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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

�L

dx sin⇣m⇡x

L

⌘sin

⇣n⇡xL

⌘=

⇢0 m 6= nL m = n

Printed: September 22, 2013 1

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� 1

3+

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� . . .


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 � 1cos(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 � 1cos(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


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 2

n⇡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.


n=0

1

(2n+ 1)2=⇡2

8.


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 2

n⇡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


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

⇡

vuut1

2

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.


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.


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 f(�k) = f ⇤(k).

2. In terms of f(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

✓� x2

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 = hk,compute the uncertainty in p, and demonstrate Heisenberg’s Uncertainty Principle,

�p�x =h

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


(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 f(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

g(k) =�4ik

4 + k4.

7. Let

ha

(x) ⌘⇢e�ax x � 00 x < 0.

(a) Show that the FT of ha

(x) is

ha

(k) =1

a+ ik.

(b) Take the FT of the equation

df

dx(x) + 2f(x) = h1(x)

and show that

f(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.


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 2

1 f(x)�(x� 3) dx

3. Evaluate Z⇡+0.1

0.1

dx

Z 4

�1

dy �(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)


(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+ !2

0z = a(t).

(a) Take the Fourier Transform of this equation (from t to !) and show that

z(!) =a(!)

!20 � !2

(b) Hence show that

z(t) =1

2⇡

Z 1

�1

a(!)

!20 � !2

ei!td!.

(c) If a(t) = sin2 ⌦t, find a(!).

(d) Hence find a solution for z(t) (ignore solutions to the homogeneous equation).


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

✓� t2

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:

h(k) =1

2⇡f(k) ⇤ g(k) =

Z 1

�1

dk0

2⇡f(k0)g(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 h(k) = f ⇤(k)g(k).

5. A signal f(x) = e�x for x > 0 and zero otherwise.(a) Show that the Fourier Transform is f(k) = (1 + ik)�1.

(b) Using Parseval’s theorem, relate the integral of the power |f(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.


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

h(!) =2a

a2 + !2.

(b) A system obeys the di↵erential equation

d2z

dt2� !2

0z = f(t).

Calculate z(!) in terms of f(!).

(c) By considering the form of z(!), 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.


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� z

as a Taylor series about z = 0, or as a power series.


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


4. Consider the equation, valid for t � 0

d2f

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).


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 Schrodinger equation inside the well (where the potential is zero) is

� h2

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)h2

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,


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.


(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 ✓.


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

�=@u(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

u(k, t) = A(k)e�ikct +B(k)eikct.

(b) Show that at t = 0,

u(k, 0) =2a

a2 + k2.

(c) Hence, applying the boundary conditions, show that

u(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).


(a) Take the Fourier Transform with respect to x, i.e.

u(k, t) =

Z 1

�1u(x, t)e�ikx dx

Note that the transform is still a function of t. Show that it obeys

@u(k, t)

@t= �k2

u(k, t).

(b) Now fix the value of k for now, and use an integrating factor to find

u(k, t) = f(k)e�k

2t/

for some (arbitrary) function f(k).

(c) From the initial condition u(x, 0) = �(x) show that

f(k) = �(k) ) u(k, t) = �(k)e�k

2t/.

(d) Using the result that the FT of e�x

2/(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


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�x

2/(2�2)

is sourced by a charge density field

⇢(x) =✏0�2

e�x

2/(2�2)

✓1� x2

�2

◆.

In doing this, you will demonstrate that

Z 1

�1

dk

2⇡k2e�k

2�

2/2eikx =

1p2⇡�3

e�x

2/(2�2)

✓1� x2

�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�x

2/(2�2) is

p2⇡� e�k

2�

2/2, and that

R1�1 e�u

2/2du =p

2⇡. (This method is of more practical use if ⇢ is known and you want �, when the directmethod 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�r

2/(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� r2

�2

◆e�r

2/(2�2).

You may assume that the Fourier Transform (w.r.t. x, ! kx

) of e�x

2/(2�2) is

p2⇡� e�k

2x

�

2/2,

and thatR1�1 e�u

2/2du =

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.


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)� 1

2gt2.

(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⇡

.


(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 f(k) = 2 cos(ka).

(b) Show that the FT of g(x) is

g(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.


(a) Solve this first by using an integrating factor to show that

y(t) =

Zt

0

e�(t�t

0)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:

y(!) =

Z 1

�1y(t)e�i!t dt

to show that

y(!) =f(!)

i! + 1

where f(!) 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.


fourier analysis workshop 1: fourier series analysis workshop 1: fourier series ... =ex,deﬁnedin0

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