Wave motion transfers energy from one place to another. There are two main types of waves: transverse waves where particles oscillate perpendicular to the wave direction, and longitudinal waves where particles oscillate parallel to the wave direction. Key wave properties include amplitude, wavelength, frequency, and phase. Waves can undergo various interactions including reflection, refraction, diffraction, interference, and superposition. Interference of waves leads to constructive and destructive interference patterns. Standing waves occur due to interference of waves traveling in opposite directions.

1. WAVE MOTION
Wave motion is a means of moving energy from place to place. A wave is a disturbance in a medium.
Waves which move energy from place to place are called Progressive waves. There are two main groups of
waves. These are:
a) A transverse wave is one in which the vibrations of the particles in the medium oscillate at right
angles to the direction in which the energy of the wave is travelling.

2. b) A longitudinal wave is one in which vibrations of the particles of the medium oscillate parallel to the
direction in which the energy of the wave is travelling.

3. DEFINITIONS
i. The displacement of a particle on a wave is its distance from its rest position.
ii. The amplitude of the wave is defined as the maximum displacement of a particle in the
wave.
iii. One wavelength (𝝀) is the distance between two neighboring peaks or two neighboring troughs, or
two neighboring points which are vibrating together in the same way (in phase). It’s the distance
moved by the wave during one oscillation of the source of the waves.
iv. The period (T) of the wave is the time for a particle in the wave to complete one vibration, or one
cycle.
v. The frequency of the wave is the number of complete vibrations (cycles) per unit time.
vi. Wavefronts are lines drawn joining crests of waves from the same source. The distance between
two adjacent wavefronts is equal to wavelength.
A term used to describe the relative positions of the crests or troughs of two waves of the same frequency
is phase. When crests and troughs of two waves are in perfect alignment, the two waves are said to be in
phase.

4. When a crest is aligned with a trough, the two waves are out of phase.
Phase difference is measured in degrees or radians. Waves that are exactly out of phase have a phase
difference of 𝝅 radians = 180o
.
Phase difference can be calculated using time delay between two points compared against period, or
distance between the points compared against wavelength. i.e:
Or
𝝀 𝝀
One of the characteristics of a progressive wave is that it carries energy. The amount of energy passing
through a unit area per unit time is called intensity of the wave. The intensity of a wave at a particular point
is proportional to the square of the amplitude of a wave. Thus doubling the amplitude of a wave increases
the intensity of the wave by a factor of four. The intensity also depends on frequency: intensity is
proportional to the square of the frequency.
For spherical waves originating from a point source, the intensity I is:

5. WAVE EQUATION
For a wave that covers a distance of 𝝀 in a time interval of T, its velocity is:
𝝀
Thus if , then
𝝀
PROPERTIES OF WAVE MOTION
Waves undergo reflection, refraction, absorption, diffraction, interference and superposition.
1. REFLECTION
This is the bouncing back of a wave as it strikes a barrier. The two laws of reflection are:
- The angle of incidence = angle of reflection
- The incident ray, normal and reflected ray all lie in the same plane.

6. During reflection, the wavelength, frequency and wave speed remain unchanged since the wave is
still in the same medium.
TYPES OF REFLECTION
There are two types of reflection:
i) Specular (regular) reflection: when waves are incident upon a smooth reflective surface. All
incident rays are parallel and all reflected rays are parallel. This is because all normal are
parallel.

7. ii) Diffuse (irregular) reflection: when waves are incident on a rough surface. Incident rays are
parallel while reflected rays are not. This is because normal are not parallel.
2. REFRACTION
This is the change in direction of a wave due to a change in wave speed. As waves move from less
dense to denser medium, its wave speed increases, frequency remains unchanged and wavelength
increases. As waves move from denser to less dense, wave speed and wavelength reduces while
frequency remains unchanged.

8. S
When a wave moves from air into a material; the refractive index of the material is:
Or
That is:
As waves move from a medium with refractive index to a medium with refractive index ,
Snell’s law is:
Critical angel: this is an angle of incidence that gives the angle of refraction = 90o
.
And thus:

9. or
NOTE: whenever there is refraction, a weaker ray is reflected back into the incident medium.
If the angle of incidence is greater than the critical angle, then total internal reflection occurs.

10. Applications of total internal reflection:
b) Reflecting prisms:
Light entering the prism on side ‘a’ is at 0o
(i.e. through the normal) is not refracted. The ray of light
reaches side ‘b’ at an angle greater than the critical angle. The ray of light is totally internally
reflected. The ray of light leaves the prism through the normal.
c) Optical fibres:
An optical fiber is a flexible, transparent fiber made of glass or plastic, slightly thicker than a human
hair. It can function as a waveguide, or “light pipe” to transmit light between the two ends of the
fiber. Optical fibres are used in endoscopes and by engineers to see places that are hard to see.
a) Periscope: light incident at 0o
to the
normal is not refracted. In a
periscope, light is incident at 0o
to
the normal (i.e through the normal)
on the glass blocks. The refracted ray
passes through the normal of the
other side of the block.

