trigonometric identity, and without
doing any calculations, write down a FourierSeries for f(x).
3. If f(x) = sin x for 0 x ⇡,(a) compute the fundamental period
for a sine series expansion
(b) compute its Fourier sine series
(c) sketch the function in (b) for �2⇡ < x < 2⇡.
(d) compute the fundamental period for a cosine series
expansion
(e) compute its cosine series and show it is
sin x =2
⇡� 4⇡
1X
m=1
1
4m2 � 1 cos(2mx)
(f) sketch the function in (e) for �2⇡ < x < 2⇡.
4. Consider f(x) = e�x, defined in 0 < x < 1. Expand it as
a Fourier sine series, and sketch
Printed: September 22, 2013 3
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the function for �2 < x < 2.
5. If f(x) = x for �1 x 1,(a) show that its Fourier Series
is
f(x) =1X
n=1
(�1)n+1 2n⇡
sin(n⇡x).
(b) Hence show that1X
k=0
(�1)k
2k + 1=⇡
4.
6. Compute the complex Fourier Series for f(x) = x, �⇡ < x
< ⇡.
7. If f(x) = |x| for �⇡ x ⇡,(a) show that its Fourier Series
is
f(x) =⇡
2� 4⇡
1X
n=0
cos[(2n+ 1)x]
(2n+ 1)2.
(b) Hence show that1X
n=0
1
(2n+ 1)2=⇡2
8.
Printed: September 22, 2013 4
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Fourier Analysis
Workshop 3: Parseval, and ODEs by Fourier Series
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/148th & 11th October
2013
1. Compute the complex Fourier Series for
f(x) =
⇢�1 �1 < x < 0+1 0 < x < 1
and show it is cn
= (cos(kn
)� 1)i/kn
.
2. Prove Parseval’s theorem for the (sine and cosine) Fourier
Series.
3. You are given that the Fourier Series of f(x) = |x| (defined
for �⇡ x ⇡) is
f(x) =⇡
2� 4⇡
1X
n=0
cos[(2n+ 1)x]
(2n+ 1)2.
State Parseval’s theorem, and prove that
1X
n=0
1
(2n+ 1)4=⇡4
96.
4. You are given that the Fourier Series of f(x) = x (defined
for �1 x 1) is
f(x) =1X
n=1
(�1)n+1 2n⇡
sin(n⇡x).
Using Parseval’s theorem, show that
1X
n=1
1
n2=⇡2
6.
5. (a) By expanding both sides as Fourier Sin Series, show that
the solution to the equation
d2y
dx2+ y = 2x
Printed: September 22, 2013 5
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with boundary conditions y(x = 0) = 0, y(x = 1) = 0 is
y(x) =4
⇡
1X
n=1
(�1)n+1
n(1� n2⇡2) sin(n⇡x).
(b) Show that the r.m.s. value of y(x) is
phy2(x)i = 4
⇡
vuut12
1X
n=1
1
n2(1� n2⇡2)2
6. If the function f(x) is periodic with period 2⇡ and has a
complex Fourier Series representation
f(x) =1X
n=�1fn
einx
then show that the solution of the di↵erential equation
dy
dx+ ay = f(x)
is
y(x) =1X
n=�1
fn
a+ ineinx.
7. An RLC series circuit has a sinusoidal voltage V0 sin!t
imposed, so the current I obeys:
Ld2I
dt2+R
dI
dt+ CI = !V0 cos!t.
(a) What is the fundamental period of the voltage?(b) Write I(t)
as a Fourier Series,
I(t) =a02
+1X
n=1
[an
cos(n!t) + bn
sin(n!t)]
and show that an
and bn
satisfy
Ca02
+1X
n=1
�an
⇥�Ln2!2 cos(n!t)�Rn! sin(n!t) + C cos(n!t)
⇤+
bn
⇥�Ln2!2 sin(n!t) +Rn! cos(n!t) + C sin(n!t)
⇤ = !V0 cos!t.
(c) Hence show that only a1 and b1 survive, with amplitudes
a1 =!V0(�L!2 + C)
(C � L!2)2 +R2!2
b1 =!2V0R
(C � L!2)2 +R2!2 .
