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Oscillations
Oscillations are movements back and forth about a mean position - e.g. the
swinging of a pendulum, the vibration of a ruler held over the edge of a table, etc.
A typical imaginary oscillator is shown below:
Displacement to one side of the mean position is taken to be positive - and to the
other side, negative.
A graph of displacement against time for a typical oscillator is shown below:
As shown above:
The amplitude A is the maximum displacement from the mean position.
The period T is the time taken to complete one oscillation.
The frequency f is the number of oscillations completed per second.
It follows from the above that: f = 1/T
If T is in s, f is in hertz Hz.
Waves
A wave can be thought of as a series of oscillators in a row, each one a little out of
phase with the one behind it.
If a wave is frozen in time, a graph of displacement against distance along the wave
will be as shown below:
The wavelength  is the distance shown in the graph.
There is also a relationship between the velocity v (in ms-1) of the wave, its
wavelength  (in m), and its frequency f (in Hz):
v = f
REMEMBER!
Phase Difference
Phase difference is illustrated by the diagram below:
A and B could be 2 different oscillators or 2 different waves - depending on whether
the "x-axis" is time or distance.
The phase difference could therefore be expressed either as:





a time difference (in s)
a fraction of a period (T/8 in the above example)
a distance (in m)
a fraction of a wavelength (/8 in the above example)
or an angle (45o or /4 radians in the above example)
This last way of expressing phase difference is based on the fact that the graphs are
sinusoidal - i.e they are of the same form as sin against  as shown below:
Radians are an alternative way of expressing angles:
45o = /4 radians
90o = /2 radians
180o =  radians
360o = 2 radians
etc etc
In the example above, A has its peak at an earlier time (or smaller value of distance)
than B's peak. Therefore A is said to be leading B by /4 radians etc.
Alternatively, one could say that B is lagging A by /4 radians etc.
"In phase" means that there is zero phase difference.
"Out of phase" means that there is a phase difference of T/2, /2, 180o or  radians.
Transverse and Longitudinal Waves
In a transverse wave the oscillations are at right angles to the direction of travel of
the wave. A "snapshot" of a transverse wave therefore "looks like a wave", as shown
below:
Light (electromagnetic) waves, water waves, waves on strings, etc are examples of
transverse waves.
In a longitudinal wave, however, the oscillations are parallel to the direction of
travel of the wave. A longitudinal wave therefore "doesn't look like a wave". It
appears to be a series of compressions and rarefactions, as shown below:
However, if you plot a graph of displacement against distance for a longitudinal
wave, you get a sinusoidal (wave) shape - just as for a transverse wave. The only
difference is that positive displacement is "to the right" rather than "up" - and
negative displacement is "to the left" rather than "down".
Sound Waves
Sound waves are an important example of longitudinal waves. In the case of a
sound wave the particles would be air molecules (or molecules of whatever material
the sound was passing through).
The graphs below show the variation of displacement and pressure for a sound
wave:
Note that the displacement and pressure variations are out of phase by 90 o (/2).
Superposition
Superposition (or interference) is what happens when 2 waves are in the same
place at the same time. Essentially they add together, as shown below:
The above assumes that the 2 waves both have the same amplitude. If their
amplitudes are different they will not completely cancel out when they are out of
phase.
If the 2 waves are not perfectly in phase (or out of phase) they will superpose to give
a wave of the same wavelength as the originals, but whose amplitude is less than
the sum of their amplitudes.
If 2 waves arrive at the same place you will only get consistent superposition effects
if they have the same frequency (or wavelength), and there is a constant phase
difference between them. Such waves are said to be coherent. In practice, this
means that it is more or less impossible to get interference effects using (say) 2
separate sources of light since, even if they had the same wavelength, there would
be no consistent phase relationship between them. To get interference effects with
light you have to split light from a single source into 2 beams.
Standing Waves
A standing wave is the result of the superposition of 2 identical waves travelling in
opposite directions, as shown below:
(The wave coming in the opposite direction is often the result of the first wave being
reflected back off some barrier or change in the medium. e.g. You get a standing
wave on a long spring when you waggle one end, and the other end is tied to a post.
The wave you send out is reflected back from the post.)
The standing wave pattern does not move either to the left or the right. There are
fixed places (nodes) where its displacement is always zero. Halfway between the
nodes are the antinodes - fixed places where the displacement varies through the
maximum range.
The distance between adjacent nodes is /2.
The distance between adjacent antinodes is /2.
