WAVES, PRODUCTION AND PROPAGATION
WAVES,
PRODUCTION AND PROPAGATION
INTRODUCTION
We make and use
waves every day in different forms. All forms of waves transfer or transmit
energy from one place to another through a medium or a vacuum. Radio and
television communications are possible because waves are produced at the
transmitting station and are transferred in form of energy to our radio and
television sets. The radio and television sets only transform the energy of the
wave to sound and pictures. All forms of communications involve the making of
waves at one end, and the use of waves at the other end. The energy of the sun
reaches us here on earth in the form of waves. One common form of wave to be
discussed in this chapter is the water wave. We will also consider some
properties of waves.
OBJECTIVES
At the end of this
section, students should be able to:
·
generate
mechanical waves;
·
state
the important characteristics of waves;
·
produce
circular and plane waves using ripple tanks;
·
generate
and demonstrate longitudinal and transverse waves using suitable materials;
·
classify
waves into longitudinal and transverse waves by using mode of vibration and
direction of propagation;
·
identify
the crest, trough, amplitude, wavelength and points in phase on a given sine
waveform;
·
explain
compression, rarefaction, period, cycle and frequency as they apply to
waves;
·
derive
and use the relationship between wave velocity, wavelength and frequency;
·
identify
light as electromagnetic waves.
What is a wave?
If a stone is
dropped into a still pond or pool of water, the following will be observed:
·
the
water is disturbed where the stone hits the water surface;
·
ripples
travel out from the point where the water is disturbed.
A wave is a disturbance which transfers
energy from the point where a medium is disturbed to other parts of the medium
without any transfer of the particles of the medium between the points.
A wave motion is a succession of ripples
transferring energy through a vacuum or a medium without transferring particles
of the medium between the points.
Two important characteristics of waves are the wave motion and the vibration of the particles of the medium. Waves transfer
energy away from the point where the medium is disturbed to other parts while
the particles of the medium vibrate about their mean positions.
Mechanical waves do not transfer particles of the medium along with energy. The particles vibrate but remain in their fixed positions.
We can demonstrate
that waves do not carry particles along with the experiment below.
·
Generate
waves on a pool of water by dipping a long stick at regular interval.
·
Place
a cork on the path of the wave moving away from the point of disturbance.
·
Observe
the movement of the cork. What do you notice about its movement as the wave
passes under it?
As the wave pass under the cork, it moves up and down (to and fro) but does not travel along with the wave. This proves that waves transfers only energy as it travels through a medium.
Types of waves
Waves are classified
as mechanical or electromagnetic.
Mechanical
waves
A
mechanical wave needs a medium to transfer its energy away from the source.
Energy is transferred
through the medium but the particles of the medium are not carried along by the
wave. They only vibrate about their mean or fixed position. Mechanical waves
include waves travelling through springs, water waves and sound waves.
Production
of mechanical waves
A stone dropped in
water makes the water particles vibrate. The result of disturbing the medium is
that waves are set up in the medium.
Mechanical waves in ropes and springs
To produce a
mechanical wave in a rope, one end of the rope is fixed and the free end is
moved up and down as shown in Figure 11.2. A pulse travels along the rope. If
the hand is moved up and down periodically, successions of pulses travel along
the rope to set up a mechanical wave in the rope.
Figure
11.3: Production of mechanical wave
in a rope
Fig. 11.2:
Producing a mechanical wave in a ropes
Figure
11.4: Mechanical waves in a rope
Mechanical waves can
be produced in springs as shown in Figure 10.5.
Figure 11.5:
Mechanical waves in springs
Water waves
The ripples produced
in Figure 10.1 are waves travelling on water. Water waves are mechanical wave.
They are produced in the school laboratory using a ripple tank. (Figure 10.6).
Figure 11.6: Ripple
tank and production of circular and plane waves
A ripple tank is a
shallow tank with a glass base, a lamp placed above the tank and a white screen
placed under it. The lamp is used to form the image of the ripples or waves
produced on the white screen.
Waves are produced
by dipping an object in and out of water at regular intervals. Circular or
spherical waves are produced by dipping a finger, the sharp point of a pencil
or a spherical dipper into the water. Plane waves are produced by using a
straight dipper like a ruler or the side of a rod.
Wave front
Any periodic
disturbance of a medium always generates waves. The waves produced spreads out
from the source of the waves to other parts of the medium. The particles A, B
and C in Figures 11.7 and 11.8 are always the same distance from the source of
the waves and also are subject to the same vibration. They are said to be in
phase.
Wave front is the
line or curve which joins all the particles vibrating in phase. There are two
types of wave fronts; the circular or spherical wave front and the plane or
straight wave front. Circular wave front spreads out in all directions as the
wave travels away the source while the plane wave front spreads out in one
direction only.
Fig.
11.7: Circular or spherical wave fronts
Figure
11.8: Straight and plane wave fronts
The lines normal or
radial to the wave fronts are called rays. Rays indicate the direction the wave
is travelling. Plane wave fronts have parallel rays while circular wave fronts
diverge in all directions as the wave spreads from the source of the wave.
