SHOWING SOLUTIONS
Tuesday, 30 April 2019
09:30 – 11:00
EXAMINATION FOR THE DEGREES OF B.SC. (DESIGNATED AND
HONOURS), M.SCI. (HONOURS) AND M.SC.
[ PHYS4031 ]
Waves & Diffraction
Candidates should answer Question 1 (16 marks)
and either Question 2A or Question 2B (24 marks each)
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.
PHYS4031 Waves & Diffraction
Fundamental constants
name symbol value
speed of light c 2.998× 108 m s−1
permeability of free space µ0 4pi × 10−7 H m−1
permittivity of free space 0 8.854× 10−12 F m−1
electronic charge e 1.602× 10−19 C
Avogadro’s number N0 6.022× 1023 mol−1
electron rest mass me 9.110× 10−31 kg
proton rest mass mp 1.673× 10−27 kg
neutron rest mass mn 1.675× 10−27 kg
Faraday’s constant F 9.649× 10−4 C mol−1
Planck’s constant h 6.626× 10−34 J s
fine structure constant α 7.297× 10−3
electron charge to mass ratio e/me 1.759× 1011 C kg−1
quantum/charge ratio h/e 4.136× 10−15 J s C−1
electron Compton wavelength λe 2.426× 10−12 m
proton Compton wavelength λp 1.321× 10−15 m
Rydberg constant R 1.097× 107 m−1
Bohr radius a0 5.292× 10−11 m
Bohr magneton µB 9.274× 10−24 J T−1
nuclear magneton µN 5.051× 10−27 J T−1
proton magnetic moment µp 1.411× 10−26 J T−1
universal gas constant R 8.314 J K−1 mol−1
normal volume of ideal gas – 2.241× 10−2 m3 mol−1
Boltzmann constant kB 1.381× 10−23 J K−1
First radiation constant 2pihc2 c1 3.742× 10−16 W m2
Second Radiation constant hc/kB c2 1.439× 10−2 m K
Wien displacement constant b 2.898× 10−3 m K
Stefan-Boltzmann constant σ 5.670× 10−8 W m−2 K−4
gravitational constant G 6.673× 10−11 m3 kg−1 s−2
impedance of free space Z0 3.767× 102 Ω
Derived units
quantity dimensions∗ derived unit
energy ML2T−2 J
force MLT−2 N
frequency T−1 Hz
gravitational field strength LT−2 N kg−1
gravitational potential L2T−2 J kg−1
power ML2T−3 W
entropy ML2T−2 J K−1
heat ML2T−2 J
capacitance M−1L−2T 4I2 F
charge IT C
current I A
electric dipole moment LTI C m
electric displacement L−2TI C m−2
electric polarisation L−2TI C m−2
electric field strength MLT−3I−1 V m−1
electric (displacement) flux TI C
electric potential ML2T−3I−1 V
inductance ML2T−2I−2 H
magnetic dipole moment L2I A m2
magnetic field strength L−1I A m−1
magnetic flux ML2T−2I−1 Wb
magnetic induction MT−2I−1 T
magnetisation L−1I A m−1
permeability MLT−2I−2 H m−1
permittivity M−1L−3T 4I2 F m−1
resistance ML2T−3I−2 Ω
resistivity ML3T−3I−2 Ω m
∗ M = mass, L = length, T = time, I = current
SHOWING SOLUTIONS
1 (a) A string of length 2 m and total mass 0.1 kg is placed under a tension of
30 N.
i. Calculate the propagation speed of transverse waves on the string. [2]
Solution: Unseen problem requiring recall of standard relations and
deduction of appropriate units.
The linear density ρ = 0.12 = 0.05 kg m
−1 .
c =
√
T
ρ
=
√
30
0.05
=
√
600 = 10
√
6 = 24.5 m s−1 .