Printed: September 22, 2013 6
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8. A simple harmonic oscillator with natural frequency !0 and no
damping is driven by a drivingacceleration term f(t) = sin t+ sin
2t.
(a) Write down the di↵erential equation which the displacement
y(t) obeys.
(b) Compute the fundamental period of the driving terms on the
right hand side, and henceT (where the solution is assumed periodic
on �T < t < T ).(c) Assuming the solution is periodic with
the same fundamental period as the driving term,find the resultant
motion.
(d) Calculate the r.m.s. displacement of the oscillator.
Printed: September 22, 2013 7
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Fourier Analysis
Workshop 4: Fourier Transforms
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1415th & 18th October
2013Handin Deadline: 4 p.m. Friday 25th October 2013
1. Prove that, for a real function f(x), its Fourier Transform
satisfies f̃(�k) = f̃ ⇤(k).
2. In terms of f̃(k), the Fourier Transform of f(x), what are
the Fourier Transforms of thefollowing?(a) g(x) = f(�x)(b) g(x) =
f(2x)(c) g(x) = f(x+ a)(d) g(x) = df/dx.(e) g(x) = xf(x)
3. Consider a Gaussian quantum mechanical wavefunction
(x) = A exp
✓� x
2
2�2
◆,
where A is a normalisation constant, and the width of the
Gaussian is �.(a) Compute the Fourier Transform ̃(k) and show that
it is also a Gaussian.(b) Noting that the probability density
function is | (x)2|, show by inspection that theuncertainty in x
(by which we mean the width of the Gaussian) is �x = �/
p2.
(c) Then, using de Broglie’s relation between the wavenumber k
and the momentum, p = h̄k,compute the uncertainty in p, and
demonstrate Heisenberg’s Uncertainty Principle,
�p�x =h̄
2.
4. Express the Fourier Transform of g(x) ⌘ eiaxf(x) in terms of
the FT of f(x).
5. The function f(x) is defined by
f(x) =
⇢e�x x > 00 x < 0
Printed: September 22, 2013 8
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(a) Calculate the FT of f(x), and, using Q4, of eixf(x) and of
e�ixf(x).
(b) Hence show that the FT of g(x) = f(x) sin x is
1
(1 + ik)2 + 1
(c) Finally, calculate the FT of h(x) = f(x) cosx.
6. (a) Show that the FT of f(x) = e�a|x| is f̃(k) = 2a/(a2 +
k2), if a > 0.
(b) Sketch the FT of the cases a = 1 and a = 3 on the same
graph, and comment on thewidths.
(c) Using the result of question 4, show that the FT of g(x) =
e�|x| sin x is
g̃(k) =�4ik4 + k4
.
7. Letha
(x) ⌘⇢e�ax x � 00 x < 0.
(a) Show that the FT of ha
(x) is
h̃a
(k) =1
a+ ik.
(b) Take the FT of the equation
df
dx(x) + 2f(x) = h1(x)
and show that
f̃(k) =1
1 + ik� 1
2 + ik.
(c) Hence show that f(x) = e�x � e�2x is a solution to the
equation (for x > 0).(d) Verify your answer by solving the
equation using an integrating factor.
(e) Comment on any di↵erence in the solutions.
8. Compute the Fourier Transform of a top-hat function of height
h and width 2a, which iscentred at x = d = a. Sketch the real and
imaginary parts of the FT.
Printed: September 22, 2013 9
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Fourier Analysis
Workshop 5: Dirac Delta Functions
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1422nd & 25th October
2013
1. By using the result that if, for all functions f(x),Z 1
�1f(x)g(x)dx =
Z 1
�1f(x)h(x)dx
then g(x) = h(x), show that
(a) �(�x) = �(x)Hint: show that Z 1
�1f(x)�(�x)dx = f(0) =
Z 1
�1f(x)�(x)dx
(b) �(ax) = �(x)|a|(c) �(x2 � a2) = �(x�a)+�(x+a)2|a| .(d) x�(x)
= 0.
