Methods of Mathematical Physics I
Assignment 1
Due in Class on Friday, September 06, 2024.
Solve all the problems. Each problem carries 12 marks.
Problem 1. The Dirac δ-function is dened under the integral as
t
0+
t
0−
f(t) δ(t − t
0
) dt =
f(t
0
) t
0−
< t < t
0+
0 otherwise
a) i. Simplify the integral
∞
−∞
dx f(x) δ(−ax + b),
where a and b are real, positive constants.
ii. The function δ(cos x) can be written as a sum of Dirac δ-functions
δ(cos x) =
n
a
n
δ(x − x
n
).
Find the range for n and the values for the a
n
and the x
n
.
b) Evaluate the following integrals which involve the Dirac δ-function:
i.
0
−3
dx δ(x − 1)
ii.
∞
−∞
dx (x
2
+ 3) δ(x − 5)
iii.
5
−5
dx x δ(x
2
− 5)
iv.
3π/4
π/4
dx x
2
δ(cos x)
v.
10
−10
dx (x
2
+ 3) δ
′
(x − 5)
c) The identity
∞
−∞
dt sin(t)/t = π is called the Dirichlet integral, and is a very important result.
Establish this result by any means.
d) Therefore, show that
∞
−∞
dt
lim
n→∞
sin nt
πt
f(t) = f(0)
and hence establish that the limiting sinc function is a valid Dirac δ-function.
e) An ideal impulse moving in space and time can be described by the function
f(x, t) = I
0
δ(x − v
0
t),
where v
0
is a constant. Make a 3-dimensional plot for f(x, t) vs x and t to show how this
impulse propagates. What are the dimensions of v
0
? How does the plot change if
f(x, t) = I
0
δ(x − a
0
t
2
/2),
2
where a
0
is a constant? What are the dimensions of a
0
?
f) A single point charge q
0
is located at (1, 1, 0) in a Cartesian coordinate system, so that its
charge density can be expressed as
ρ
c
(x, y, z) = q
0
δ(x − 1)δ(y − 1)δ(z).
i. What is ρ
c
(r, θ, z), its charge density in cylindrical coordinates?
ii. What is ρ
c
(r, θ, φ), its charge density in spherical coordinates?
Problem 2. For a periodic signal f(t) with period T
0
, the Fourier series expansion is given by
f(t) =
∞
n=0
[a
n
cos ω
n
t + b
n
sin ω
n
t]
.
a) Show that f(t) = f(t + mT
0
), m = 1. A Fourier series for f(t) can be constructed that uses
ω
n
= 2πn/mT
0
. Show that this is the same Fourier series that is generated using ω
n
= 2πn/T
0
.
b) Consider the function
f(t) =
∞
n=−∞
δ(t − nT
0
),
where n is an integer and δ(x) is the Dirac δ-function.
i. Make a labelled plot of f(t) vs t.
ii. Determine the fundamental frequency and the coecients a
n
and b
n
of the Fourier series
dened above.
iii. The δ-function is highly discontinuous. We should suspect some odd behaviour with the
Fourier series representation of this function. What is unusual about the coecients de-
termined in (ii)?
Problem 3. A string of length L, xed at its two ends, is plucked at its mid-point by an amount
A and then released.
a) Prove that the subsequent displacement is given by
φ(x, t) =
∞
n=0
8A
π
2
(2n + 1)
2
sin
(2n + 1)πx
L
cos
(2n + 1)πct
L
,
where, in the usual notation, c
2
= T/ρ.
b) Find the total kinetic energy of the string when it passes through its unplucked position, by
calculating it in each mode and summing, using the result
∞
n=0
1
(2n + 1)
2
=
π
2
8
.
c) Conrm that the total energy is equal to the work done in plucking the string.
3
Problem 4. Consider the one-dimensional wave equation
∂
2
φ
∂x
2
−
1
c
2
∂
2
φ
∂t
2
= 0,
a) Obtain a solution φ(x, t) with units and wave speed=1, of the wave equation above, which has
∂φ/∂t = 0 and φ = δ(x − x
0
) at t = 0, by determining the functions C and D in D’Alembert’s
general solution
φ(x, t) = C(t − x/c) + D(t + x/c).
b) Obtain, by any method, a solution which satises the above wave equation and has
φ(x = 0, t) = e
−iωt
, i.e. for a string harmonically shaken at x = 0. The equation should be
satised at x > 0 and x < 0, all t.
c) What happens at x = 0?
Problem 5. Dene the Heaviside step function
Θ(x) =
0 x < 0
1 x > 0
1
2
x = 0
a) Justify the statement
˜
Θ(k) =
1
i(k − iϵ)
(ϵ → 0 from above).
b) Then nd the Fourier transform of Θ(x − a)e
−bx
, where a, b > 0, using the above result and
taking ϵ → 0 and applying the shift/translation property of Fourier transform.