Teacher's Notes
P
W
(N
ON
L
IGHT
)
Duration: 25 mins
Years: 10-12 page 2
SUMMARY
1. Mechanisms
The point made here is that although waves may look similar their mechanisms may be entirely different.
Types of Waves
Waves are generally described as transverse or longitudinal. These terms are mentioned in this video but as they are descriptions of appearance only, they are of little use when analysing the mechanisms of different waves. It is tempting to see all transverse waves as being the same, yet this is not the case. In this video we concentrate on the force causing the particles of matter to move.
A wave will occurr if we have an oscillating movement transmitted to neighbouring particles with a delay.
In wave movement energy travels while the matter stays more or less where it is.
Tension Waves
We use a chain to show tension waves as there can be no doubt that a chain can exert a force in only one way, tension. The same with a waving flag. Surface tension carries waves known as ripples.
Gravity Waves
To distinguish these from gravity waves through space we refer to them as gravity waves in water. Large ocean waves are gravity waves. Tides are gravity waves driven by the moon's gravity. We also have tides in our atmosphere.
Push Waves or Compression Waves
Sound waves are longitudinal. The force driving the wave is compression.
Shear Waves
A line of trolleys joined with elastic makes an excellent way of demonstrating a shear wave in slow motion.
Another way is to use a large block of gelatine. It should really be floating in water or zero gravity. The examples we show are affected by the table-top and are not pure shear waves.
page 3
Shear waves cannot cross liquid because liquid has an insignificant shear strength.
Because shear waves from earthquakes do not pass through the centre of the earth, it was postulated that the centre of the earth was surrounded by liquid (the centre is solid).
Torsion Waves
A series of weighted rods hanging on a spring steel strap make a torsional pendulum. The restoring force is assumed to be all due to the torsional force in the spring steel strap. (In fact, if the strap is vertical, part of the force will be due to gravity.)
Other types of Waves
This section raises the question:
With every force you should be able to imagine and produce a wave. The idea is to remove inhibitions to thinking about waves.
2. Characteristics of Waves
Although the mechanisms of waves are entirely different, there are characteristics common to all: l Amplitude l Wave length l Velocity l Period l Frequency
The velocity of a wave can be described by v = f
λ when v = velocity f = frequency Hz l = wave length (lambda)
We demystify this formula using a passing train.

Page 4
3. Wave Behaviour
Velocity through a medium. When waves travel through matter their velocity is affected by the restoring force, and the intertia of the matter.
Velocity in a stretched string: v = tension in string mass per length of string
Velocity of sound in gas: v = gas pressure density
Velocity of sound in solids: v = bulk modulus of elasticity density
Velocity of gravity waves in water: v = g
λ tanh 2
π
depth
2
π λ
Velocity of surface tension waves - ripples: v = 2
π surface tension
λ density
Refraction
Refraction is the changing of velocity due to the changing of the medium characteristics. The bending of sound can be due to refraction, but it is not refraction itself.
Reflection
Normally reflection is taught before refraction but refraction fits next to velocity and reflection introduces superposition. Once again, reflection is a property of all waves no matter their type.
Page 5
Superposition
We look at superposition of gravity waves in water and torsional waves.
We show the graphical adding of two waves and test these in a wave flume at Manly Vale Water Research Laboratory.
Wave period 4 sec & 0.5 sec
Wave period 1 sec & 1.2 sec
This leads to beats with a period of 6 sec.
Diffraction
Water passing to headland.
Each particle is a source of waves.
Waves passing through a gap, diffracting.
Interference patterns caused by two gaps.
Stationary or Standing Waves
Rope bridge, phone cord.
Sheet of gelatine - free ends.
Lines of trolleys joined with elastic - free ends.
First and second harmonic - we ignore fixed ends and look at free ends only.
Shear wave standing.
Torsional wave standing.
Gravity wave in water standing as a syche;
in a river.
Main Misconceptions of Students
From a paper by Joseph Snir.
1. Students confuse frequency and period.
2. Phase is poorly understood
3. Most students are unable to distinquish between the motion of the substance and the motion of the disturbance through the medium.
4. Independence of amplitude, velocity and frequency.
Students often believe that rapid oscillation will produce faster waves and will produce larger amplitude. Some students think small amplitude produces slow waves.
Page 6
5. Many students assume a collision of two waves will be similar to a mechanical collision and the waves will cancel each other.
6. In complex phenomena, such as standing waves or beats, students see a stable or static phenomenon rather than the combination of two dynamic waves.
Exercises
1. Invent new waves, look at different forces, and different oscillating motions and different planes.
2. Invent a wave that is a combination of two forces.
3. How many different waves could you make out of a line of cans joined by elastic?
4. Could you make a wave out of cans joined by rope?
Clue: Consider gravity.
5. Would gravity waves in water be faster or slower on the moon?
6. Will air pressure affect the speed of sound in air?
v = p (p = pressure; d = density)?
d
7. How would you find out if the moon had a liquid centre?
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Phone: 1800 644 254 - Fax: 1800 282 129