[. . . 2]
ii. Define the impedance of a stretched string. Calculate the impedance of
the string with parameters given above. Ensure you quote appropriate
units for your result. [3]
Solution: The impedance of a stretched string is the ratio of transverse
force to transverse velocity. The transverse force is −T ∂y∂x . For
non-dispersive propagation ∂y∂x = −1c ∂y∂t , so the transverse force is Tc ∂y∂t . Thus
Z = Tc =
T√
T/ρ
=
√
Tρ . [. . . 2]
So for the problem given, Z =
√
30× 0.05 = √1.5 = 1.22 N s m−1 . [. . . 1]
iii. The ratio of reflected amplitude to incident amplitude for waves inci-
dent from a medium of impedance Z1 on a boundary with a medium of
impedance Z2 is
Z1 − Z2
Z1 + Z2
.
What is the ratio of reflected energy to incident energy? Hence deduce an
expression for the ratio of transmitted energy to incident energy. Finally,
deduce an expression for the ratio of the reflected wave amplitude to the
incident amplitude. [4]
Solution: Bookwork, with a small unseen element.
The ratio of reflected energy to incident energy = Z1Z1
[
Z1−Z2
Z1+Z2
]2
where
the first term is included here – though the numerator and denominator
cancel – as a reminder that the impedance of the string in which a wave
propagates must be taken into account when moving from wave amplitude to
energy. This will not be required in a student’s solution. [. . . 1]
Conservation of energy then dictates that the ratio of transmitted
energy to incident energy must be
1− Z1
Z1
[
Z1 − Z2
Z1 + Z2
]2
PHYS4031 Waves & Diffraction 3/16 Q 1 continued over. . .
Q 1 continued
SHOWING SOLUTIONS
=
(Z1 + Z2)
2 − (Z1 − Z2)2
(Z1 + Z2)2
=
4Z1Z2
(Z1 + Z2)2
.
[. . . 1]
The ratio of transmitted energy to incident energy is Z2Z1
C2
A2
, where we
denote the incident wave amplitude by A and the transmitted wave
amplitude by C . [. . . 1]
The ratio of reflected amplitude to incident amplitude is
C
A
=
√
C2
A2
=
√
Z1
Z2
[
4Z1Z2
(Z1 + Z2)2
]
=
√
4Z1
2
(Z1 + Z2)2
=
2Z1
Z1 + Z2
.
[. . . 1]
(b) Waves propagating in a particular medium follow a dispersion relation
given by
ω = ak3 − bk2 ,
where a and b are constants and k is the wavevector. Obtain expressions for
the phase and group velocities of the propagating waves and show that they
are equal for k = b/(2a) . [3]
Solution: Standard knowledge plus application to an unseen problem.
Phase velocity = ωk = ak
2 − bk . [. . . 1]
Group velocity = ∂ω∂k = 3ak
2 − 2bk . [. . . 1]
Apart from the trivial solution when k = 0 these velocities are equal when
ak2 − bk = 3k2 − 4bk ⇒ ak − b = 3ak − 2b⇒ k = b2a . [. . . 1]
Sketch the 2-dimensional Fraunhofer diffraction pattern that would arise
from plane wave illumination of a circular aperture. Referring to your diagram,
give an expression for the angle subtended at the aperture by the first signif-
icant feature in the diffraction pattern. Without detailed calculation, sketch
the intensity profile along a straight line through the midpoint of the pattern. [4]
Solution:
Bookwork
2 marks for circular ring pattern plus formula, and 2 marks for a rough sketch
of the radial intensity distribution.
PHYS4031 Waves & Diffraction 4/16 Q 1 continued over. . .
Q 1 continued
SHOWING SOLUTIONS
[. . . 4]
PHYS4031 Waves & Diffraction 5/16 Paper continued over. . .
SHOWING SOLUTIONS
2A A string of length L and linear density ρ lies along the x-axis and is stretched
under tension T . The ends of the string are fixed at x = 0 and x = L .
(a) Sketch the first four normal modes of vibration of the string. [2]
Solution: Standard knowledge
Mode shapes for n = 1...4 are plotted below. Half a mark will be gained for
each of the four plots.