Electromagnetic
waves
Electromagnetic
waves do not need a medium to transfer its energy from one point to another.
Electromagnetic
waves consist of electric and magnetic fields vibrating at right angles to each
other. The family or group of waves called electromagnetic waves are radio
waves, infrared waves, visible light, ultra-violet rays, X-rays and gamma rays.
Electromagnetic waves can pass a vacuum travelling at a constant speed of 3.0 ×108
m.
There are basically two major ways of classifying waves.
·
Requirement
or non-requirement of medium of propagation, (i.e. mechanical wave and
electromagnetic wave).
·
Wave
direction compared with the direction of vibration of the particle of the
medium, (i.e. transverse or longitudinal wave.)
A wave which travels
along a medium transferring energy from one part of the medium to another, is a
progressive wave. A progressive wave
is also called travelling wave
because it transfers energy outward from the source.
Figure 11.9:
Progressive or travelling wave
A standing or stationary wave is formed
when two waves travelling in the opposite direction meets or by superimposition
of incident wave and its reflection. The amplitude of a standing or stationary
wave varies along the wave.
Figure
11.10: Standing or stationary wave
Progressive or
travelling waves are divided into transverse and longitudinal waves.
Transverse
waves
A
transverse wave is a wave in which the vibration of the medium transmitting the
wave is at right angles (perpendicular) to the direction the wave is
travelling.
Figure
11.11: A transverse wave
As the wave passes
through the medium, its particles vibrate (move up and down). Vibration of the
particles in a transverse wave is perpendicular to the direction the wave is
travelling. The peak of the wave is called crest
while its lowest depression is called trough.
Crest
is the region of maximum upward displacement of the particles from the
equilibrium position.
Trough
is the region of maximum downward displacement of the particles from the
equilibrium position.
Examples of transverse waves are: all electromagnetic
waves, water waves and waves in ropes and stings.
Longitudinal
waves
In a longitudinal
wave, the vibration of particles of the medium are parallel to the direction
the wave is travelling.
A
longitudinal wave is a wave whose direction of propagation in a medium is
parallel to the vibration of the particles of the medium transmitting it. c = compression
(region of squeezed particles)
Figure
11.12: A longitudinal wave
r = rarefaction (region of dispersed
particles)
As the particles
vibrate, they are squeezed or compressed in some parts of the medium. The
region where the particles are squeezed is called compression (c). A region of the compressed particles is followed
by a region of spaced out (dispersed) particles called rarefaction (r).
Sound waves are
waves formed in a compressed spiral spring as shown above are examples of
longitudinal waves.
Describing
waves
The transverse and
longitudinal waveforms are shown below. They are used to define the terms used in describing waves.
Amplitude (A)
Amplitude
is the maximum upward or downward displacement of the particles from the
equilibrium position.
The amplitude is represented by (A) in the transverse progressive waveform
above and the unit is metre (m).
Cycle: A cycle is one complete
vibration or oscillation of a particle.
Wavelength (
Wavelength
is the distance between two points on the waveform, which are vibrating in
phase.
The points may be two successive crests or two successive troughs. We can then
define wavelength as the distance between two successive crests or
troughs.
The unit of wavelength is metre (m).
Frequency (f)
Frequency
is the number of vibrations completed in one second. The unit of
frequency is hertz (Hz).
Period (T)
Period
is the time it takes to complete one vibration or cycle. The unit of period
is seconds (s).
Wave velocity (v)
Wave
velocity is the distance the wave travelled in one second. The unit of wave
velocity is metre per second (ms-1).
Relationship
between velocity, frequency and wavelength
Wave velocity,
wavelength and frequency are connected by the equation:
Mathematical
representation of wave motion
A progressive wave
can be represented graphically by a sine or cosine curve as shown in Figure
11.9. Such waves are formed by causing the medium to vibrate. The displacement
of the particles of the medium at any time is given by:
Phase
In
phase
Phase is an
important characteristic of a wave. Wave is transmitted through a medium by
vibrating particles. These particles are at different positions from the source
of the wave and move in different directions. Phase is the term used to
describe the position of the particle on the waveform. Particles which are in
the same relative position and moving in the same direction, are in phase. Two
waves are travelling in phase if they arrive at the same spot with their crests
or troughs. Such waves reach their maximum points at the same time.
Figure
11.14a: Waves in phase
The waves X and Y
are in phase because their crests and troughs arrive at the same spot at the
same time. The points Q and T on the wave Y are in phase because they are in
the same relative position and are moving in the same direction.
Out of phase
Particles which are
in different positions on the waveform are out of phase. Two waves are out of
phase if their crests or troughs arrive at the same spot at different times.
