View source for FHSST Physics Waves:Properties of Waves

{{FHSST_Physics_Waves}}

= Properties of Waves =

We have discussed some of the simple properties of waves that we need
to know. These have just been describing the characteristics that
waves have. Now we can progress onto some more interesting and,
perhaps, less intuitive properties of waves.

== Properties of Waves : Reflection ==


When waves strike a barrier they are reflected. This means that waves
bounce off things. Sound waves bounce off walls, light waves bounce
off mirrors, radar waves bounce off planes and how bats can fly at
night and avoid things as small as telephone wires. etc. The property
of reflection is a very important and useful one.

(NOTE TO SELF: Get an essay by an air traffic controller on radar)
(NOTE TO SELF: Get an essay by on sonar usage for fishing or for submarines)

When waves are reflected, the process of reflection has certain
properties. If a wave hits an obstacle at a right angle to the surface
(NOTE TO SELF: diagrams needed) then the wave is reflected directly
backwards.

[[Image:fhsst_waves9.png|372px|]]

[[Image:fhsst_waves10.png|372px|]]

If the wave strikes the obstacle at some other angle then it is
not reflected directly backwards. The angle that the waves arrives
at is the same as the angle that the reflected waves leaves at.
The angle that waves arrives at or is incident at equals the angle
the waves leaves at or is reflected at. Angle of incidence equals
angle of reflection


{| 
|-
|<math>\begin{matrix}\theta_{i}=\theta_{r}\end{matrix}</math>
|- 
| class="eqno" align="RIGHT" | 
(2.1)
|}

<blockquote style="background: white; border: 1px solid black; padding: 1em;">
{| cellpadding="3"
|-
| align="CENTER" colspan="2" | <math>\begin{matrix}\theta_{i}=\theta_{r}\end{matrix}</math>
|- 
| align="RIGHT" | &nbsp; 
| align="LEFT" | &nbsp; 
|- 
| align="RIGHT" |<center><math>\theta_{i}</math></center>
| align="LEFT" | : angle of incidence
|- 
| align="RIGHT" |<center><math>\theta_{r}</math></center>
| align="LEFT" | : angle of reflection
|}
</blockquote>

[[Image:fhsst_waves11.png|372px|]]

In the optics chapter you will learn that light is a wave. This means
that all the properties we have just learnt apply to light as well. Its
very easy to demonstrate reflection of light with a mirror. You can also
easily show that angle of incidence equals angle of reflection.

If you look directly into you see yourself ....

'''Need to mention that the incident wave, normal to the surface and the reflected wave all lie in the same plane. The same also holds for refraction at a surface.'''

=== Phase shift of reflected wave ===

When a wave is reflected from a more dense medium, it undergoes a phase
shift. That means that the peaks and troughs are swapped around.

The easiest way to demonstrate this is to tie a piece of string to
something. Stretch the string out flat and then flick the string
once so a pulse moves down the string. When the pulse (a single peak
in a wave) hits the barrier that the string is tied to, it will be
reflected. The reflected wave will look like a trough instead of a
peak. This is because the pulse had undergone a phase change. The
fixed end is like reflection off a more dense medium.

[[Image:fhsst_waves12.png|236px|]]

[[Image:fhsst_waves13.png|236px]]

[[Image:fhsst_waves14.png|236px|]]

If the end of the string was not fixed, i.e. it could move up and
down then the wave would still be reflected but it would 
not undergo a phase shift.

[[Image:fhsst_waves15.png|236px|]]

[[Image:fhsst_waves16.png|236px|]]

[[Image:fhsst_waves17.png|236px|]]

== Properties of Waves : Refraction ==

Sometimes waves move from one medium to another. The medium is the
substance that is carrying the waves. In our first example this was
the water. When the medium properties change it can affect the wave.

Let us start with the simple case of a water wave moving from one
depth to another.
The speed of the wave depends on the depth. If the wave moves directly
from the one medium to the other then we should look closely at the
boundary. When a peak arrives at the boundary and moves across it must
remain a peak on the other side of the boundary. This means that the
peaks pass by at the same time intervals on either side of the
boundary. The period and frequency remain the same! But we said the
speed of the wave changes, which means that the distance it travels in
one time interval is different i.e. the wavelength has changed.

