CONTENT

• Progressive waves
• Transverse and longitudinal
waves
• Energy and intensity of
waves
• Polarisation
• Determination of frequency
and wavelength of sound
waves
 Progressive Waves
Wave
▪ is a propagation of a disturbance which transfers energy from one point
in space to another without the physical transfer of matter.
▪ the source of any wave is a vibration or an oscillation.

Progressive Wave ▪ a.k.a. travelling wave

▪ a wave which results in a net transfer of energy from one place to
another.
▪ energy is transferred from the source outwards, along the direction of
propagation of the wave

Stationary Wave ▪ a.k.a. standing wave

▪ has a waveform that does not move.
▪ it is formed from superposition of two similar progressive waves travelling in
opposite directions.
 Types of Waves
Waves can be classified in different ways according to
1. Mechanical or Non-mechanical
2. Transverse or Longitudinal

Mechanical
▪ requires a medium for their propagation.
e.g., water or sound waves

Non-mechanical

▪ does not require a medium for their propagation.

e.g., electromagnetic, gravitational, and quantum mechanical
waves
 Types of Waves
Waves can be classified in different ways according to
1. Mechanical or Non-mechanical
2. Transverse or Longitudinal
Transverse
▪ has a direction of oscillation perpendicular to its direction of propagation.
e.g., waves on a rope, electromagnetic waves

Longitudinal
▪ has a direction of oscillation parallel to its direction of propagation.
e.g, slinky spring and sound waves
 Displacement-distance graph
Terminologies

Displacement x/y
▪ Position of an oscillating particle from its equilibrium position

Amplitude A
▪ The maximum distance (magnitude of displacement) of an oscillating
particle from its equilibrium position

Wavelength 𝝀
▪ For a progressive wave, it is the distance between any two successive
particles that are in phase. (e.g. the distance between 2 adjacent maximum
displacements)
 Terminologies Displacement-time graph

Period T

▪ Time taken for a particle to undergo one complete cycle of oscillation

▪ Time for the wave to travel through one wavelength.
1
Frequency f 𝑓=
𝑇
▪ Is the number of complete cycle performed by a particle per unit time
▪ Number of wavelengths that pass a given point per unit time.
Terminologies
Wave speed v

▪ Distance travelled by the waveform/wave

profile per unit time
𝑣 = 𝑓𝜆

Wavefront

▪ is a locus or imaginary line joining all the points of

the wave that have the same phase.
▪ It is useful to draw the wavefront by joining all the
crests of a wave, and then seeing it from a bird’s
eye view.

▪ 1 wavelength = distance between 2 successive

wavefronts
Terminologies

Ray
▪ Indicates the path taken by the
wave
▪ Always perpendicular to the
wavefronts
 Terminologies

Phase 𝝓

▪ of a particle gives a measure of the fraction

of a cycle that has been completed by an
oscillating particle.
▪ One cycle corresponds to 360° or 2𝜋 rad.

Displacement-distance graph of two waves with the same 𝝀

Displacement-time graph of two waves with the same T

 Terminologies

Phase Differrence
▪ is a measure of how much one wave is out of step with another or one
particle in a wave is out of step with another particle in the same wave.

Δ𝑥 Δ𝑡
Δ𝜙 = (2𝜋) Δ𝜙 = (2𝜋)
𝜆 𝑇
where Δ𝑥 = distance between maximum/minimum distance
Δ𝑡 = time interval between maximum/minimum distance
𝜆 = wavelength
𝑇 = period

In phase: 𝛥𝜙 = 0
Out of phase: 𝛥𝜙 ≠ 0
Antiphase: 𝛥𝜙 =180° or 𝜋 rads
Note

1. In order to compare phase or finding phase difference, amplitudes of

oscillating particles need not be the same but they must have the same
frequency and wavelength
2. Two particles or two waves are said to be in phase when their Δ𝜙 = 0.
▪ Waves that are n apart (where n is a positive integer) are also in
phase, e.g. , 2, 3, etc.
3. Two particles or two waves are said to be in anti-phase when their
Δ𝜙 = radian.
1
▪ Waves that are 𝑛 + 𝜆 apart (where n is a positive integer) are
2
also exactly out of phase, e.g. /2, 3/2, 5/2, etc.
Practice Example 1

With a frequency of 440 Hz, a sound wave is travelling with a speed of

330 m/s. What is the phase difference between two points on the
wave 0.25 m apart in the direction of travel?