11. 3. ABSORPTION
Waves undergoing absorption, are absorbed by a material and may disappear.

12. 4. DIFFRACTION
Diffraction is the spreading of a wave into regions where it would not be seen if it moved only in a
straight line. It may be defined as the ability of a wave to curve around obstacles or pass through a
gap. The narrower the gap, the more the waves spread out. The longer the wavelength, the more
the waves spread out.
When diffraction occurs:
• Wavelength is unchanged
• Frequency is unchanged
• Speed is unchanged
• Wave direction changes hence
• Wave velocity changes

13. HUYGEN’S WAVELET EXPLANATION
Huygen’s suggested that at any instant, all points on a wavefront could be regarded as secondary
disturbances giving rise to their own outspreading circular wavelets.
This constant production of wavelets makes it possible for diffraction to occur as points on
wavefronts produce wavelets whose tangents join up forming the next wavefront.

14. 5. INTERFERENCE AND SUPERPOSITION
 If two or more coherent waves overlap, the resultant displacement is the sum of the of individual
displacements. This overlapping is called interference of waves. Wave interference is the
phenomenon which occurs when two waves from 2 coherent sources meet while travelling along
the same medium.
 2 waves are said to be coherent if
 They produce waves of the same frequency
 They produce waves of the same phase (or with constant phase difference)
 The interference of waves causes the medium to take on a shape which results from the net effect
of the two individual waves upon the particles of the medium
 When waves are produced on the surface of water, the wave crests will act like a convex lens while
the troughs will act like a concave lens causing bright and dark fringes
 Waves interference can be constructive or destructive
 A wave-front is a line that joins all the points vibrating in-phase and is represented by the bright and
dark fringes or maxima and minima respectively, collectively called the interference pattern

15. The principle of superposition
 The task of determining the shape of the resultant demands that the principle of superposition is
applied.
 The principle of superposition is generally stated as follows:
When two or more waves interfere i.e. meet at the same point, the resulting displacement
is the algebraic sum of the displacements of the individual waves at that same point
• The principle applies to all types of waves
Interference can be constructive or destructive.
a) Constructive interference
This is the superposition of 2 waves which are in phase to produce a resultant wave of
maximum amplitude of the same original frequency
In this example, amplitude A + amplitude A = 2A
b) Destructive interference
This is the superposition of 2 waves which are in anti-phase to produce a resultant wave of zero
amplitude
In this example, amplitude A – amplitude A = 0

16. Parth difference
• Constructive interference occurs when the wave amplitudes reinforce each other, building a wave
of even greater amplitude.
• This happens when the waves are in phase i.e. path difference is a whole number of wavelengths,
nλ
• Destructive interference occurs when the wave amplitudes oppose each other, resulting in waves of
reduced amplitude i.e. path difference is an odd number of wavelengths, (n + ½) λ
Interference can be demonstrated in the ripple tank by using two point sources.

17. Producing an interference pattern
The diagram below illustrates the interference of waves from two point sources A and B. the point C is
equidistant from A and B: a wave travelling to C from A moves through the same distance as a wave
travelling to C from B: the path difference is zero. If the waves started from A and B at the same time and
are in phase (Phase difference = zero), they arrive at C in phase. They combine constructively producing a
large disturbance.
At other places, such as D, the waves have travelled different distances from the two sources. There is a
path difference between the waves arriving at D. if this path difference is a whole number of wavelengths
𝝀 𝝀 𝝀 𝝀 the waves arrive in phase and interfere constructively producing a maximum
disturbance. Thus the general equation for path difference of constructive interference is:
𝝀 ; m = 0, 1, 2, 3, 4,…

18. However, at places such as E, the path difference is = 𝝀 𝝀 𝝀 𝝀 . The waves arrive at E out of
phase, and interfere destructively, producing a minimum resultant disturbance. The general equation for
path difference of destructive interference is:
( ) 𝝀 ; m = 0, 1, 2, 3, 4,…
This collection of maxima and minima produced by the superposition of overlapping waves is called an
interference pattern.
An interference pattern can be produced using sound waves or light as shown below:

19. To produce an observable interference pattern, the two wave sources must have the same single
frequency and not a mixture of frequencies as is the case for light from car headlamps. Wave sources
which maintain a constant phase relationship, producing waves of the same frequency, wavelength and
wave speed, are said to be coherent sources.
YOUNG’S DOUBLE SLIT EXPERIMENT
Originally performed by Young (1801) to demonstrate the wave-nature of light. Has now been done with
electrons, neutrons, atoms among others.

