Tuesday, 10 December 2019
2 Questions: 09:30 – 10:50
3 Questions: 09:30 – 11:30
4 Questions: 09:30 – 12:10
PHYSICS 3 CLASS EXAM
Physics 3 – Theoretical Physics 3 – Chemical Physics 3
Physics with Astrophysics 3 – Combined Physics 3
[ PHYS4011, PHYS4031, PHYS4003, PHYS4017 ]
Mathematical methods I — Waves &
diffraction — Circuits & systems —
Numerical methods
Answer the question for each lecture course you have enrolled in
this semester.
Answer each question in a separate booklet
Candidates are reminded that devices able to store or display text or images
may not be used in examinations without prior arrangement.
Approximate marks are indicated in brackets as a guide for candidates.
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-986
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SECTION I – Mathematical methods I
1 A damped, driven, harmonic oscillator is described by the differential equation
d2y(t)
dt2
+
dy(t)
dt
+ y(t) = F (t) ,
where F (t) is the time-dependent driving force. For a harmonic driving force
F (t) = A sin(ωt+ φ) ,
the steady-state solution is given by
y(t) =
A
Z(ω)
sin [ωt+ φ+ ∆φ(ω)] ,
where Z(ω) and ∆φ(ω) are functions of ω .
(a) Using a simple argument involving φ, or otherwise, verify that, for a driv-
ing force
F (t) = A cos(ωt+ φ) ,
the steady-state solution is given by
y(t) =
A
Z(ω)
cos [ωt+ φ+ ∆φ(ω)] .
[1]
(b) Show that the differential equation is linear by showing that, if y1(t) and
y2(t) are solutions for the driving forces F1(t) and F2(t) , respectively, then
ay1(t) + by2(t) is a solution for the driving force aF1(t) + bF2(t) . [4]
(c) What is the steady-state solution for the driving force
F (t) = 4 sin(2t) + 3 cos(t) ?
Do not attempt to find and/or evaluate Z(...) and ∆φ(...) . [1]
For the remainder of this question, the driving force is given by a sawtooth
function:
F (t) =
{
2t for − 1 < t ≤ 1 ,
periodically repeating elsewhere .
(d) Sketch a plot of F (t). What is its period? [2]
(e) Define even and odd functions. Does F (t) fall into either of these symme-
try categories? [2]
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9863/8 Q 1 continued over. . .
Q 1 continued
(f) Outline how one can obtain the displacement of the oscillator, y(t), in
response to the sawtooth driving force by using the solution given in part (a)
in response to a sinusoidal driving force. [2]
(g) The current density J associated with a magnetic field B is given by the
equation
∇×B = µ0J ,
where µ0 is the magnetic permeability.
For the magnetic field (in Cartesian coordinates)
B = B0z
−1
 y−x
z2
 ,
where B0 is a constant, show that the associated current density is
J = −B0
µ0
z−2
 xy
2z
 .
Determine the component of the current density J in the direction (1,1,0) at
the position x = y = z = 0.5 m for a magnetic field constant B0 = 1.2 mT. [5]
(h) Show that the magnetic field above can be written as
B = B0
(
−ρ
z
eˆφ + zeˆz
)
,
where ρ, φ and z are cylindrical coordinates. [3]
[Total: 20]
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9864/8 Paper continued over. . .
SECTION II – Waves & diffraction
2 (a) Show that the wave-equation for a transverse wave oscillating in the y-
direction and travelling in the x-direction along a stretched string is given
by
∂2y
∂t2
=
T
ρ
∂2y
∂x2
,
where y is the wave amplitude, T is the tension in the string and ρ is the mass
per unit length. Make sure that you justify the steps of your proof, explaining
clearly any assumptions made. [6]
(b) Demonstrate that y(x, t) = A ei(ωt−kx) is a possible travelling wave solu-
tion of the wave equation of part a) and show that the velocity of waves on
the string is given by c =
√
T/ρ . [4]
(c) A stretched string lying along the negative x-axis is terminated at x = 0
by a “damper” with impedance Zd. The damper provides a force proportional
in magnitude to the transverse velocity of the end point of the string but acting
in the opposite direction. The damper thus absorbs some of the wave energy
incident on it.