2. Evaluate(a)
R1�1 f(x)�(2x� 3) dx
(b)R 21 f(x)�(x� 3) dx
3. Evaluate Z⇡+0.1
0.1
dx
Z 4
�1dy �(sin x)�(x2 � y2)
4. Show that the derivative of the Dirac delta function has the
property thatZ 1
�1
d�(t)
dtf(t) dt = � df
dt
����t=0
5. What are the Fourier Transforms of:(a) �(x)(b) �(x� d)
Printed: September 22, 2013 10
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(c) �(2x)?By writing �(x) as an integral (i.e. as an Inverse
Fourier Transform) show that(d) �⇤(x) = �(x)
6. Evaluate Z 1
�1
Z 1
�1t2e�iaxeitx dxdt
where a is a constant.
7. Compute Z 1
�1
Z 1
�1
Z 1
�1
Z 1
�1eixye�ixzeibzeiyte�t
3dxdydzdt
where b is a constant.
8. A one-dimensional harmonic oscillator with natural frequency
!0 is driven with a drivingacceleration a(t), so obeys
d2z
dt2+ !20z = a(t).
(a) Take the Fourier Transform of this equation (from t to !)
and show that
z̃(!) =ã(!)
!20 � !2
(b) Hence show that
z(t) =1
2⇡
Z 1
�1
ã(!)
!20 � !2ei!td!.
(c) If a(t) = sin2 ⌦t, find ã(!).
(d) Hence find a solution for z(t) (ignore solutions to the
homogeneous equation).
Printed: September 22, 2013 11
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Fourier Analysis
Workshop 6: Convolutions, Correlations
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1429th October & 1st
November 2013Handin Deadline: 4 p.m. Friday 8th November 2013
1. Show that convolving a signal f(t) with a Gaussian smoothing
function
g(t) =1p2⇡�
exp
✓� t
2
2�2
◆
results in the Fourier Transform being ‘low-pass filtered’ with
a weight exp(��2!2/2).
2. Show that the FT of a product h(x) = f(x)g(x) is a
convolution in k-space:
h̃(k) =1
2⇡f̃(k) ⇤ g̃(k) =
Z 1
�1
dk0
2⇡f̃(k0)g̃(k � k0).
3. Show that the convolution of a Gaussian of width �1 with a
Gaussian of width �2 gives an-other Gaussian, and calculate its
width. (A Gaussian of width � has the formN exp[�x2/(2�2)]).
4. Show that the FT of the cross-correlation h(x) of f(x) and
g(x),
h(x) =
Z 1
�1f ⇤(x0)g(x0 + x)dx0 is h̃(k) = f̃ ⇤(k)g̃(k).
5. A signal f(x) = e�x for x > 0 and zero otherwise.(a) Show
that the Fourier Transform is f̃(k) = (1 + ik)�1.
(b) Using Parseval’s theorem, relate the integral of the power
|f̃(k)|2 to an integral of |f(x)|2.
(c) Hence show that Z 1
�1
dk
1 + k2= ⇡.
(d) If the signal is passed through a low-pass filter, which
sets the Fourier transform coe�-cients to zero above |k| = k0,
calculate k0 such that the filtered signal has 90% of the
originalpower.
Printed: September 22, 2013 12
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6. (a) Compute the Fourier Transform of
h(t) =
⇢e�bt t � 00 t < 0.
(b) A system obeys the di↵erential equation
dz
dt+ !0z = f(t)
By using Fourier transforms, show that a solution of the
equation is a convolution of f(t)with
g(t) =
⇢e�!0t t � 00 t < 0
i.e.
z(t) =
Z 1
�1f(t0)g(t� t0) dt0,
and write down the full expression for z(t).
7. (a) Show that the FT of h(t) = e�a|t|, for a > 0 is
h̃(!) =2a
a2 + !2.
(b) A system obeys the di↵erential equation
d2z
dt2� !20z = f(t).
Calculate z̃(!) in terms of f̃(!).
(c) By considering the form of z̃(!), show using the convolution
theorem that a solution ofthe equation is the convolution of f(t)
with some function g(t).
(d) Using your answer to (a), find the function g(t) and write
down explicitly a solution tothe equation.
8. Compute the Fourier Transform of
h(x) =
⇢1 |x| 10 otherwise
Show that the convolution H(x) ⌘ h(x)⇤�(x�a) is 1 if a�1 < x
< a+1 and zero otherwise,and compute its Fourier transform
directly, and via the convolution theorem.