1 (a) What is the wavelength of the sound wave produced by a tuning fork of frequency 384 Hz played in air? Take the speed of sound as 330 ms
-1
.
(b) An experiment determines that the wavelength of the sound produced by a 256 Hz tuning fork is
1.30 m. What is the speed of sound as determined by the experiment?
2 Why is it that the frequency of a wave remains constant when it travels through different materials?
3 If two sounds have the same frequency but one is louder than the other. How do the two sound waves differ?
4 An echo sounder on a ship receives a pulse reflected from the sea bed after 0.06 s. What is the
-1 depth of the sea bed? The speed of sound in water is 1440 ms .
5 What is the frequency of X-rays of wavelength 33
×
10 -9 m. The wave speed is 3
×
10 8 ms -1 .
6 The following graph shows the displacement against time for an object undergoing wave motion.
1.0
0.5
0.0
-0.5
-1.0
0.0
2.0
4.0
6.0
Time (s)
8.0
10.0
12.0
(a) Use the graph to determine the amplitude of this motion.
(b) Determine the period of the motion.
(c) If the velocity of the wave is 30 ms -1 what is the wavelength of the wave?
7 What happens to the speed of a pulse in a spring when the tension is increased?
8 What is the difference between a transverse wave and a longitudinal wave?
9 A student blows across the mouth of a closed tube of length 0.322 m. What is the frequency of the fundamental note emitted?
10 An open tube of length 66 cm resonates with a tuning fork of frequency 256 Hz.
(a) What is the wavelength of the vibration in the tube?
(b) Use the data to determine a value for the speed of sound in air.
11 A stretched string is tuned to a frequency of 384 Hz. Indicate two ways of producing a note of
768 Hz from the string.
12 A stretched string of length 0.60 m is tuned to resonate with a 256 Hz tuning fork. The string vibrates in its fundamental mode.
(a) What is the wavelength of the vibration in the spring?
(b) What is the wavelength of the sound note emitted? Take the he speed of sound as 333 ms
-1
.
13 A stroboscope is used to freeze the wave pattern in a ripple tank. The highest frequency at which the waves ‘freeze’ is 40 Hz. At this point there are 10 wavelengths in a distance of 0.80 m.
Determine the wave speed.
14 The following graph shows the displacement against time for an object undergoing wave motion.

1.25
1.00
0.75
0.50
0.25
time (s)
0.00
-0.25
0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144
-0.50
-0.75
-1.00
-1.25
(a) Use the graph to determine the amplitude of this motion.
(b) Determine the period of the motion.
(c) If the velocity of the wave is 10 ms -1 what is the wavelength of the wave?
1 The following diagram represents a side view of water waves.
Scale
0 10 20 30 40 50 60 m
What is the wavelength of the water waves.
2 The following diagram represents a waveform, whose frequency is 50 Hz, at a certain instant of time.

2
0
1
1 2 3 4 5 6 7 8 distance (mm)
-1
-2
(a) What is the amplitude of the wave?
(b) What is the wavelength of the wave?
(c) What is the period of the wave?
(d) What is the wave velocity?
3 The following graph shows the displacement against time for an object moving with simple wave motion.
2.0
1.0
0.0
-1.0
-2.0
0.0
1.0
2.0
3.0
Time (s)
4.0
5.0
6.0
(a) Determine the amplitude of this motion.
(b) Determine the period of this motion.
4 The following graph shows the displacement against time for an object moving with simple harmonic motion.