Practice Example 2

Consider a displacement-time graph of two

waves detected by a sensor shown on the
left. The waves have the same frequency.
What is the phase difference between the
two waves?
Practice Example 3

Shown below are the wavefronts of a wave travelling to the right. It has a
speed of 2.25 m/s and a frequency of 0.80 Hz.
Determine the following:
a. Distance between points A and B
b. Phase difference between points B and C
c. Phase difference between points B and D
 Transverse Waves
▪ is a wave in which its particles oscillates in a direction
perpendicular to its direction of propagation.
e.g., waves on a rope, electromagnetic waves

Electromagnetic Waves
▪ the EM wave spectrum is divided into radio waves,
microwaves, infra-red, visible light, ultraviolet, X-rays,
and gamma rays.
▪ the frequency of an EM wave does not change when
the waves go from one medium to another.
▪ They travel at the speed of 𝑐 = 3.00 × 108 𝑚/𝑠 in
vacuum
Graphical Representation of Transverse Wave
 Longitudinal Waves
▪ is a wave in which its particles oscillate parallel to its direction of
propagation.
e.g, slinky spring and sound waves

▪ Compression occurs where the air molecules are closest together.

▪ Rarefaction occurs where the air molecules are furthest apart from each other.
▪ The distance between successive compressions or successive rarefactions
is equal to the wavelength.
▪ The compressions and rarefactions occur at points of zero displacement.

Graphical
Representation
Displacement-time graph of transverse and longitudinal wave

All the particles move in a similar

manner with the same amplitude and
frequency as the wave. That is,
▪ frequency of particle = frequency of
the wave
▪ amplitude of particle = amplitude of
the wave

From the graph, t0 to t8 signifies the

completion of one oscillation, and is the
period of the wave.
 Displacement-distance graph of transverse and longitudinal wave

The displacement-distance graph shows how the displacements of the particles (from their individual equilibrium
position) vary with the distance from the source at a particular instant in time.

▪ For transverse waves, this is similar to a snapshot of the actual wave travelling through the medium.
▪ For longitudinal waves, however, unlike transverse waves, the displacement-distance graph is not a snapshot of the
actual wave travelling through the medium and has to be found by finding the displacement of individual particles.
Practice Example 4

A displacement-distance graph of a transverse progressive wave is

shown below. It travels to the right along a rope. In which direction are
P and Q moving?
Practice Example 5

The figures below show a progressive transverse and a progressive longitudinal

wave respectively,

On each wave, identify a point or points at which

a. the velocity is zero
b. the acceleration is zero
c. the velocity is in the same direction as the displacement
d. the acceleration is in the same direction as velocity
 Energy and Intensity of Waves
• Wave motion involves the transportation of energy from one place to another.

Symbol: 𝐼
Intensity SI unit: Watts per meter squared [W/m2]

▪ is the energy delivered per unit area per unit time

𝐸 𝑃 where 𝐸 = energy
𝐼= = 𝑃 = power
𝐴𝑡 𝐴 𝐴 = area

Sinusoidal Waves
1
▪ wave vibrates in simple harmonic motion 𝐸 = 𝑚𝜔2 𝐴2
2
𝐼 ∝ 𝐴2
 Energy and Intensity of Waves

Spherical Waves
▪ waves coming from a point source have spherical wavefronts
▪ Surface area: = 4𝜋𝑟 2

𝑃
𝐼=
4𝜋𝑟 2
1
𝐼∝ 2
𝑟
1
Amplitude ∝
𝑟
Practice Example 6

A person is initially 5.0 m from a point source which emits energy

uniformly in all directions at a constant rate. If the power of the source
is to be doubled but the sound is to be as loud as before, at what
distance should the person be from the source?

Practice Example 7

Suppose a loudspeaker operating at 35 W is producing sound waves

in all directions. Calculate the following:
a) the intensity of sound at a distance of 12 m away
b) the power received by a square microphone of length 4.0 cm
placed at a distance 8.0 m away from the loudspeaker
c) the amplitude of the vibrations at 8.0 m, given that at 4.0 m, the
amplitude of the vibrations is 3.0 cm.
Polarisation
▪ is a phenomenon whereby the oscillation of transverse waves
are restricted to a single plane.
*does not apply to longitudinal waves

Illustration of polarization of light wave using a polariser

Polarisation of transverse wave
▪ the transmitted wave through polarizer A is said to
be plane-polarised or polarized in the vertical
plane.
▪ This vertically polarized wave is able to pass
through polariser B and the polarizer B has the
same transmission (polarization axis) as polariser A.