Going from one medium to another the period or frequency does not
change only the wavelength can change.

Now if we consider a water wave moving at an angle of incidence not 90
degrees towards a change in medium then we immediately know that not
the whole wave will arrive at once. So if a part of the wave arrives
and slows down while the rest is still moving faster before it arrives
the angle of the wavefront is going to change. This is known as
refraction. When a wave bends or changes its direction when it goes
from one medium to the next.

If it slows down it turns towards the perpendicular.

[[Image:fhsst_waves18.png|372px|]]

If the wave speeds up in the new medium it turns away from the
perpendicular to the medium surface.

When you look at a stick that emerges from water it looks like it
is bent. This is because the light from below the surface of the
water bends when it leaves the water. Your eyes project the light
back in a straight line and so the object looks like it is a
different place.

[[Image:fhsst_waves19.png|372px|]]

== Properties of Waves : Interference ==

If two waves meet interesting things can happen. Waves are basically
collective motion of particles. So when two waves meet they both try
to impose their collective motion on the particles. This can have
quite different results.

If two identical (same wavelength, amplitude and frequency) waves are
both trying to form a peak then they are able to achieve the sum of
their efforts. The resulting motion will be a peak which has a height
which is the sum of the heights of the two waves. If two waves are
both trying to form a trough in the same place then a deeper trough is
formed, the depth of which is the sum of the depths of the two
waves. Now in this case the two waves have been trying to do the same
thing and so add together constructively. This is called
constructive interference.

[[Image:fhsst_waves20.png|761px|]]

[[Image:fhsst_waves21.png|761px|]]

[[Image:fhsst_waves22.png|761px|]]

If one wave is trying to form a peak and the other is trying to
form a trough then they are competing to do different things. In
this case they can cancel out. The height of the peak less the
depth of the trough will be the resulting effect. If the depth of
the trough is the same as the height of the peak nothing will
happen. If the height of the peak is bigger than the depth of the
trough a smaller peak will appear and if the trough is deeper then
a less deep trough will appear. This is destructive
interference.

[[Image:fhsst_waves23.png|761px|]]

[[Image:fhsst_waves24.png|761px|]]

[[Image:fhsst_waves25.png|761px|]]

== Properties of Waves : Standing Waves ==

When two waves move in opposite directions, through each other,
constructive interference happens. If the two waves have the same
frequency and wavelength then a specific type of constructive
interference can occur: standing waves can form.

Standing waves are disturbances which don't appear to move, they
stand in the same place. Lets demonstrate exactly how this comes
about. Imagine a long string with waves being sent down it from
either end. The waves from both ends have the same amplitude,
wavelength and frequency as you can see in the picture below:

[[Image:fhsst_waves26.png|460px|]]

To stop from getting confused between the two waves we'll draw the
wave from the left with a dashed line and the one from the right
with a solid line. As the waves move closer together when they touch
both waves have an amplitude of zero:

[[Image:fhsst_waves27.png|460px|]]

If we wait for a short time the ends of the two waves move past each
other and the waves overlap. Now we know what happens when two waves
overlap, we add them together to get the resulting wave.

[[Image:fhsst_waves28.png|460px|]]

Now we know what happens when two waves overlap, we add them
together to get the resulting wave. In this picture we show the two
waves as dotted lines and the sum of the two in the overlap region
is shown as a solid line:

[[Image:fhsst_waves29.png|460px|]]

The important thing to note in this case is that there are some
points where the two waves always destructively interfere to zero.
If we let the two waves move a little further we get the picture
below:

[[Image:fhsst_waves30.png|460px|]]

Again we have to add the two waves together in the overlap region to
see what the sum of the waves looks like.

[[Image:fhsst_waves31.png|460px|]]

In this case the two waves have moved half a cycle past each other
but because they are out of phase they cancel out completely. The
point at 0 will always be zero as the two waves move past each
other.

When the waves have moved past each other so that they are
overlapping for a large region the situation looks like a wave
oscillating in place. If we focus on the range -4, 4 once the waves
have moved over the whole region. To make it clearer the arrows at
the top of the picture show peaks where maximum positive
constructive interference is taking place. The arrows at the bottom
of the picture show places where maximum negative interference is
taking place.