0.0
-4.0
8.0
4.0
-8.0
0.0
0.50
1.00
1.50
2.00
2.50
3.00
time (s)
(a) Determine the amplitude of this motion.
(b) Determine the period of this motion.
(c) If the velocity of the wave is 10 ms-1 what is the wavelength of the wave?
5 The following graph shows the displacement against time for a point on a wave.
2
1
0
-1
-2
0 1 2 3 4 5 6 7 8 9 10 time (ms)
(a) Determine the amplitude of this motion.
(b) Determine the period of this motion.
(c) If the velocity of the wave is 200 ms-1 what is the wavelength of the wave?
6 Electromagnetic waves such as light travel at a speed of 3.0 × 108 ms-1. The wavelengths of visible light range approximately from 400 nm (violet) to 760 nm (red). What is the frequency range of visible light?
7 The following diagram shows the pattern of water-wave crests in the shallow end of a large pool.
The pattern in the deep end is not shown though the wave speed is double that in the shallow end.
The waves are produced by a vibrator of frequency 50 Hz.

shallow deep
0 50 mm scale
(a) What is the period of the waves in the shallow end?
(b) What is the wavelength of the waves in the shallow end?
(c) What is the speed of the waves in the shallow end?
(d) What is the frequency of the waves in the deep end?
(e) What is the speed of the waves in the deep end?
(f) What is the wavelength of the waves in the deep end?
8 The diagram represents a wave in the same piece of string at two different instances of time. y
(I) 0 t = 0.00 s x
(II) y
1 2 3 4 5 6 7 8 mm
0 t = 1.0 s x
(a) What is the wavelength of the wave?
(b) What is the minimum wave velocity?
(c) What is the frequency of the wave based on the minimum velocity?
9 The following diagram shows the positions of air molecules before and at an instant when a longitudinal wave passes through the air.

A B C D E F G H I J K L M N O P
A B C DE F G H I JK L M N O P
0 1 2 3 4 5 6 7 8 9 mm
What is the wavelength of the longitudinal wave?
10 The following diagram represents waves travelling across a boundary between two media. The lines represent the crests of waves. boundary medium 1 medium 2
8 cm 20 cm
(a) What is the value of the ratio: frequency of waves in medium 1: frequency of waves in medium
2?
(b) What is the value of the ratio: wavelength of waves in medium 1: wavelength of waves in medium 2?
(c) What is the value of the ratio: speed of waves in medium 1: speed of waves in medium 2
11 An observant Physics student at the beach notices that the crests of the waves coincide with two buoys 12.0 m apart and that there is a 8.0 s interval between successive wave crests. What is the speed of the waves?
12 The following diagram represents the screen of a dual trace cathode ray oscilloscope (CRO). The grid is in cm. It shows two waves, equal in size and frequency. The time base setting (X-axis) is 50 ms/div and the vertical display setting is 5.0 volts/div for both waves.

(a) What is the period of each wave?
(b) What is the frequency of each wave?
(c) What is the amplitude of the wave in volts?
(d) It is possible for the CRO to add the two waveforms. Sketch the appearance of this wave as it would appear on the screen. The time base and vertical display settings remain unchanged.
13 The following diagram represents the screen of a dual trace cathode ray oscilloscope (CRO). The grid is in cm and it shows two waves. The time base setting (X-axis) is 10 ms/div and the vertical display setting is 1.0 volts/div for both waves.

(a) What is the period of each wave?
(b) What is the frequency of each wave?
(c) What is the amplitude of the wave in volts?
(d) It is possible for the CRO to add the two waveforms. Sketch the appearance of this wave as it would appear on the screen. The time base and vertical display settings remain unchanged.
14 A buoy on a wave takes 4.0 s to move between its highest and lowest points. The distance between successive wave crests is 12.0 m. What is the speed of the wave?
15 The following diagram represents a transverse wave travelling to the right at 20 ms-1.
0 1.0
2.0
time (s)
3.0
(a) What is the period of the wave?
(b) What is the frequency of the wave?
(c) What is the wavelength of the wave?

16 The speed of sound in air is 330 ms-1 and in water is 1500 ms-1. A sound wave of frequency 50
Hz travels from air into water.
(a) What will be the value of the ratio - frequency of sound wave in water: frequency of sound wave in air?
(b) What will be the value of the ratio - wavelength of sound wave in water: wavelength of sound wave in air?
17 The following diagram represents a single pulse moves along a string to the right. A, B, C, D, E,
F and G are points on the string.
C
Α
B D
E
G
F
At the instant represented by the diagram indicate the point or points that are:
(a) moving up
(b) moving down
18 The following diagram represents the pattern produced by two in-phase sources S1 and S2 in a ripple tank. The lines represent the position of wave crests.
X
Z Y
S S
2
Which point represents the constructive interference of a crest with a crest?
Which point represents the destructive interference of a crest with a trough?
Which point represents the constructive interference of a trough with a trough?
19 The frequency of a wave is 50 Hz. Two points, 40 cm apart, on the wave are observed to be 90° out of phase. What is the wave velocity?
20 A sinusoidal wave moving along a string at 10.0 ms-1 has a wavelength of 2.0 m and amplitude of 10.0 cm.
(a) Calculate the wave frequency and period.