9. A triple slit experiment consists of slits which each have a
Gaussian transmission with Gaus-sian width �, and they are
separated by a distance d � �. Compute the intensity
distributionfar from the slits, and sketch it.
Printed: September 22, 2013 13
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Fourier Analysis
Workshop 7: Sampling, and Green’s Functions
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/145th & 8th November
2013
1. (a) Expand1
1� zas a Taylor series about z = 0, or as a power series.
(b) Hence show that1X
j=0
eijk�x =1
1� z
where z = exp(ik�x).(c) Similarly, show that
0X
j=�1
eijk�x =1
1� 1/z .
(d) Finally, show that if z 6= 1,1X
j=�1
e�ijk�x = 0.
2. Letting p = dy/dt, and then using an integrating factor, show
that the general solution to
d2y
dt2+
dy
dt= 0
is y(t) = A+Be�t, where A and B are constants.
3. Show that the Green’s function for the range x � 0,
satisfying
@2G(x, z)
@x2+G(x, z) = �(x� z)
with boundary conditions G(x, z) = @G(x, z)/@x = 0 at x = 0
is
G(x, z) =
⇢cos z sin x� sin z cos x x > z
0 x < z
Printed: September 22, 2013 14
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4. Consider the equation, valid for t � 0d2f
dt2+ 5
df
dt+ 6f = e�t,
subject to boundary conditions f = 0, df/dt = 0 at t = 0. Find
the Green’s function G(t, z),showing is is zero for t < z, and
for t > z it is
G(t, z) = e2z�2t � e3z�3t.
(You may find the complementary function (homogeneous solution)
by using a suitable trialfunction).
Hence show that the solution to the equation is
f(t) =1
2e�t � e�2t + 1
2e�3t.
5. (a) Show that the Green’s function for the equation, valid
for t � 0
d2y
dt2+
dy
dt= f(t),
with y = 0 and dy/dt = 0 at t = 0, is
G(t, T ) =
⇢0 t < T
1� eT�t t > T .
(b) Hence show that if f(t) = Ae�2t, the solution is
y(t) =A
2
�1� 2e�t + e�2t
�.
6. The equation for a driven, damped harmonic oscillator is
d2y
dt2+ 2
dy
dt+ (1 + k2)y = f(t)
(a) If the initial conditions are y = 0 and dy/dt = 0 at t = 0,
show that the Green’s function,valid for t � 0, is
G(t, T ) =
⇢A(T )e�t cos kt+B(T )e�t sin kt 0 < t < TC(T )e�t cos
kt+D(T )e�t sin kt t > T
(b) Show that A = B = 0 and so G(t, T ) = 0 for t < T .
(c) By matching G(t, T ) at t = T , and requiring dG/dt to have
a discontinuity of 1 there,show that, for t > T
G(t, T ) =eT�t
k(� sin kT cos kt+ cos kT sin kt) .
(d) Hence if f(t) = e�t, find the solution for y(t).
Printed: September 22, 2013 15
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Fourier Analysis
Workshop 8: Partial Di↵erential Equations
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1412th & 15th November
2013Handin Deadline: 4 p.m. Friday 22nd November 2013
1. Find solutions u(x, y) by separation of variables to(a)
x@u
@x� y@u
@y= 0
(b)@u
@x� xy@u
@y= 0
2. Consider a particle of mass m which is confined within a
square well 0 < x < ⇡, 0 < y < ⇡.The steady-state 2D
Schrödinger equation inside the well (where the potential is zero)
is
� h̄2
2mr2 = E .
The walls have infinite potential, so = 0 on the boundaries.(a)
Find separable solutions (x, y) = X(x)Y (y) and show that they
are
(x, y) = A sin(rx) sin(ny)
for integers r, n.
(b) The wavefunction is normalised so thatR
| (x, y)|2 dx dy = 1. For given r, n, find A.(c) Show that the
energy levels corresponding to the quantum numbers m,n are
E = (r2 + n2)h̄2
2m.
3. Show by direct substitution into the equation that
u(x, t) = f(x� ct) + g(x+ ct)
where f and g are arbitrary functions, is a solution of the 1D
wave equation,
@2u
@x2=
1
c2@2u
@t2,
Printed: September 22, 2013 16
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where the sound speed c is a constant. You may recall that the
partial derivative of f(y)with respect to x (where y may be a
function of several variables y(x, t, . . .)) is
@f
@x=@y
@x
df
dy.