(Note that you could choose to model the “damper” by a suitably-chosen
second string. In that case the second string would convey away some of the
energy from the first string rather than dissipate it. The effect on the first
string would be the same.)
A single frequency transverse wave propagates along the string from the
negative x-direction. Apply an appropriate boundary condition at x = 0 to
show that the amplitude reflection coefficient of the wave is given by
r =
Zs − Zd
Zs + Zd
,
where Zs and Zd are the impedances of the string and the damper respectively. [7]
(d) Is the stretched string a dispersive or non-dispersive medium? Justify
your answer. Suggest an alteration to the stretched string case that would
change its dispersive behaviour. [3]
[Total: 20]
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9865/8 Paper continued over. . .
SECTION III – Circuits & systems
3 (a) Show that the s-domain transfer function for the circuit shown below is
given by
G(s) =
1
1 + sτ
,
where τ = RC .
[3]
(b) Sketch a pole-zero diagram for the transfer function of part (a). [2]
In an effort to design an oscillator, an engineer proposed the following
circuit in which three copies of the building block of part (a) are cascaded and
a feedback loop created using an inverting operational amplifier circuit.
The engineer expected that the gain of this amplifier could be adjusted (by
choice of opamp feedback resistor) so that the circuit oscillated.
(c) What condition would have to be achieved for the circuit to oscillate? By
considering the phase response of a single block, find the expected frequency
of oscillation in terms of the time constant τ . [5]
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9866/8 Q 3 continued over. . .
Q 3 continued
(d) Defining K = Rf/R1, write down the s-domain form of the open-loop
transfer function. [1]
(e) Set down the Characteristic Equation for the feedback system. Explain
briefly the significance for feedback loop stability of the positions on the com-
plex plane of the roots of the Characteristic Equation.
[3]
(f) By using the technique of equating real and imaginary parts, find the value
of K for which the circuit just starts to oscillate. Find also the frequency of
oscillation. Does this result agree with your prediction from part (c)? [6]
[Total: 20]
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9867/8 Paper continued over. . .
SECTION IV – Numerical Methods
4 (a) State briefly what Gaussian elimination is used for. [1]
Describe how the Newton-Raphson method works and how it can be used
to find the root of a continuous function f . State any assumption that is
necessary. [5]
When this method is used, one finds
xn+1 = xn − f(xn)
f ′(xn)
.
Show that the error in the Newton-Raphson method converges quadratically. [4]
(b) The diffusion equation is given by:
∂u
∂t
= D
∂2u
∂x2
.
What class of second order partial differential equation is the diffusion equa-
tion?
[1]
The finite difference formulas for space and time are
∂2u
∂x2
=
unj+1 − 2unj + unj−1
(∆x)2
,
∂u
∂t
=
un+1j − unj
∆t
,
where x is denoted by index j, t is denoted by n, ∆x is the step size in x, and
∆t is the step size in t.
Show that
un+1j = R(u
n
j+1 + u
n
j−1) + (1− 2R)unj ,
where
R = D
∆t
(∆x)2
.
[3]
For the finite difference formula presented above, what is the condition for
the maximum time step at which numerical stability is assured? Evaluate this
maximum time step for a case in which ∆x = 0.06 m and D = 10−6 m2 s−1 [3]
The method used above is an explicit method. Name another type of
finite difference method that can be used to solve the diffusion equation that
is stable for all values of ∆t. Why is this method always stable? [3]
[Total: 20]
End of Paper
Mathematical methods I — Waves & diffraction — Circuits & systems — Numerical methods/312-9868/8 END













































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































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