(b) Sketch the waveform for a cycle of this wave.
(c) Indicate on this sketch where the velocity of individual points on the string is (i) a maximum and
(ii) a minimum.
(d) Indicate on this sketch where the acceleration of individual points on the string is (i) a maximum and (ii) a minimum.
(e) Calculate the maximum (i) displacement, (ii) velocity and (iii) acceleration of a individual point on the string.
21 The speed of a wave in a spring (v) is related to the tens ion in the spring (F) and the mass per unit length (
µ
) of the spring by the relation: v =
F
µ
A spring of mass per unit length (
µ
) 3.00 kg m-1 is kept under a tension of 27 N.
(a) What will be the velocity of a wave in the spring?
(b) If the tension is reduced to 12 N what will be the velocity of a wave in the spring?
(c) In an experiment the velocity of a wave in a spring was recorded at different tensions. The results are recorded in the following table. Complete the table and graph force versus (velocity)2 to determine a value of
µ
for the spring. force
(N) velocity
(ms-1) velocity2
0.0 0.00
(ms-1)2
10.0
20.0
30.0
40.0
50.0
1.41
2.00
2,45
2.83
3.16
22 A string is vibrating in its fundamental mode of 200 Hz. The length of string is 60 cm.
(a) What is the wavelength of the fundamental frequency?
(b) If the string were clamped at distances of 10 cm and 20 cm from one end what would be the fundamental frequency now?
23 What is the minimum length of closed pipe which will resonate with a tuning fork of frequency
384 Hz. The speed of sound in air is 333 ms-1.
24 A tuning fork of frequency 320 Hz produces a resonance with a closed pipe of length 25.0 cm.
What is the speed of sound at this temperature?
25 A tuning fork of unknown frequency produces a resonance with a closed pipe of length 31.25 cm. The speed of sound at the temperature of the experiment is 320 ms-1. What is the frequency of the tuning fork?
26 A signal generator is connected to a loudspeaker held above a measuring cylinder 30 cm high.
The lowest resonant frequency is found to be 200 Hz. What is the next resonant frequency?
27 A group of students investigated the relationship between the angle of incidence and the angle of refraction for a beam of light through a prism. They varied the angle of incidence and measured the angle of refraction. Their results are shown below.

incident ray in air i normal r
θ refracted ray in glass
φ i
10.0°
15.0°
20.0°
25.0°
30.0°
45.0° r
6.05°
9.03°
12.0°
14.9°
17.6°
25.4° sin i
0.174
0.259
0.342
0.423
0.5
0.707 sin r
0.105
0.157
0.207
0.256
0.303
0.428
60.0° 31.7° 0.866 0.525
(a) Plot i versus r. Is there a simple linear relation between them. If so determine the slope of the line.
(b) Plot sin i versus sin r. Is there a simple linear relation between them.
(c) For a value of i = 45° in the diagram suggest the values of the angles
θ
and
φ
.
28 A modified form of Kundt’s method of determining the speed of sound in various gases is shown below. It consists of a long glass tube fitted with a piston containing fine sawdust is adjacent to the speaker of a frequency generator as shown. When the frequency generator is on the piston is adjusted until sawdust collects at regular intervals (d) along the tube. frequency generator
d piston fine sawdust
(a) What sort of waves are set up in the glass tube?
(b) What is the significance of the points where the sawdust collects?
(c) Show that the velocity of the waves in the glass tube is given by: v = 2df.
(d) If the frequency generator is set at 1665 Hz and the dust collects at intervals of 10.0 cm, determine the speed of the wave in the tube.
(e) If the air was replaced by he lium at the same temperature what would happen to the intervals?
29 How can resonance be set up in an open pipe when the wave must be reflected from an open end in the same medium (air)?