▪ The polarized wave is completely blocked by

polarizer C which has a transmission axis to A and
no light is able to pass through
Applications (polarization)

▪ Polaroid used in sunglasses to reduce glare

▪ 3D glasses used to watch 3D movies
▪ Reduction of haziness of pictures
Malus Law
▪ states that the intensity of a beam of plane-polarised light after
passing through a polariser varies with the square of the cosine
of the angle through which the polariser is rotated from the
position that gives maximum intensity.

𝐼 = 𝐼0 cos2 𝜃 where 𝐼0 = intensity of unpolarized light

When polarised visible light is now incident on a second

polariser (usually called an analyser) placed with its
polarising axis at an angle 𝜃 to the polarising axis of the
first polariser, only the electric field component parallel to
the polarising axis of the analyser will be transmitted,
while the component perpendicular to the polarising axis
will be absorbed by analyser.
Practice Example 8

Explain why it would not be possible to polarize sound waves.

Practice Example 9

Two polarisers P and Q are placed next to each other such that their
polarization axes are parallel and vertical, as shown below. If intensity
of the emergent light is 𝐼0 , through what angle must Polaroid Q be
1
rotated so that the intensity of the emergent light decreases to 𝐼 ?
4 0
Determination of frequency and wavelength of sound waves

▪ As discussed earlier, soundwaves produce regions of compressions and rarefactions as they travel trough air.
This gives rise to pressure variations as the wave travels.
▪ Hence, by placing a microphone in front of a loudspeaker connected to a signal generator, the microphone will
detect a continuous series of compressions and rarefactions over time.
▪ If the microphone is connected to a cathode ray (CRO), the CRO will be able to display a variation of the
pressure experienced by the microphone with respect to time.A
Determination of frequency and wavelength of sound waves

When a signal is viewed on the CRO display, the period of the signal can be measured by
counting the number of horizontal divisions a complete waveform covers and multiplying it by the
scale of each division. This is known as the time base.
An interpretation of a CRO signal is shown below

The horizontal distance from peak to peak (1 wave) is 8 divisions. As time-base is set
to 50 ms/div, the period, T, is therefore 𝑇 = 8 × 50 = 400 𝑚𝑠
1
Using frequency, 𝑓 = 𝑇
1
𝑓= = 2.5 𝐻𝑧
400 × 10−3
Practice Example 10

A sinusoidal sound wave of unknown frequency is fed into a C.R.O.

and the waveform on C.R.O is shown below. The length of each
division for the time-base is 1 cm. Find the frequency of the sound.
Suggested
Solutions to
Practice
Examples
Practice Example 1

With a frequency of 440 Hz, a sound wave is travelling with a speed of

330 m/s. What is the phase difference between two points on the
wave 0.25 m apart in the direction of travel?

Solution:
𝑣 330 𝑚/𝑠
𝑣 = 𝑓𝜆 → 𝜆 = =
𝑓 440 𝐻𝑧
𝜆 = 0.75 𝑚

Δ𝑥 0.25
Δ𝜙 = 2𝜋 = 2𝜋
𝜆 0.75
2𝜋
Δ𝜙 =
3
 Practice Example 2

Consider a displacement-time graph of two waves

detected by a sensor shown on the left. The waves
have the same frequency. What is the phase
difference between the two waves

Solution:
Δ𝑡
Δ𝜙 = 2𝜋
𝑇
1
4𝑇
= (2𝜋)
𝑇
𝜋
= 𝑟𝑎𝑑
2
 Practice Example 3 Shown below are the wavefronts of a wave travelling to the
right. It has a speed of 2.25 m/s and a frequency of 0.80 Hz.
Determine the following:
a. Distance between points A and B
b. Phase difference between points B and C
c. Phase difference between points B and D

a. distance between A and B = 2𝜆 b. Since B and C are 2𝜆 apart, they are in phase. Hence
𝑣 their phase difference is 0 rad.
𝑣 = 𝑓𝜆 → 𝜆 =
𝑓 Alternatively,
Δ𝑥 2𝜆
2.25𝑚/𝑠 Δ𝜙 = 2𝜋 = (2𝜋)
𝜆= = 2.81 𝑚 𝜆 𝜆
0.8 𝐻𝑧
Δ𝜙 = 4𝜋 = 0 𝑟𝑎𝑑
distance between A and B = 2𝜆 = 2 2.81𝑚 = 5.62 𝑚

1
c. Since B and D are 2 𝜆 apart, they are exactly out of phase. Hence their phase difference is 𝜋 𝑟𝑎𝑑. Alternatively,
1
Δ𝑥 𝜆
Δ𝜙 = 2𝜋 = 2 (2𝜋)
𝜆 𝜆
Δ𝜙 = 𝜋 𝑟𝑎𝑑
Practice Example 4

A displacement-distance graph of a transverse progressive wave is

shown below. It travels to the right along a rope. In which direction are
P and Q moving?

