Properties of Waves
Introduction


Many fields of science and engineering require a thorough understanding of different types of waves. Movement within the Earth's crust can cause devastating seismic waves. Medical scientists use ultrasound and other types of waves to scan the insides of the human body, while civil engineers need to know how vibrations affect buildings and other structures.



Waves are used for communicating information and for carrying energy. When sound waves, water waves, or mechanical waves pass through a substance, the particles in the substance vibrate in a specific order. Electromagnetic waves do not need a material medium to travel through, so they can travel through a vacuum.

Nomenclature
In earlier courses you may have been introduced to waves by considering the type of motion shown in Fig.1.

Figure 1.   A stadium or Mexican wave.
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Complete the following statements for the wave in Fig.1.

  • The hats move as the disturbance moves from . Each of the hats moves up and down so the overall effect is that the moves from left to right. In this type of wave each part of the material through which the wave travels moves at to the direction in which the wave travels. This behaviour is typical of . The disturbance moves along even though the individual parts only move up and down.
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The particles in the medium through which a transverse wave passes move at right angles to the direction in which the disturbance travels.

Continuous transverse waves can be produced in Fig.2 by moving the end point up and down.

Figure 2.   Transverse waves.
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Another type of wave motion is shown in Fig.3. Move the point at the left-hand side of the stretched spring as indicated and note how the disturbance travels.

Figure 3.   Longitudinal waves.
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Complete the following statements for the wave in Fig.3.

  • The coils in the stretched spring move to and fro as the disturbance moves the spring. Each of the coils moves so the overall effect is that the moves along the stretched spring. In this type of wave each part of the material through which the wave travels moves along the direction in which the wave travels. This behaviour is typical of .
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Sound
energy
A system has energy when it has the capacity to do work. The scientific unit of energy is the joule.
energy
is transmitted by the movement of the air molecules to and fro along the direction in which the energy is moving. Sound waves are longitudinal waves. Sound, ultrasound, and some seismic (shock) waves are the most common examples of longitudinal waves.

So far we have seen that waves can be divided into two categories, transverse and longitudinal. However, we need to define a few more quantities before we can describe waves more fully:
The material through which a wave travels is called the medium. However, electromagnetic waves are exceptional. As their name suggests, they are disturbances in either an electric or a magnetic field. But, because a changing electric field creates a changing magnetic field and vice versa, they are self-sustaining. Thus they can pass through a vacuum.
The wave frequencyf is the number of complete waves passing any point each second. Frequency is measured in hertz, Hz. Once a wave leaves the source, its frequency cannot be altered.
The words crest and trough are used to describe the top and bottom respectively of a transverse wave.
The amplitude of the wave is the distance between the peak of the crest and the undisturbed position. The energy associated with a wave of a certain frequency depends on its amplitude.
     The wavelength is the distance from one point on a wave to the identical point on the next wave. This can be stated as the distance from a crest on a wave to the next crest on the wave.



Figure 4.   A wave.
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Complete the following statements to summarize the terminology used to describe waves.

  • The material through which a wave travels is called the . The wave is the number of complete waves passing any point each . Frequency is measured in . Once a wave leaves the source its frequency be altered. The words and trough are used to describe the and respectively of a wave. The of a transverse wave is the distance between the of the crest and the undisturbed position.The associated with a wave of a certain frequency depends on its amplitude. The is the distance from one point on a wave to the identical point on the next wave. This can be stated as the distance from a crest on a wave to the next crest on the wave.
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Some of these terms can also be used to describe longitudinal waves, but additional descriptions must also be introduced.

Figure 5.   Longitudinal wave.
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The slider control in Fig.5 allows you to control the
amplitude
The amplitude of a wave is the distance between the peak of the crest and the undisturbed position.
amplitude
of the
longitudinal wave
A wave is said to be longitudinal when it causes the particles of the medium through which it passes to move parallel to the direction in which the wave is moving.
longitudinal wave
. Press the 'pause' button to reveal the labels.

Longitudinal sound waves passing through air produce a series of compressions and rarefactions.

Complete the following statement to summarize longitudinal wave terminology.

  • Sound waves are . Before sound waves pass through air, the air molecules have an average equilibrium separation. Passing sound waves produce a series of and rarefactions. The distance between the centres of successive compressions gives the . Similarly the distance between the centres of successive rarefactions also gives the wavelength.
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Sound waves need a medium to travel through. This can be shown using the apparatus shown in Fig.6.

Figure 6.   Experiment to show that sound waves need a medium.
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Click the button in Fig.6 to activate the bell and again to remove air from the jar. As air is removed from the bell jar the level of sound outside the jar reduces. The bell inside the jar can still be seen to be working properly but, since there is no air, no sound waves are being transmitted.