30 The following diagram represents waves moving towards a barrier with a small opening in a ripple tank. The wave crests are shown. The wavelength of the waves is a little less than the width of the opening. wave motion
(a) What process occurs to produce the curved waves originating from the opening?
(b) If the wavelength was much smaller than the opening (see following diagram) what would be the observed appearance of the waveform originating from the opening?
31 When a beam of light passes through a triangular prism a spectrum of colours is produced as illustrated. beam of white light glass prism red orange yellow green blue violet
Define the term dispersion? Why does a spectrum of colours form when a beam of light passes through a triangular prism?
32 How is it that you can hear around corners but not see around corners?
33 The ear canal in an adult is about 2.1 cm long. What is its fundamental frequency? Take the speed of sound as 340 ms-1. Why is human hearing most acute for frequencies of 3 000 - 4 000 Hz.
Suggest why children can more readily hear higher frequencies than adults.
34 What is the minimum length of open pipe which will resonate with a tuning fork of frequency
256 Hz. The speed of sound in air is 333 ms-1.
35 A tuning fork of frequency 384 Hz produces a resonance with an open pipe of length 41.7 cm.
What is the speed of sound at this temperature?
36 A tuning fork of unknown frequency produces a resonance with an open pipe of length 47.1 cm.
The speed of sound at the temperature of the experiment is 345 ms-1. What is the frequency of the tuning fork?

37 A slinky is stretched a distance of 20 m and held rigidly at point B. A pulse is observed to take
5.0 s to travel from A to B and back to A.
The slinky is then shaken at point A to generate a standing wave as shown in the following diagram.
A
20 cm
B
20 m
(a) What is the speed of the pulse in the slinky?
(b) What is the wavelength and amplitude of the standing wave?
(c) With what frequency does point A have to be shaken to produce the standing wave shown?
(d) What will be the shape of the waveform at time T/4 after the diagram shown. (T = period)
38 A 30 cm string is vibrating in its fundamental mode. The speed of waves in the string is 990 cm s1 . The speed of sound is 330 ms1 for the conditions of the experiment.
(a) What is the wavelength of the wave in the string?
(b) What is the frequency of the wave in the string?
(c) What is the wavelength of the emitted wave in the air?
(d) What is the wavelength of the emitted wave in the air?
39 (a) What are the four longest possible wavelengths on a15 cm length of string stretched between two supports?
(b) What are the four longest possible wavelengths in a15 cm closed pipe?
40 A 30 cm string is stretched between two supports. It is observed to have resonant frequencies of
750 Hz and 1000 Hz with none between.
(a) What is the lowest resonant frequency of the string?
(b) What is the wave speed in the string?
41 Why do different musical instruments playing the same note have different distinctive sounds?
42 Waves are not reflected appreciably from objects smaller than their wavelength. If a hunting bat emits and can detect a frequency of 120 000 Hz what is the minimum size of the object they can detect? Assume a speed of sound of 340 ms
-1
.
43 Two points 0.332 m apart are 90° out of phase in a wave of frequency 256 Hz.
(a) What is the wavelength of the wave?
(b) What is the speed of the wave?
44 The speed of a wave in a wire (v) is related to the tension in the spring (F) and the mass per unit length (
µ
) of the spring by the relation:
v =
F
µ
A 0.50 m length of wire of mass 5.0 g is kept under a tension of 25.0 N.

(a) What will be the velocity of a wave in the wire?
(b) What are the three lowest frequencies of standing waves produced on the wire?
45 The velocity of a wave in a wire of mass per unit length (
µ
) 0.010 kg m-1 is found to be 80 ms-1 at a certain tens ion in the wire. The wire is 1.0 m long.
(a) What is the tension in the wire?
(b) What is the fundamental frequency of standing waves produced on the wire?
46 When light of wavelength 4.50 × 10
-7
m enters a slab of flint glass its speed changes from 3.00 ×
10 8 ms-1 to 1.50 × 10 8 ms -1 .
(a) What is the frequency of the light in the air?
(b) What is the frequency of the light in the Crown glass?
(c) What is the wavelength of the light in the Crown glass?
47 A 512 Hz tuning fork is used to set up standing waves in a string clamped at both ends. The wave speed for the string is 128 ms-1. The string is observed to have four loops as illustrated in the following diagram.
(a) Calculate the wavelength of the standing wave.
(b) What is the length of the string?
48 The speed of sound in helium is 2.9 times that in air. What is the frequency of a man saying “oo” normally about 440 Hz when the man has inhaled helium?
49 A 1.0 m pipe will resonate at frequencies of 260 Hz, 430 Hz and 600 Hz.
(a) What is the fundamental frequency of the pipe?
(b) Is the pipe open or closed?
50 A stretched string is 0.50 m long and has a fundamental frequency of 384 Hz. How could it be made to vibrate at 768 Hz?
Answers Set 1
Exercises
1 (a) 0.86 m (b) 333 ms
-1
2 The frequency depends on the source.
3 Amplitude
4 43 m
5 9.1 x 10 15 s -1
6 (a) 0.63 m, (b) 2.3 s, (c) 69 m
7 The speed increases.
8 In a transverse wave (e.g, water wave) the particles move up and down across the direction of motion of the wave. In a longitudinal wave the p[articles vibrate along the direction of motion of the wave.