(In the plot, the horizontal axis extends from x = 0 to x = 1, corresponding to
the 0 to L points along the string length. The vertical axis is the transverse
displacement. All modes are illustrated with the same amplitude.) [. . . 2]
(b) The string is initially undisplaced and at time t = 0 is given a velocity
profile
v(x) =
4v0
L2
x(L− x)
for a positive constant v0. Sketch this velocity profile. Without calculation,
discuss the expected excitation of normal modes of vibration of the string that
would result from this initial perturbation. [4]
Solution: Unseen problem; related problems have featured in examples sheet
and in past papers.
The initial velocity profile is a quadratic in x. The velocity is 0 at x = 0 and
x = L and has a maximum value of v0 at x = L/2.˙
PHYS4031 Waves & Diffraction 6/16 Q 2A continued over. . .
Q 2A continued
SHOWING SOLUTIONS
(In the plot, the horizontal axis extends from x = 0 to x = 1, corresponding to
the 0 to L points along the string length. The vertical v(x) axis is scaled so that 1
corresponds to the maximum initial transverse velocity v0 .) [. . . 2]
The shape of the velocity profile is similar to that of the n = 1 mode. It is not
precisely the same, however, so the expectation would be that the fundamental
mode would be the dominant one excited, with smaller amplitudes of higher order
modes. The initial profile has a large value at the midpoint of the string, so even
order modes – which would have a node at the midpoint – cannot be excited. [. . . 2]
(c) The motion of such a string can, in general, be described by a Fourier
decomposition of the form
y(x, t) =
∞∑
n=1
sin
(npix
L
)[
An cos
(npict
L
)
+Bn sin
(npict
L
)]
.
Explaining carefully each step of your calculation, show that for the given
initial condition the subsequent string motion can be expressed as
y(x, t) =
∞∑
n=1,3,5,...
32v0L
n4pi4c
sin
(npix
L
)
sin
(npict
L
)
,
where c is the propagation speed of transverse waves on the string. [10]
PHYS4031 Waves & Diffraction 7/16 Q 2A continued over. . .
Q 2A continued
SHOWING SOLUTIONS
Solution: Standard type of problem, but not seen before in this form.
Note that for educational purposes the solution given below includes even
relatively minor steps in the argument. A typical student-presented solution could
be expected to be significantly more concise.
y(x, t) =
∞∑
n=1
sin
(npix
L
)[
An cos
(
npict
L
)
+Bn sin
(
npict
L
)]
.
The string is initially undisplaced, so y(x, 0) = 0 for all x along the string.
y(x, 0) =
∞∑
n=1
sin
(npix
L
)
[An] = 0 ⇒ An = 0
for all n . [. . . 2]
y(x, t) =
∞∑
n=1
Bn sin
(npix
L
)
sin
(
npict
L
)
.
We are given an initial velocity profile so we need
y˙(x, t) =
∞∑
n=1
Bn
(npic
L
)
sin
(npix
L
)
cos
(
npict
L
)
⇒ y˙(x, 0) =
∞∑
n=1
Bn
(npic
L
)
sin
(npix
L
)
.
Thus,
4v0
L2
x(L− x) =
∞∑
n=1
Bn
(npic
L
)
sin
(npix
L
)
.
[. . . 2]
We solve this by multiplying both sides by sin mpixL and then integrating both
sides from 0 to L , making use of the orthogonality of the sine function to evaluate
the right hand side. Here are the steps:
4v0
L2
x(L− x) sin
(mpix
L
)
=
∞∑
n=1
Bn
(npic
L
)
sin
(npix
L
)
sin
(mpix
L
)
∫ L
0
4v0
L2
x(L− x) sin
(mpix
L
)
dx =
∫ L
0
∞∑
n=1
Bn
(npic
L
)
sin
(npix
L
)
sin
(mpix
L
)
dx
∫ L
0
4v0
L2
x(L− x) sin
(mpix
L
)
dx =
∞∑
n=1
∫ L
0
Bn
(npic
L
)
sin
(npix
L
)
sin
(mpix
L
)
dx
4v0
L2
∫ L
0
x(L− x) sin
(mpix
L
)
dx =
∞∑
n=1
Bn
(npic
L
)∫ L
0
sin
(npix
L
)
sin
(mpix
L
)
dx
PHYS4031 Waves & Diffraction 8/16 Q 2A continued over. . .