Polarization and phase
The amplitude and
wavelength
The wavelength is the distance from one point on a wave to the identical point on the next wave. This can be stated as the distance from a crest on a wave to the next crest on the wave.
wavelength
of a
transverse wave
A wave is said to be transverse when it causes the particles of the medium through which it passes to move at right-angles to the direction in which the wave is moving.
transverse wave
can be represented as shown in Fig.7.

Figure 7.   Amplitude and wavelength of a transverse wave.
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The slider controls in Fig.7 allow you to observe the effect on a progressive transverse wave of altering its amplitude and/or wavelength.

The slider control labelled 'Phase' in Fig.8 turns on a second wave of the same wavelength. When corresponding points on the two waves reach maximum or minimum displacements at the same instant, the waves are said to be in phase. Otherwise there is a phase difference (measured in degrees or radians) between them.

Figure 8.   Phase difference.
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Set the
phase difference
The phase difference between two waves describes how the motion of particles in one wave compares with their motion in the other wave at any particular instant.
phase difference
in Fig.8 to 0°, 90°, and 180° and complete the following statements.

  • When the waves are in phase, the of one wave will pass the same point or line at the same time as the of the other.

    When there is a phase difference of 90° between the waves, the of one wave will correspond with the of the other.

    When there is a phase difference of 180° between the waves, the of one wave will correspond with the of the other.
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So far we have only considered waves in two dimensions. The oscillations have been in one plane, (x, y). Looking end-on at an approaching wave you would simply see a vertical line.

Figure 9.   Waves vibrating in one plane.
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Waves vibrating in just one plane are described as plane polarized.

Unpolarized waves such as those shown in Fig.10 cause vibrations in more than one plane.

Figure 10.   Waves vibrating in many planes.
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Light waves emitted by a lamp or arriving at the Earth's surface from the sun are unpolarized. They can be polarized by passing the light through a polaroid filter. The filter only allows through light rays which oscillate in one preferred direction, determined by the filter itself. If two filters have their planes of polarization at 90° to each other, they are referred to as crossed polaroids. The light getting through the first filter will have no component capable of passing through the second, therefore crossed polaroids will appear dark.

Radiation intensity
The planets of our solar system can be represented as shown in Fig.11. Planets closer to the sun than the Earth are too warm to sustain human life while those further away are too cold. The Earth is special because it is the correct distance from the sun to receive just the right amount of energy.

Figure 11.   Representation of the solar system.
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Common sense suggests that the energy collected by each planet in our solar system depends on its distance from the sun. Since the energy collected also depends on the time for which a planet is irradiated, we must consider the energy received over equal time intervals to achieve a fair comparison. We can do this by measuring the
power
The power of system is a measurement of the rate at which energy is transferred from one form to another. The scientific unit of power is the watt.
power
, defined as the energy received per second.



The quantity of energy collected also depends upon the size of the planet so when comparing the effect that the distance from the sun has on the energy received by planets, it is necessary to consider equal areas. The
intensity
The intensity at a particular point is described as the power per unit area.
intensity
at a particular point is described as the power per unit area.



To calculate the intensity at a point some distance from a light source such as a lamp or a star, we assume that the source is a point radiating equally in all directions. Light travels, in all directions, away from this point source with a speed of 3 × 108 ms−1.

If we consider a quantity of light leaving the point source at a particular time. One nanosecond (1 × 10−9 s) later, the light will have travelled a distance given by:
Therefore, 1 nanosecond after leaving the source, the energy is smeared out over a sphere of radius 0.3 m. The surface area of a sphere is given by the formula, .
If the power of the source is P watts, the intensity at any point 1 nanosecond after the radiation leaves the source is given by:
A further nanosecond after leaving the source the radiation will have travelled twice as far, i.e. 0.6 m.
If the power of the source is P watts, the intensity at any point 2 nanoseconds after the radiation leaves the source is given by:
    



As the distance from the point source doubles, the intensity of the radiation reduces by a factor of four. The relationship between distance and intensity can be summarized mathematically as follows:



This relationship can be confirmed by the following experiment.

Figure 12.   Take readings for the intensity at different distances from the source.
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The intensity of radiation from the sun arriving at the top of the Earth's atmosphere is called the solar constant, S. It has a value of 1370 Wm−2. The Earth's atmosphere absorbs some of this radiation as it passes through, so the atmosphere warms up and the intensity at the Earth's surface is lower.

Luminosity
While radiation intensity allows us to compare the quantity of radiation arriving at any point,
luminosity
The luminosity of a light source is defined as the rate at which it radiates energy. In other words, the luminosity is the total energy radiated per second.
luminosity
tells us the quantity of energy leaving a radiating object.

Figure 13.   Stars are luminous objects.
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The luminosity, L, of a star such as the sun is defined as the rate at which it radiates energy. In other words, the luminosity is the total energy radiated per second.