9 256 Hz
10 (a) 1.32 m (b) 338 ms -1
11 Half the length or increase the tension by a factor of four.
12 1.2 m (b) 1.3 m
13 3.2 ms -1
14 (a) 1.00 m (b) 36 s (c) 360 m
Answers - Set 2
1 20 m
2 (a) 2.0 mm
(b) 4.1 mm
(c) 0.020 s
(d) 0.205 ms-1
3(a) 1.25 cm
(b) 1.1 s
4 (a) 5.0 cm
(b) 2.0 s
(c) 20 cm
5 (a) 1.0 cm
(b) 2.0 ms
(c) 0.4 m
6 3.9 × 1014 Hz to 7.5 × 1014 Hz
7 (a) 0.020 s
(b) 10 mm
(c) 0.50 ms-1
(d) 50 Hz
(e) 1.0 ms-1
(f) 20 mm
8 (a) 4.0 mm
(b) 3.0 mm s-1
(c) f = 0.75 s-1
9 3.7 mm
10 (a) 1:1 = 1
(b) 8:20 = 0.4
(c) 8:20 = 0.4
11 1.5 ms-1
12 (a) 2 cycles of the wave takes 3.0 cm (3 × 50 ms) so T
=
150 ms
2
= 75 ms
(b) f = 1/T= 13.3 s-1
(c) 1.9 div x 5.0 volts/div = 9.5 V

(d)
13 (a) 20 ms
(b) 50 s-1
(c) 2 V
(d)
14 T = 8.0 s , f = 0.125 s-1 v = f
λ
= 0.125 s-1 × 12.0 m = 1.5 ms-1
15 (a) 1.0 s
(b) 1.0 s-1
(c) 20 m
16 (a) 1:1

(b) 1500:330 = 4.55
17 (a) D, E and F
(b) B, C and G
18 At point X a crest meets a crest. The will be a maxima of constructive interference.
At point Y a crest meets a trough, there will be maximum destructive inteference.
At point Z a trough meets a trough. The will be a maxima of constructive interference.
19 The points are
λ
/4 apart
∴ λ
= 160 cm v = f
λ
= 50 Hz x 1.6 m = 80 ms-1
20 (a) f = v/
λ
= 10 ms-1/2.0 m = 5.0 Hz, T = 1/f= 0.20 s
(b), (c), (d) v = 0, a = max
10
1.0
2.0 m v = max, a = 0
-10 v = 0, a = max
(e) (i) maximum displacement = ½ × amplitude = 5.0 cm
(ii)
ω
=2
π
/T = 2
π
/0.2 s = 10
π
rad s-1 maximum velocity =
ω
A = 10
π
rad s-1 × 0.10 m =
π
ms-1
(iii) maximum acceleration = -
ω 2A = -(10 π
rad s-1)2 × 0.10 m = -10
π 2 ms-2
(directed towards the centre)
21 (a) 3.0 ms-1
(b) 2.0 ms-1
(c)

Graph of Force versus the square of Velocity for a Spring
60
50
40
30
20
10
0
0 2 4 6
Velocity Squared (ms
-1
)
2
8
µ
= slope of line =
10
50
−
0
−
0
( ms
N
−
1
)
2
= 5.0 kg m-1
22 (a) 1.20 m
(b) 600 Hz
23 minimum length of closed pipe =
λ/4
λ f = v
λ
× 384 Hz = 333 ms-1
λ
= 0.867 m l =
λ/4
= 21.6 cm
24 v = f
λ
= 320 Hz × (4 × 0.25 m) = 320 ms-1
25
λ
= 4 × 0.3125 m = 1.25 m f = v
λ
=
320
1 .
ms
25 m
−
1
= 256 Hz
26 f = 3 × 200 Hz = 600 Hz
10 12