Q 2A continued
SHOWING SOLUTIONS
The set of integrals under the sum each evaluate to 0 for m 6= n ; for m = n the
integral evaluates to L/2 . [. . . 1]
To find the Bm coefficients we then have to solve
Bm
(mpic
L
) L
2
L2
4v0
= Bm
(
mpicL2
8v0
)
=
∫ L
0
x(L− x) sin
(mpix
L
)
dx.
[. . . 1]
Using the given integrals (which are readily obtained by integration by parts)
the integral on the right hand side becomes
L
∫ L
0
x sin
mpi
L
xdx−
∫ L
0
x2 sin
mpi
L
xdx
= L
[
− Lx
mpi
cos
(mpix
L
)
+
L2
m2pi2
sin
(mpix
L
)]L
0
−
[
2L3
m3pi3
cos
(mpix
L
)
− Lx
2
mpi
cos
(mpix
L
)
+
2L2x
m2pi2
sin
(mpix
L
)]L
0
= L
[
− L
2
mpi
cosmpi + 0− (−0)− 0
]
−
[
2L3
m3pi3
cosmpi − L
3
mpi
cosmpi + 0− 2L
3
m3pi3
+ 0− 0
]
= − L
3
mpi
cosmpi − 2L
3
m3pi3
cosmpi +
L3
mpi
cosmpi +
2L3
m3pi3
=
2L3
m3pi3
[1− cosmpi] .
[. . . 2]
This equals 0 for m even - as expected. For m odd, we then have the result
Bm
(
mpicL2
8v0
)
=
2L3
m3pi3
[1− (−1)]
⇒ Bm =
(
8v0
mpicL2
)
2L3
m3pi3
[1− (−1)]
⇒ Bm = 32v0L
m4pi4c
.
[. . . 2]
(d) The energy associated with the nth normal mode of vibration of the string
can be expressed as
ρL
4
ωn
2
(
An
2 +Bn
2
)
,
where ωn is the angular frequency of the mode.
Sum the energies of all the excited normal modes to show that the total
energy of vibration is 415Mv0
2 , where M is the total mass of the string. [4]
PHYS4031 Waves & Diffraction 9/16 Q 2A continued over. . .
Q 2A continued
SHOWING SOLUTIONS
Solution: Standard type of calculation, but a new example.
Since, by earlier analysis, or by inspection of the result to be proved in part
(c), An = 0 for all n , [. . . 1]
then En =
ρL
4 ωn
2Bn
2 . Substituting
ωn =
npic
L
,
[. . . 1]
we then have for n odd (Bn = 0 for n even)
En =
ρL
4
(npic
L
)2( 32v0L
m4pi4c
)2
= 256ρLv0
2 1
pi6
1
n6
.
Since M = ρL, we have, for odd n ,
En = Mv0
2 256
pi6
1
n6
and the total energy, summed over all modes, is
Etotal = Mv0
2 256
pi6
∑
n odd
1
n6
.
[. . . 1]
The summation required is the one given, so the final result is
Etotal = Mv0
2 256
pi6
pi6
960
= Mv0
2 4
15
.
[. . . 1]
(e) Calculate the total kinetic energy given to the string by the initial velocity
profile and show that this agrees with the sum of normal mode energies. [4]
Solution: The initial energy is all kinetic. We start with an expression for
the KE in a segment dx of the string:
Eelement =
1
2
(ρdx) v2 =
1
2
(ρdx)
(
4v0
L2
x(L− x)
)2
[. . . 1]
=
8ρv0
2
L4
[x(L− x)]2
and the total KE is then obtained by evaluating the integral
Etotal =
8ρv0
2
L4
∫ L
0
x2(L− x)2dx
PHYS4031 Waves & Diffraction 10/16 Q 2A continued over. . .