4. Consider the wave equation (with sound speed unity) for t
> 0
@2u
@x2=@2u
@t2
with initial conditions u(x, 0) = h(x) and @u/@t|t=0 = v(x).
(a) Write down d’Alembert’s solution for u(x, t).(b)If h(x) and
v(x) are known only for 0 < x < 1, then find the regions in
the x, t plane forwhich the solution for u can be determined, and
sketch it.
5. (a) Consider separable solutions for the temperature u(x, t)
= X(x)T (t) of the 1D heatequation
@2u
@x2=@u
@tand find the di↵erential equations which X and T must satisfy,
giving your reasoning.
(b) Solving for T , show that the separable solutions which are
finite as t ! 1 are of theform
[A cos(kx) + B sin(kx)] exp(�k2t).where k2 > 0.
(c) There is one more (rather simple) permitted solution. What
is it?
(d) Following on from the last question, find all solutions for
which u(0, t) = u(⇡, t) = 0 atall times. Hint: the answer is not a
single term, but rather a sum.
(e) If the initial temperature (at t = 0) is u(x, 0) = sin x cos
x, what is the full solutionu(x, t)?
6. This is a question which looks hard, because it uses polar
coordinates, but you can solve itin exactly the same way as the
cartesian equations.
Laplace’s equation in polar coordinates r, ✓ is
@2u
@r2+
1
r
@u
@r+
1
r2@2u
@✓2= 0
(a) Show that for solutions which are separable, u(r, ✓) =
R(r)⇥(✓),
⇥00(✓) = �k2⇥; r2R00(r) + rR0(r)� k2R(r) = 0
for some constant k2.
(b) Argue that the solution must be periodic in ✓, and say what
the period must be.
Printed: September 22, 2013 17
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(c) As a consequence, what values of k are permitted?
(d) By trying power-law solutions R(r) / r↵, find the general
solution which is finite at theorigin.
(e) Find the solution for a situation where u is fixed on a
circular ring at r = 1 to be
u(r = 1, ✓) = sin2 ✓ + 2 sin ✓ cos ✓.
Printed: September 22, 2013 18
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Fourier Analysis
Workshop 9: Fourier PDEs
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1419th & 22nd November
2013
1. If we have a function u(x, t), we may do a partial Fourier
Transform, changing x to k butleaving t in the equations. We have
used the result that
FT
@u(x, t)
@t
�=@ũ(k, t)
@t
Show this (you can probably fit it on one line).
2. Consider the 1D wave equation@2u
@x2=
1
c2@2u
@t2,
with boundary conditions at t = 0 that u(x, t) = e�a|x| for some
a > 0, and @u(x, t)/@t = 0.(a) By applying a Fourier Transform
with respect to x, show that the FT of the generalsolution is of
the form
ũ(k, t) = A(k)e�ikct +B(k)eikct.
(b) Show that at t = 0,
ũ(k, 0) =2a
a2 + k2.
(c) Hence, applying the boundary conditions, show that
ũ(k, t) =a
a2 + k2�e�ickt + eickt
�.
Note that you will need to argue that the boundary condition on
@u/@t also applies to eachFourier component individually.
(d) Finally deduce that
u(x, t) =1
2
�e�a|x�ct| + e�a|x+ct|
�
3. Consider the 1D heat equation for the temperature u(x,
t),
@2u
@x2=
@u
@t
where the initial condition is that u(x, t = 0) = �(x).
Printed: September 22, 2013 19
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(a) Take the Fourier Transform with respect to x, i.e.
ũ(k, t) =
Z 1
�1u(x, t)e�ikx dx
Note that the transform is still a function of t. Show that it
obeys
@ũ(k, t)
@t= �k
2
ũ(k, t).
(b) Now fix the value of k for now, and use an integrating
factor to find
ũ(k, t) = f̃(k)e�k2t/
for some (arbitrary) function f̃(k).
(c) From the initial condition u(x, 0) = �(x) show that
f̃(k) = �̃(k) ) ũ(k, t) = �̃(k)e�k2t/.