20. In Young’s double slit experiment, fringes are formed due to interference of light from the two slits:
- Where a bright fringe if formed, the light from one slit reinforces the light from the other slit. In
other words, the light waves from each slit arrive in phase with each other.
- Where a dark fringe is formed, the light from one slit cancels the light from the other slit. In
other words, the light waves from two slits arrive out of phase.
- The distance from the centre of a bright fringe to the centre of the next bright fringe is called the
fringe separation (fringe width), . This depends on the slit spacing , and the distance D from
the slits to the screen, in accordance with the equation:
𝝀
Where 𝝀 is the wavelength of light.
The equation shows that the fringes become more widely spaced if:
The distance from the slits to the screen is increased;
The wavelength 𝝀 of the light used is increased;
The slit spacing (distance from the center of one slit to the center of the other slit) is reduced.
From the diagram above;
 For constructive interference (bright fringes):
𝝀
 For destructive interference (dark fringes):
𝝀

21. WHITE LIGHT FRINGES
When white light is used in the double slit experiment,
 The central fringe is white, because every colour contributes at the centre of the pattern.
 The inner fringes are tinged with blue on the inner side and red on the outer side. This is because
the red fringes are more spaced out than the blue fringes, and the two fringe patterns do not
overlap exactly.
 The outer fringes merge into an indistinct background of white light. This is because, where the
fringes merge, different colours reinforce and therefore overlap.
DIFFRACTION GRATING
A diffraction grating has many closely-spaced parallel slits ruled on it. When monochromatic light (light of a
single frequency/wavelength) is shown on a diffraction grating;
o Light passing through each grating is diffracted
o Constructive and diffractive interference occurs

22. The angle of diffraction between each transmitted beam and central beam (zero order), increase if
 Light of longer wavelength is used.
 Grating with closer slits is used.
For white light:

23. The diffraction grating equation
Consider the diagram below:

24. For a given order and wavelength, the smaller the value of , the greater the angle of diffraction is.
In other words, the larger the number of slits per meter is, the bigger the angle of diffraction is.
Fractions of a degree are usually expressed as a decimal or in minute:
To find maximum number of orders produced, substitute and thus
𝝀
.
The maximum number of orders is given by the value of
𝝀
round down to the nearest whole
number.
NOTE: to get the number of orders a diffraction grating can give is , because there are
orders on either side of the zero order plus the zero order itself.
6. POLIRISATION
When transverse waves are generated, the vibrations are said to be plane-polarised in either a
vertical plane of a horizontal plane.
The condition for a wave to be plane-polarized is for the vibrations to be in just one direction normal
to the direction in which the wave is travelling.

25. Some transparent materials, such as a Polaroid sheet, allow vibrations to pass through in one
direction only. A polaroid sheet contains long chains of organic molecules aligned parallel to each
other. When unpolarised light arrives at the sheet, the component of the electric field of the
incident radiation which is parallel to the molecules is strongly absorbed, whereas radiation with its
electric field perpendicular to the molecules is transmitted through the sheet. The polaroid sheet
acts as a polarizer, producing plane-polarized light from light that was originally unpolarised.

26. STANDING WAVES
 A stationary wave is the result of interference between two waves of equal frequency and
amplitude, travelling along the same line with the same speed but in opposite directions.
 This is the phenomenon when 2 progressive waves of equal amplitude and frequency travel along
the same line with the same speed but in opposite directions [recap: Waves which move energy
from place to place are called progressive waves. Waves that do not are called stationary waves]
 The waves interfere producing a wave pattern in which the crests and troughs do not move, unlike
progressive waves
 They travel along with the same speed e.g. strings in musical instruments
 Within a stationary/standing wave, regions of constructive interference are called antinodes and
regions of destructive interference are called nodes.
 Nodes and antinodes do not move along the string

27. COMPARISON BETWEEN STATIONARY AND PROGRESSIVE WAVES:
Stationary waves Progressive waves
Frequency All particles, except at the
nodes, vibrate at the same
frequency
All particles vibrate at the
same frequency
Amplitude The amplitude varies from
zero at the nodes to a
maximum at the antinodes.
The amplitude is the same
for all particles
Phase difference between
two particles
m𝝅,
where m is the number of
nodes between the two
particles.
𝝀
Where =distance apart and
𝝀 is the wavelength.

28. i) STRING WAVES
 If a string is plucked and left to vibrate freely, there are certain frequencies at which it will vibrate
with large amplitudes. This is called resonance.
 The simplest way in which a stretched string can vibrate is a single loop. This is called the
fundamental mode of vibration, or the first harmonic.
The points of no vibrations are called nodes.
The points of maximum amplitude are called antinodes.
In this case,

30. For a wave on a string to travel along and back on the string, the time taken is:
The time taken for the vibrator to pass through a whole number of cycles is:
e.g. for 1 cycle, m =1 and
for 2 cycles, and
thus;
Implying that:
And:
𝝀
And the length of the string is:
𝝀
Thus stationary waves are formed only at frequencies of

31. ii) STATIONARY WAVES IN A PIPE CLOSED AT ONE END
For stationary waves in a closed pipe, the air cannot move at the closed end, so it is always a
node, N
The open end is a position of maximum disturbance and hence is an antinode, A
The particular frequencies at which stationary waves are obtained in a pipe are the resonant
frequencies of the pipe. These frequencies are as below:

34. iii) STATIONARY WAVES IN A PIPE OPEN AT BOTH ENDS