Q 2A continued
SHOWING SOLUTIONS
[. . . 1]
=
8ρv0
2
L4
∫ L
0
(L2x2 − 2Lx3 + x4) dx = 8ρv0
2
L4
[
L2x3
3
− 2Lx
4
4
+
x5
5
]L
0
=
8ρv0
2
L4
∫ L
0
(L2x2 − 2Lx3 + x4) dx = 8ρv0
2
L4
[
L5
3
− L
5
2
+
L5
5
]
= 8ρLv0
2
[
1
3
− 1
2
+
1
5
]
= 8ρLv0
2
[
10− 15 + 6
30
]
= 8ρLv0
2
[
1
30
]
=
4
15
Mv0
2,
as expected. [. . . 2]
You may find the following relations useful:∫
x sin ax dx = −x
a
cos ax+
1
a2
sin ax
∫
x2 sin ax dx =
2
a3
cos ax− x
2
a
cos ax+
2x
a2
sin ax
∞∑
n=1,3,5,...
1
n6
=
pi6
960
PHYS4031 Waves & Diffraction 11/16 Paper continued over. . .
SHOWING SOLUTIONS
2B The Fresnel-Kirchhoff diffraction formula for the field at a point P due to
diffraction at an aperture Q can be expressed as
ψP = − i
λ
∫
Q
eik r
r
ψQ
(
n + n′
2
)
· dΣ ,
where ψQ is the field at the aperture and other symbols have their usual
meanings.
(a) With the aid of a diagram, explain the meaning of the symbols that appear
in the Fresnel-Kirchhoff diffraction formula. [4]
Solution: General understanding
Figure: From Applications of Classical Physics, by
Roger D. Blandford and Kip S. Thorne, available at
http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1208.1.K.pdf.
Formally the integral to calculate the field at an interior point P starts as
being evaluated over a closed surface around P that includes the aperture Q .
Taking most of the surface to be far away from the aperture (plus some other -
irrelevant here - mathematical tricks near the aperture) makes only the
contribution to the integral coming from evaluation over the aperture significant. [. . . 1]
The i term gives a phase offset and e
ikr
r gives the amplitude of a
spherically-expanding wavelet of wavevector k at a distance r. [. . . 1]
The vector area element dΣ, pointing inwards, is the element of the surface of
integration. It’s shown here for a generic location on the closed surface, however,
PHYS4031 Waves & Diffraction 12/16 Q 2B continued over. . .
Q 2B continued
SHOWING SOLUTIONS
for the reason given above, a student may well jump to showing it only as an
element of the surface within the aperture.
Regardless, r is the distance from the area element to the point of interest, P . [. . . 1]
The unit vector n′ is in the direction of propagation of the illuminating plane
waves incident on the aperture, and the unit vector n is in the direction from the
aperture to the point P . [. . . 1]
When evaluating the Fraunhofer case, integrating over the aperture the
obliquity factor
(
n+n′
2
)
· dΣ becomes dΣ . This factor suppresses
backward-propagating rays. [. . . 1]
NB. Full marks will be awarded for solutions containing a majority of the
above information.
(b) Discuss briefly the ways in which this result was an improvement over the
previous Huygens-wavelet approach to modeling diffraction. [3]
Solution: General understanding
The Fresnel-Kirchhoff treatment correctly includes the wavelength dependence
of the diffraction process. It also provides the correct prediction of phase. The
inclusion of the obliquity factor removes what would otherwise be a backward
propagating wave from the spherical Huygens wavelets. [. . . 3]
(c) When applied to the diffraction of light, the Fresnel-Kirchhoff result is
not complete in its description of the diffraction process. Why is this? [2]
Solution: General understanding
The Fresnel-Kirchhoff treatment calculates diffraction of a scalar field. Light,
being an electromagnetic wave, is a vector quantity. The principal omission is that
the scalar treatment cannot include consideration of polarisation effects. [. . . 2]
(d) A diffraction apparatus is set up with parallel illumination incident on
a rectangular aperture – of width a in the x-direction and height 3a in the
y-direction – followed by a lens of focal length f that focuses the resulting
diffraction pattern onto a screen.
Explain why this setup can produce a Fraunhofer diffraction pattern on
the screen.
By taking the Fourier Transform of the diffracting aperture, show that
the intensity in the Fraunhofer diffraction pattern as a function of spatial
frequencies kx and ky is given by
I(kx, ky) = I0 sinc
2
[
3aky
2
]
sinc2
[
akx
2
]
,
PHYS4031 Waves & Diffraction 13/16 Q 2B continued over. . .