[[Image:fhsst_waves32.png|369px|]]

As time goes by the peaks become smaller and the troughs become
shallower but they do not move.

[[Image:fhsst_waves33.png|369px|]]

For an instant the entire region will look completely flat.

[[Image:fhsst_waves34.png|369px|]]

The various points continue their motion in the same manner.

[[Image:fhsst_waves35.png|369px|]]

Eventually the picture looks like the complete reflection through
the x-axis of what we started with:

[[Image:fhsst_waves36.png|369px|]]

Then all the points begin to move back. Each point on the line is
oscillating up and down with a different amplitude.

[[Image:fhsst_waves37.png|369px|]]

If we superimpose the two cases where the peaks where at a maximum
and the case where the same waves where at a minimum we can see the
lines that the points oscillate between. We call this the envelope
of the standing wave as it contains all the oscillations of the
individual points. A node is a place where the two waves cancel out
completely as two waves destructively interfere in the same place.
An anti-node is a place where the two waves constructively
interfere.

[[Image:fhsst_waves38.png|453px|]]

To make the concept of the envelope clearer let us draw arrows
describing the motion of points along the line.

[[Image:fhsst_waves39.png|453px|]]

Every point in the medium containing a standing wave oscillates up
and down and the amplitude of the oscillations depends on the
location of the point. It is convenient to draw the envelope for the
oscillations to describe the motion. We cannot draw the up and down
arrows for every single point!

=== Reflection from a fixed end ===

If waves are reflected from a fixed end, for example tieing the end
of a rope to a pole and then sending waves down it. The fixed end
will always be a node. Remember: Waves reflected from a
fixed end undergo a phase shift.

The wavelength, amplitude and speed of the wave cannot affect this,
the fixed end is always a node.

=== Reflection from an open end ===

If waves are reflected from end, which is free to move, it is an
anti-node. For example tieing the end of a rope to a ring, which can
move up and down, around the pole. Remember: The waves sent
down the string are reflected but do not suffer a phase shift.


=== Wavelengths of standing waves with fixed and open ends ===

[[Image:fhsst_waves40.png|392px|]]

[[Image:fhsst_waves41.png|392px|]]

== Beats ==

If the waves that are interfering are not identical then the waves
form a modulated pattern with a changing amplitude. The peaks in
amplitude are called beats. If you consider two sound waves
interfering then you hear sudden beats in loudness or intensity of
the sound.

The simplest illustration is to draw two different waves and then
add them together. You can do this mathematically and draw them
yourself to see the pattern that occurs.

Here is wave 1:

[[Image:fhsst_waves42.png|761px|]]

Now we add this to another wave, wave 2:

[[Image:fhsst_waves43.png|761px|]]

When the two waves are added (drawn in coloured dashed lines) you
can see the resulting wave pattern:

[[Image:fhsst_waves44.png|761px|]]

To make things clearer the resulting wave without the dashed lines
is drawn below. Notice that the peaks are the same distance apart
but the amplitude changes. If you look at the peaks they are
modulated i.e. the peak amplitudes seem to oscillate with another
wave pattern. This is what we mean by modulation.

[[Image:fhsst_waves45.png|761px|]]

The maximum amplitude that the new wave gets to is the sum of the
two waves just like for constructive interference. Where the waves
reach a maximum it is constructive interference.

The smallest amplitude is just the difference between the
amplitudes of the two waves, exactly like in destructive
interference.