Answer:
Movement of P is upwards while the movement of Q is downwards.
 Practice Example 5

The figures below show a progressive transverse

and a progressive longitudinal wave respectively,

On each wave, identify a point or points at which

a. the velocity is zero
b. the acceleration is zero
c. the velocity is in the same direction as the displacement
d. the acceleration is in the same direction as velocity

Answer:
Transverse Longitudinal
a. D A
a. B C
a. C D
a. A B
Practice Example 6

A person is initially 5.0 m from a point source which emits energy uniformly in all
directions at a constant rate. If the power of the source is to be doubled but the sound
is to be as loud as before, at what distance should the person be from the source?

Solution:
𝑃 𝑃
Use 𝐼 = 4𝜋𝑟 2 and note that intensity is the same. Let the primed variables be the variables
4𝜋𝑟 2
taken at a distance from the source.
𝐼 = 𝐼 ′′
𝑃 𝑃′ 𝑃 𝑃′
= → =
4𝜋𝑟 22 4𝜋𝑟′22 𝑟 22 𝑟′22
2
𝑟 ′′ 2 𝑃′′ ′′
𝑃′′
= →𝑟 = 𝑟
𝑟 𝑃 𝑃
Power of the source is doubled
(2𝑃)
𝑟 ′′ = 𝑟 = 2𝑟 = 2 (5)
𝑃
𝑟 ′′ = 7.07 m
 Practice Example 7
Suppose a loudspeaker operating at 35 W is producing sound waves in all directions. Calculate
a) the intensity of sound at a distance of 12 m away
b) the power received by a square microphone of length 4.0 cm placed at a distance 8.0 m away from
the loudspeaker
c) the amplitude of the vibrations at 8.0 m, given that at 4.0 m, the amplitude of the vibrations is 3.0 cm.

Solution (a):
Solution (c):
Use 𝐼 = 𝑃/𝐴
𝑃 35 Let 𝐴𝑚 = amplitude. Primed variables =
𝐼= = measured at 8.0 m
4𝜋𝑟 2 4𝜋 12 2
𝐼 = 0.019 𝑊/𝑚2 1
𝐴𝑚 ∝
𝑟
𝐴𝑚 ′ 𝑟
Solution (b): ⇒ =
𝐴𝑚 𝑟′
′
𝑟
Use 𝐼 = 𝑃/𝐴 𝐴𝑚 = ′ 𝐴𝑚
𝑟
𝑃 = 𝐼𝐴 4
𝐴𝑚′ = 3 × 10−2
8
𝑃 = 0.019 4 × 10−2 2
𝐴𝑚′ = 0.015 𝑚
𝑃 = 3.04 × 10−5 𝑊
Practice Example 8

Explain why it would not be possible to polarize sound waves.

Answer:
Sound waves in a gas or liquid do not have polarization because the
medium vibrates only along the direction in which the waves are
travelling.
Practice Example 9

Two polarisers P and Q are placed next to each other such that their polarization axes are parallel
and vertical, as shown below. If intensity of the emergent light is 𝐼0 , through what angle must
1
Polaroid Q be rotated so that the intensity of the emergent light decreases to 𝐼0 ?
4

Solution:
𝐼 ∝ 𝑥02
2
𝐼′ 𝑥0′
=
𝐼 𝑥0
1 2
𝐼0 𝑥 cos 𝜃
4 =
0
𝐼0 𝑥0
1
= cos 2 𝜃
4
1 𝜋
𝜃 = cos −1 = 60° = 𝑟𝑎𝑑
2 3
Practice Example 10

A sinusoidal sound wave of unknown frequency is fed into a C.R.O.

and the waveform on C.R.O is shown below. The length of each
division for the time-base is 1 cm. Find the frequency of the sound.

Solution:
From C.R.O., 8 divisions correspond to 3.5
wavelengths,

2.5𝑇 = 8(2 × 10−3 )

𝑇 = 0.0064 𝑠
1 1
𝑓= = = 156.25
𝑇 0.0064
𝑓 = 160 𝐻𝑧