Figure
11.14b: Waves out of phase by
The waves A and B
above are out of phase by
General progressive wave equation
Figure
11.15
For the progressive
waveform above, the particle, O is at the source while the particle P has
travelled a distance (x) from the source. The phase angle difference between O
and P is
Taking simple ratios
of the angles (
The letters have
their usual meanings. A,
Worked
examples
1. A radio station
broadcast at a wavelength of 20 m. Calculate the frequency of transmission if
the velocity of the wave is 3.0 × 108ms-1.
Solution
2. The diagram below
represents a wave profile. Calculate the wavelength, frequency, wave velocity, period and amplitude
Solution
3. A progressive transverse wave profile in a stretched
rope is given by:
The units of x and y
are in centimetres and t is time measured in seconds. Calculate the:
i.
amplitude
of the wave;
ii.
wavelength
of the wave;
iii.
wave
velocity;
iv.
frequency
of the wave.
Solution
Solution
To solve this
problem, we will compare the given wave equation with the general wave
equation.
(d) frequency of the
wave.
Summary
·
A
wave is a disturbance which transfers energy from the point where the medium is
disturbed to other parts of the medium without any transfer of the particles
between the points.
·
Waves
transfer energy only as they travel through a medium while the particles of the
medium vibrate about their mean positions.
·
Wave
front is the line or curve which joins all the particles vibrating in phase.
Two types of wave fronts are the circular or spherical wave front and the plane
or straight wave front.
·
Rays
are the lines normal or radial to the wave fronts. Rays indicate the direction
of the wave is travelling.
·
A
transverse wave is one in which the particles of the medium transmitting the
wave vibrate at right angle (perpendicular) to the direction the wave is
travelling.
·
Crest
is the region of maximum upward displacement of the particles from the
equilibrium position.
·
Trough
is the region of maximum downward displacement of the particles from the equilibrium
position.
·
A
longitudinal wave is a wave whose direction of propagation in a medium is
parallel to the vibration of the particles of the medium transmitting it.
·
The
region in a wave where the particles are squeezed is called compression (c).
The region of the compressed particles is followed by a region of spaced out
(dispersed) particles called rarefaction (r).
·
A
mechanical wave needs a medium of propagation to transfer its energy to other
parts of the medium.
·
Electromagnetic
waves do not need a medium to transfer its energy from one point to another.
·
Amplitude
is the maximum upward or downward displacement of the particles from the
equilibrium position.
·
Wavelength
is the distance between two successive crests or troughs.
·
Wave
velocity is the distance the wave travelled in one second. Frequency is the
number of vibrations completed in one second.
·
A
progressive wave is a wave which is free to travel outward from the point of
disturbance to other parts of the medium.
·
A
standing or stationary wave is a wave which is trapped in the medium. It is not
free to travel away from medium.
·
Phase
refers to particles which are in the same relative position and are moving in
the same direction. Two waves are travelling in phase if they arrive at the
same spot with their crests or troughs. Particles, which are in different
positions on the waveform, are out of phase. Two waves are out of phase if
their crests or troughs arrive at the same spot at different times.
Practice
questions 11a
1. (a) What is a
wave?
(b) In an experiment
to find the wave velocity of water wave, a girl used a vibrating dipper of
frequency 20 Hz to produce a wave of wavelength 5 cm. What is the speed of the
wave?
2. (a) What is a
wave motion?
(b) How will you
demonstrate that a mechanical wave transmits only energy?
(c) Describe the
motion of the particles of the medium.
3. State two:
(i) differences
between a mechanical wave and an electromagnetic wave;
(ii) examples of
each for mechanical and electromagnetic waves.
4. (a) Define the following:
transverse wave and longitudinal wave.
(b) State the
differences between transverse and longitudinal waves.
(c) Give two
examples each of transverse and longitudinal waves.
5. (a) Define the
terms frequency, wavelength and wave velocity;
(b) Derive a
mathematical equation linking the three terms in (a) above.
(c) A radio station
broadcasts at a frequency of 1.062 ×108Hz. At what wavelength is the
station broadcasting if the speed of wave is 3.0 ×108ms-1?
6. (a) Explain the
terms: phase, in phase and out of phase as applied to waves.
(b) Make a sketch of
a transverse waveform and indicate on your sketch:
(i) two points
vibrating in phase;
(ii) two points
vibrating out of phase;
a particle whose
relative positions is
7. What is a
progressive wave?
A
progressive wave profile is given by:
(i)
What do the symbols A, x and
(ii)
Sketch the wave and indicate the positions of A, x and
(iii)
A particular wave profile is given by: y = 0.08sin (10
8. (a) Define
wavelength, frequency and wave velocity;
(b) Radio waves transmitted from certain a radio station is represented
by the wave equation
y = 0.75sin (0.67
Calculate the:
(i) wavelength of
the wave;
(ii) frequency of
the wave;
(iii) velocity of
the wave.
y, x are in metres
while t is in seconds
9. (a) Define
frequency and period of a wave,
(b)
(i) What is the wavelength of the wave? (ii) What is the period and frequency of the wave? (iii) Calculate the speed of the wave in ms-1.
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