(d) Using the result that the FT of e�x2/(4t) is
p4⇡t/e�k
2t/, show using the convolution
theorem that the general solution for u(x, t) in terms of �(x)
is
u(x, t) =
pp4⇡t
Z 1
�1e�(x�x
0)2/(4t)�(x0) dx0
(e) If �(x) = �(x� 1), what is u(x, t)?
4. Using d’Alembert’s method, show that the solution to the wave
equation
@2u
@t2= c2
@2u
@x2
with the boundary conditions
u(x, t = 0) = h(x) = 0@u
@t(x, t = 0) = v(x) = x
is
u(x, t) =1
4c
⇥(x+ ct)2 � (x� ct)2
⇤.
If v(x) = x only for 0 < x < 1 (and is zero otherwise),
what is the solution? Note - you willhave to consider many di↵erent
combinations depending on the values of x� ct and x+ ct -be guided
by the spacetime diagram which is in the notes.
5. The charge density ⇢ and the electrostatic potential � are
related by Poisson’s equation
r2�(x) = ⇢(x)✏0
Printed: September 22, 2013 20
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where we assume that there is no time-dependence. Treating this
as a one-dimension problem(so r2 ! d2/dx2), show using a Fourier
Transform that a Gaussian potential
�(x) = �e�x2/(2�2)
is sourced by a charge density field
⇢(x) =✏0�2
e�x2/(2�2)
✓1� x
2
�2
◆.
In doing this, you will demonstrate that
Z 1
�1
dk
2⇡k2e�k
2�
2/2eikx =
1p2⇡�3
e�x2/(2�2)
✓1� x
2
�2
◆.
You can use this result without proof in handin question 6.
Verify the solution by direct di↵erentiation of �(x).
You may assume that the Fourier Transform of e�x2/(2�2) is
p2⇡� e�k
2�
2/2, and that
R1�1 e
�u2/2du =p2⇡. (This method is of more practical use if ⇢ is
known and you want �, when the direct
method here cannot be employed).
6. (Hint: do all of this in cartesian coordinates - do not be
tempted to use spherical polars,despite the symmetry of the
problem).
The charge density ⇢ and the electrostatic potential � are
related by Poisson’s equation
r2�(r) = ⇢(r)✏0
where we assume that there is no time-dependence. Treating this
now as a 3D problem (sor2 ! @2/@x2+@2/@y2+@2/@z2), show using a
Fourier Transform that a Gaussian potential
�(r) = �e�r2/(2�2)
(where r2 = x2 + y2 + z2) has a charge density FT given by
⇢̃(k) = (2⇡)3/2�3✏0 k2e�k
2�
2/2
where k2 = k2x
+ k2y
+ k2z
.
and so the potential is sourced by a charge density field
⇢(r) =✏0�2
✓3� r
2
�2
◆e�r
2/(2�2).
You may assume that the Fourier Transform (w.r.t. x, ! kx
) of e�x2/(2�2) is
p2⇡� e�k
2x
�
2/2,
and thatR1�1 e
�u2/2du =p2⇡. You can also assume the inverse FT of k2e�k
2�
2/2 which you
proved in question 5. Hint: you will be faced with an integral
with 3 terms in it (involvingk2x
+ k2y
+ k2z
). Do one of them only, and engage brain to write down the
answer for the othertwo without doing more algebra.
Printed: September 22, 2013 21
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Fourier Analysis
Workshop 10: Revision: Green’s functions and convo-lutions
Professor John A. PeacockSchool of Physics and
[email protected]: 2013/1426th & 29th November
2013
Marks out of 25 (like exam paper). There are no hand-ins from
this revision workshop.
1. On the mysterious and enigmatic planet Pendleton, a planetary
explorer vehicle falls o↵ acli↵ at t = 0. The acceleration due to
gravity is a constant �g, and the vehicle attemptsto slow down its
motion by applying an upward acceleration f(t). Unfortunately the
fuelin the vehicle rapidly runs out, so the upward acceleration
decays with time according tof(t) = ae�t, for a constant a.
The equation of motion for the height z(t) is evidently
d2z(t)
dt2= f(t)� g ⌘ F (t).
(a) If the top of the cli↵ is at z = 0, then evidently z(t = 0)
= 0. What is dz/dt at t = 0?