30
20
10
0
0
70
60
50
40 fundamental
λ l =
4 f = v
λ
= v
4l
= f
1 first overtone l =
3
λ
4 f = v
λ
=
3v
4l
= 3f
1
l
27 (a) This graph is not linear.
Graph of the angle of incidence versus the angle of refraction
5 10 15 20
Angle of Refraction
25 30
(b)
35

Graph of the sin of the angle of incidence versus the sin of the angle of refraction
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.000
0.100
0.200
0.300
0.400
0.500
0.600
sin r
This is a linear relationship. sin i = ksin r, where k is the slope. The slope is
0 .
866
0 .
525
−
−
0
0
= 1.65.
∴
sin i = 1.65 × sin r
(c)
θ
= 25.4° and
φ
= 45°
28 (a) longitudinal standing waves
(b) The sawdust collects at the nodes of the standing wave as there is little movement of the air at these points in the wave.
(c) The wavelength of a standing wave = 2 × distance of separation of nodes
Since v = f
λ then v = f × 2d, or v = 2df
(d) v = 2df = 2 × 0.100 m × 1665 Hz = 333 ms-1
(e) They would increase as d is proportional to v and v is higher for helium.
29 A change in the properties of the air occurs at the open end because the air molecules outside the pipe are not constrained and can move in all directions. The open end thus becomes a boundary.
30(a) diffraction
(b)
31 Dispersion is the separation of different wave frequencies by refraction. It occurs with light because light is composed of a spectrum of waves of different frequencies. These frequencies have a spectrum of speeds in a medium like glass.
32 Diffraction is most apparent when the wavelength is similar in magnitude to the object. Light waves have very small wavelengths 400 nm to 700 nm and are not diffracted to any significant

extent by large obstacles. Sound waves have wavelengths of similar magnitude 0.5 -2.0 m to doors, windows etc and are readily diffracted by them.
33 About 4 000 Hz. These frequencies resonate most readily in the ear canal. Children have a shorter ear cana l. It will resonate at higher frequencies.
34 65 cm
35 320 ms-1
36 366 Hz
37 (a) 8 ms-1
(b)
λ
= 10 m, A = 10 cm
(c) 0.80 Hz
(d)
A B
38 (a) 60 cm
(b) 33 Hz
(c) 33 Hz
(d) 10 m
39 (a) The string will resonate at l =
λ
/2,
λ
, 3
λ
/2, 2
λ
etc
∴ λ
= 30 cm, 15 cm, 10 cm, 7.5 cm etc
(b) The pipe will resonate at l =
λ
/4, 3
λ
/4, 5
λ
/4, 7
λ
/4 etc
∴ λ
= 60 cm, 20 cm, 12 cm, 8.6 cm etc
40 (a) 250 Hz (The resonant frequencies of a string are simple multiples of the fundamental)
(b) v = f
λ
= 250 Hz × 0.60 m = 150 ms-1
41 The different instruments vary in their shape and material composition and how they are played.
They will consequently differ in the number and the intensity of the overtones produced.
42 2.8 mm
43 (a)
λ
= 4 × 0.332 m = 1.33 m
(b) v = f
λ
= 256 Hz × 1.33 m = 340 ms-1
44
(a) v =
F
µ v =
0 .
25 .
010
0 kg
N m
−
1
= 50 ms-1
(b) l = ½
λ
for fundamental so
λ
= 1.0 m f (fundamental) = v/
λ
= 50 Hz

Overtones = 100 Hz and 150 Hz
45 (a) 64.0 N
(b) 40 Hz
46 (a) f = v/
λ
= 3.00 × 108 ms-1/4.50 × 10-7 m = 6.67 × 1014 Hz
(b) 6.67 × 1014 Hz
(c)
λ
= v/f = 1.50 × 108 ms-1/6.67 × 1014 Hz = 2.25 × 10-7 m
47 (a)
λ
= v/f= 128 ms-1/512 Hz = 0.25 m
(b) l = 2
λ
= 0.50 m
48 1280 Hz
49 (b) The increments in frequency are 170 Hz. 260 Hz - 170 Hz = 90 Hz
The fundamental frequency of the pipe is 90 Hz.
(a) There are only odd harmonics so the pipe is closed.
50 The string should be clamped in the middle and plucked at a point ¼ from one end.