The beats have a frequency which is the difference between the
frequency of the two waves that were added. This means that the
beat frequency is given by

{| 
|-
|<center><math>f_B=|f_1-f_2|</math></center>
|- 
| class="eqno" align="RIGHT" | 
(2.2)
|}


<blockquote style="background: white; border: 1px solid black; padding: 1em;">
{| cellpadding="3"
|- 
| align="CENTER" colspan="2" | <center><math>f_B=|f_1-f_2|</math></center>
|- 
| align="RIGHT" | &nbsp; 
| align="LEFT" | &nbsp; 
|- 
| align="RIGHT" | ''f''<sub>B</sub>
| align="LEFT" | : beat frequency (Hz or s<sup>-1</sup>)
|- 
| align="RIGHT" | ''f''<sub>1</sub>
| align="LEFT" | : frequency of wave 1 (Hz or s<sup>-1</sup>)
|- 
| align="RIGHT" | ''f''<sub>2</sub>
| align="LEFT" | : frequency of wave 2 (Hz or s<sup>-1</sup>)
|}
</blockquote>

== Properties of Waves : Diffraction ==

One of the most interesting, and also very useful, properties of
waves is diffraction. When a wave strikes a barrier with a
hole, only part of the wave can move through the hole. If the hole
is similar in size to the wavelength of the wave diffractions
occurs. The waves that comes through the hole no longer looks like
a straight wave front. It bends around the edges of the hole. If
the hole is small enough it acts like a point source of circular
waves.

This bending around the edges of the hole is called diffraction.
To illustrate this behaviour we start with Huygen's principle.

=== Huygen's Principle ===

Huygen's principle states that each point on a wavefront acts like
a point source or circular waves. The waves emitted from each
point interfere to form another wavefront on which each point
forms a point source. A long straight line of points emitting
waves of the same frequency leads to a straight wave front moving
away.

To understand what this means lets think about a whole lot of
peaks moving in the same direction. Each line represents a peak of
a wave.

[[Image:fhsst_waves46.png|734px|]]

If we choose three points on the next wave front in the direction of
motion and make each of them emit waves isotropically (i.e. the same in all
directions) we will get the sketch below:

[[Image:fhsst_waves47.png|734px|]]

What we have drawn is the situation if those three points on the
wave front were to emit waves of the same frequency as the moving
wave fronts. Huygens principle says that every point on the wave
front emits waves isotropically and that these waves interfere to
form the next wave front.

To see if this is possible we make more points emit waves
isotropically to get the sketch below:

[[Image:fhsst_waves48.png|734px|]]

You can see that the lines from the circles (the peaks) start to
overlap in straight lines. To make this clear we redraw the sketch
with dashed lines showing the wavefronts which would form. Our
wavefronts are not perfectly straight lines because we didn't draw
circles from every point. If we had it would be hard to see clearly
what is going on.

[[Image:fhsst_waves49.png|734px|]]

Huygen's principle is a method of analysis applied to problems of
wave propagation. It recognizes that each point of an advancing
wave front is in fact the center of a fresh disturbance and the
source of a new train of waves  and that the advancing wave as a
whole may be regarded as the sum of all the secondary waves
arising from points in the medium already traversed. This view of
wave propagation helps better understand a variety of wave
phenomena, such as diffraction.

=== Wavefronts Moving Through an Opening ===

Now if allow the wavefront to impinge on a barrier with a hole in
it, then only the points on the wavefront that move into the hole
can continue emitting forward moving waves - but because a lot of
the wavefront have been removed the points on the edges of the
hole emit waves that bend round the edges.

[[Image:fhsst_waves50.png|736px|]]

The wave front that impinges (strikes) the wall cannot continue
moving forward. Only the points moving into the gap can. If you
employ Huygens' principle you can see the effect is that the
wavefronts are no longer straight lines.

[[Image:fhsst_waves51.png|736px|]]

'''Riaan Note: still cant find this image, have to get it from the pdf'''

For example, if two rooms are connected by an open doorway and a
sound is produced in a remote corner of one of them, a person in
the other room will hear the sound as if it originated at the
doorway. As far as the second room is concerned, the vibrating air
in the doorway is the source of the sound. The same is true of
light passing the edge of an obstacle, but this is not as easily
observed because of the short wavelength of visible light.

This means that when waves move through small holes they appear to
bend around the sides because there aren't enough points on the
wavefront to form another straight wavefront. This is bending
round the sides we call diffraction.

== Properties of Waves : Dispersion ==

Dispersion is a property of waves where the speed of the
wave through a medium depends on the wavelength. So if two waves
enter the same dispersive medium and have different wavelengths
they will have different speeds in that medium even if they both
entered with the same speed.

[[Category:FHSST Physics]]