Q 2B continued
SHOWING SOLUTIONS
where I0 is the intensity on the optic axis. [5]
Solution: General understanding applied to unseen but standard type of
problem.
Fraunhofer diffraction is observed in the far-field where the diffracted waves
are essentially parallel plane waves. The focusing lens converges these to form a
more local image in the focal plane of the lens – which is where the screen should
be placed. [. . . 2]
With the aperture centred about the origin, it is described by an aperture
function A(x, y) that is unity for −a/2 ≤ x ≤ a/2 and −3a/2 ≤ y ≤ 3a/2 and zero
elsewhere. Then the transmitted amplitude becomes
A′ (kx, ky) =
3a
2∫
− 3a
2
a
2∫
−a
2
e−i
[
x kx+ y ky
]
dxdy
=
3a
2∫
− 3a
2
e−iy ky dy ×
a
2∫
−a
2
e−ix kx dx
[. . . 1]
=
[
− 1
iky
] [
e−iy ky
]+ 3a
2
− 3a
2
×
[
− 1
ikx
] [
e−ix kx
]+a
2
−a
2
=
[
− 1
iky
] [
−2i sin 3aky
2
]
×
[
− 1
ikx
] [
−2i sin akx
2
]
[. . . 1]
=
[
3a
sin
3aky
2
3aky
2
]
×
[
a
sin akx2
akx
2
]
= 3a2 sinc
[
3aky
2
]
sinc
[
akx
2
]
⇒ I(kx, ky) = I0 sinc2
[
3aky
2
]
sinc2
[
akx
2
]
[. . . 1]
(e) Sketch this diffraction pattern intensity, making clear in your sketch the
relative orientations of the aperture and the diffraction pattern and the relative
spatial frequency values of intensity nulls. [5]
Solution: General understanding applied to unseen but standard type of
problem.
A plot along the lines of the picture below is expected. There should be an
indication of the 3:1 (kx : ky) difference in null spacing along the two axes, the
rectangular shape of the maxima should be oriented orthogonally to the the axes
of the rectangular slit, and there should be an indication that the intensity in the
secondary maxima diminishes as spatial frequency increases (e.g. by sketching of
rough contours). [. . . 5]
PHYS4031 Waves & Diffraction 14/16 Q 2B continued over. . .
Q 2B continued
SHOWING SOLUTIONS
(f) If the illuminating light has a wavelength of 500 nm, the width of the
aperture in the x-direction is 30µm and the focal length of the lens is 0.6 m, find
the position coordinates on the screen of the first two nulls in each dimension
of the diffraction pattern. [5]
Solution: Unseen numerical problem
The nulls are when the sinc functions first become zero. These will occur at
3aky
2 = ±pi and akx2 = ±pi , ⇒ ky = ±2pi3a and kx = ±2pia . [. . . 2]
The coordinates on the screen are determined by the angle corresponding to
these spatial frequencies and the focal length f of the image-forming lens. The lens
converges parallel rays originating at an angle θ, to a point that is displaced from
the (0, 0) point on the screen by a distance fθ, provided θ is small. [. . . 1]
So in this case, the first two nulls in the x-direction are located at
X = ±fθX = ±f
kx
k
= ±f
[
2pi
a
] [
1
k
]
= ±f λ
a
= ±0.6×
[
500× 10−9
30× 10−6
]
= ±10 mm .
Similarly, for the y-direction, the first two nulls are located at
Y = ±fθY = ±f
ky
k
= ±f
[
2pi
3a
] [
1
k
]
= ±f λ
3a
= ±0.6×
[
500× 10−9
90× 10−6
]
= ±3.3 mm .
[. . . 2]
PHYS4031 Waves & Diffraction 15/16 Paper continued over. . .
SHOWING SOLUTIONS
End of Paper
NOTE: SHOWING SOLUTIONS
PHYS4031 Waves & Diffraction 16/16 END































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































学霸联盟