Luminosity has units of watts.

We can use the values of the solar constant and the distance between the sun and the Earth to calculate the luminosity of the sun as follows:
The solar constant S = 1370 Wm−2 is the rate at which energy arrives per square metre at the Earth's distance from the sun.
Since the sun radiates the same amount of energy in all directions, the total rate at which it radiates energy will be given by
solar constant × surface area of a sphere with radius equal to the Earth–sun distance
For the mean Earth–sun distance (rE = 1.496 × 1011 m),
     Therefore, the sun's luminosity L is 3.853 × 1026 W. A more sophisticated analysis can use this value to derive the sun's surface temperature, 5780 K.



A star's luminosity depends upon both the temperature at its surface and its size. In the early twentieth century, scientists reasoned that the luminosity of a star could be related to its surface temperature. Two astronomers, Hertzprung and Russell independently experimented with plotting luminosity versus surface temperature. The resulting diagram came to be known as a Hertzprung-Russell diagram, or H-R diagram for short.

The position of a particular star on an H-R diagram can be used to indicate its type and give an estimate of its age.

Summary


When a transverse wave travels through a medium, the particles in the medium move at right angles to the direction in which the disturbance travels.

When a longitudinal wave travels through a medium, the particles in the medium move to and fro along the direction in which the wave travels.

Sound waves need a medium to travel through.

The phase relationship between two waves describes how the motion of particles in one wave compares with their motion in the other wave at any particular instant.

Waves vibrating in just one plane are described as plane polarized.

Intensity of radiation and the distance of the source are connected by the mathematical relationship,

The luminosity of a star such as the sun is defined as the rate at which it radiates energy.

Hertzprung-Russell diagrams plot luminosity versus surface temperature.
Exercises
1. The electronics systems on a space probe which is 300 million km from the Sun are powered from solar panels with an area of 4 m2. The space probe moves closer to the sun. What happens to the power output from the solar panels?
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2. What area of solar panels will be required to power the probe's electronic system when it is positioned 1.5 × 1011 m from the sun?
  •  m2   (to the nearest whole number)
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3. A student makes the following statements about intensity of radiation. Decide whether they are true or false.
  • The intensity of radiation arriving on a surface is defined as the power acting on unit area.
    The intensity at a surface varies inversely with the square of the distance away from the point source.
    The relationship between intensity of the radiation from a point source I and distance d can be stated as .
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4. Engineers decide that electricity to power a communications satellite is to be provided by solar cells. The payload of the launching shuttle will only allow for cells with a maximum area of 14 m2.

At a point 1.5 × 1011 m from the sun the intensity of the radiation from the sun is approximately 1500 Wm−2.
Calculate the intensity of the radiation at a point 1.45 × 1011 m from the sun.
  • Wm−2   (to the nearest whole number)
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5. If the
efficiency
The efficiency of a system is a measurement of the ratio of the useful output power to the total input power.
efficiency
of the solar cells is 16%, what is the maximum
electrical power
The electrical power dissipated by a component is the energy transferred per second when a current passes through the component.
electrical power
produced by the solar cells in the previous question?
  • Wm−2
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6. A pupil makes the following statements about transverse waves. Decide whether they are true or false.
  • Interference is a true test for the wave nature of electromagnetic waves.
    Transverse waves can be used to show reflection, refraction, and diffraction.
    Light, sound, and X-rays are all transverse waves.
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Figure 14.  
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7. A pupil studies sound waves by connecting a microphone to a cathode ray oscilloscope. The CRO traces produced by two different frequencies are drawn in Fig.14. The CRO settings are the same for each trace.

Decide whether the following statements about these sources are true or false.
  • Both sources have the same amplitude.
    Source X emits more energy per second than source Y.
    Source X has a lower pitch.
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8. A pupil makes the following statements about a continuous wave. Decide whether they are true or false.
  • It transfers energy in a regular manner.
    Points in the wave separated by one whole wavelength are in phase.
    Points in a wave separated by half a wavelength have a phase difference of 90°.
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Figure 15.  
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9. A pupil investigating the behaviour of a solar panel connects a small panel (1 cm × 1 cm) to a
joule
The joule is the unit of energy.
joule
meter and places this panel 1 m away from a 100 W light bulb, as shown in Fig.15. In his attempt to estimate the energy collected by the solar panel he assumes that the light bulb is radiating equally in all directions. Which of the following shows his estimate of the energy collected by the solar panel in 4 minutes?
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10. A mass of 100 g attached to the end of a spring is pulled down a little and released so that it oscillates. If, in the next five seconds it passes through its rest position 10 times, the
periodic time
The periodic time is the time for one repetition of an oscillation
periodic time
for the oscillation is approximately …
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