(b) Write down the equation for the Green’s function G(t, T
).
(c) Hence show that G(t, T ) = 0 for t < T .
(d) Show that the solution for G(t, T ) for t > T is
G(t, T ) = t� T t > T.
(e) Hence show that the general solution for the vertical height
as a function of time is
z(t) = a(e�t � 1 + t)� 12gt2.
(f) Verify your answer by directly integrating the equation,
with the appropriate boundaryconditions.
(g) Show that the early time behaviour is z(t) = (a� g)t2/2
+O(t3).
2. The e↵ects of magnet errors in a synchrotron require the
solution of the equation
d2y(✓)
d✓2+ !2y(✓) = g(✓)
where ✓ is an angle which lies in the range 0 ✓ 2⇡ and ! is
fixed. The solution isperiodic, so the boundary conditions are
y(0) = y(2⇡);dy
d✓
����✓=0
=dy
d✓
����✓=2⇡
.
Printed: September 22, 2013 22
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(a) Write down the equation which the Green’s function G(✓, z)
satisfies.
(b) Write down the solutions for ✓ < z and ✓ > z in terms
of complex exponentials.
(c) Applying the supplied boundary conditions and continuity of
G(✓, z) at ✓ = z, simplifythe Green’s function to
G(✓, z) = A(z)�ei!✓ � e�i!✓+2i!z�2⇡i!
�
for some arbitrary function A(z).
(d) Similarly show that
G(✓, z) = A(z)(e�2⇡i!+i!✓ + e2i!z�i!✓) ✓ > z
(e) Applying the boundary condition for the derivative of G at ✓
= z, show that
A(z) =1
2i!(e�2⇡i! � 1)
Hence show that the Green’s function is
G(✓, z) =e�i!z
2i!(e�2⇡i! � 1)�ei!✓ + e�i!✓+2i!z�2⇡i!
�✓ < z
G(✓, z) =e�i!z
2i!(e�2⇡i! � 1)�e�2⇡i!+i!✓ + e2i!z�i!✓
�✓ > z
3. Find the Fourier transform of the ‘aperture function’ of a
double slit. This consists of afunction which is two top hats,
centred on x = a and x = �a. Each has a width 2b andheight 1. i.e.
the function h(x) is equal to zero, except for �a � b < x <
�a + b anda� b < x < a+ b, where it is equal to unity.Do this
using convolutions. i.e. note that h(x) is the convolution of the
sum of two deltafunctions, f(x) = �(x�a)+ �(x+a) with a top hat
g(x) = 1 if |x| < b, and g = 0 otherwise.(a) Show that the FT of
f(x) is f̃(k) = 2 cos(ka).
(b) Show that the FT of g(x) is
g̃(k) =2 sin(kb)
k.
(c) Hence write down (assuming the convolution theorem) the
Fourier transform of h(x).
4. In this question, the logic is the important part, not the
algebra so much, so write plenty ofwords so it is clear that you
understand what is going on.
Consider the equationdy
dt+ y = f(t)
for some as yet unspecified function f(t), for t > 0. The
function f(t) = 0 for t < 0.
Printed: September 22, 2013 23
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(a) Solve this first by using an integrating factor to show
that
y(t) =
Zt
0
e�(t�t0)f(t0)dt0 + Ae�t
where A is a constant.
(b) This is a convolution of f with what function (answer
carefully)?
(c) Now solve the equation using a Fourier Transform w.r.t.
t:
ỹ(!) =
Z 1
�1y(t)e�i!t dt
to show that
ỹ(!) =f̃(!)
i! + 1
where f̃(!) is the FT of f(t).
(d) We see that this is a multiplication in Fourier space. What
does this mean for thesolution in real (t) space?
(e) By inspecting the solution in (a) obtained with an
integrating factor, can you guess whatfunction has the Fourier
Transform (1+i!)�1? Support your answer by explicitly
calculatingthe FT of the function.
(f) Hence write down the general solution obtained by the
Fourier Transform method. Itis not quite the same as in part (a).
Pay careful attention to the limits, which you shouldjustify.
(g) Why are we allowed to add an extra term Ae�t to the FT
solution?
(h) If f(t) = e�2t, and y(0) = 1, find the full solution.
Printed: September 22, 2013 24