Standing Waves in Pipes
In wind instruments standing waves are set up in pipes of various kinds. Some
instruments (e.g. the flute and the recorder) behave like pipes which are open at
both ends, and others (e.g. the clarinet) behave like pipes with one end open and
one end closed - the closed end being the reed end.
At a closed end the air cannot move - so we always get a node there. At an open
end, however, the air can vibrate - so there is always an antinode at an open end.
The diagrams below show the different possible standing waves for open and closed
pipes. It is important to bear in mind, however, that these are longitudinal standing
waves, so the lines shown within the pipes are just graphs of displacement against
distance - the standing waves actually consist of compressions and rarefactions, and
the pressure nodes and antinodes are 90o out of phase with the displacement ones
shown in the graphs.
Pipe open at both ends
Pipe closed at one end
In order to understand the above it is important to bear in mind that v = f and v (the
velocity of sound) is a constant. Thus, if  decreases by a factor of 3 (say), then f
must increase by a factor of 3 to compensate.
Note also that the possible frequencies for an open pipe are: f o, 2fo, 3fo, 4fo, 5fo .....
And for a pipe closed at one end: fo, 3fo, 5fo, 7fo .....
..... where fo is the frequency of the fundamental mode.
Standing Waves on Strings
In stringed instruments transverse standing waves are set up on the strings. The 2
ends of the string are normally fixed in some way - so they have to be nodes. Some
of the the possible modes of vibration are shown below:
Thus, for a string fixed at both ends, the possible frequencies are: fo, 2fo, 3fo, 4fo, 5fo
etc .....
..... where fo is the fundamental frequency.
There is a formula for the velocity v (in ms-1) of the waves on a string (or wire):
v = (T/)1/2
..... where T is the tension (in N) in the string, and  is the mass per unit length of
the string (in kgm-1).
For the fundamental mode:
 = 2l
Also: f = v/
It therefore follows that:
f = (1/2l)(T/)1/2
Thus:
 f is proportional to 1/l
 f is proportional to T1/2
 f is proportional to -1/2
Superposition and Path Difference
When 2 waves are out of phase it may be because they have have come from the
same source, but travelled via 2 different routes - one longer than the other. Thus the
path difference between the 2 routes will determine the phase difference.
The CD is a good example of this phenomenon. The tracks on a CD consist of a
sequence of pits and bumps which, in cross-section, might look as follows:
Laser light is used to read a CD, and the bumps are a quarter of a wavelength of
the light (/4) high. If the laser beam hits the edge of a bump as shown above, part
of the beam (A) will hit the top of the bump and part (B) will hit the bottom. Thus
there is a path difference of /2 between waves following path A and those following
path B. It follows that these 2 waves will be out of phase after reflection, and will
superpose (interfere) destructively. i.e. Zero light intensity will be reflected from the
edge of a bump.
The reason for using laser light is that it is monochromatic (of a single wavelength)
and coherent (random phase changes occur far less frequently than with "normal"
light).
Digital and Analogue Signals
The diagram in the previous section also shows the digital signal - consisting of 0's
and 1's - that would be encoded on the CD. The 0's are on the edges of the bumps
for the reason explained. At other points on the CD there is no phase difference
between parts A and B of the beam, so the interference is constructive, and a high
intensity of light is reflected - corresponding to a "1".
Electronically the 0's and 1's would be represented as low and high voltages
respectively, as shown below:
An analogue signal (such as that produced by a microphone) consists of a voltage
which varies with time and can have any value, as shown below:
Refraction of Light
A ray of light is an line with an arrow on it showing the direction in which the light
waves are travelling, as shown below:
When a ray of light passes from medium 1 into another medium 2 in which its
velocity is lower (e.g. from air to glass) the ray is bent towards the normal, as shown
below:
The above phenomenon is known as refraction.
Snell's Law states that sin i / sin r = v1/v2 = a constant
..... where v1 is the velocity of light in medium 1, and v2 is the velocity of light in
medium 2.
The above constant is known as "the refractive index for light travelling from
medium 1 to medium 2" - or 12.
Thus: 12 = sin i / sin r = v1/v2
Note that refractive index is a ratio - and therefore has no units.
N.B. If medium 1 is air (or vacuum), 12 can simply be referred to as "the
refractive index of medium 2" or 2.
Lenses
Lenses are of 2 main types, converging and diverging, as shown below:
Converging lenses cause rays of light to converge inwards towards the axis, and
diverging lenses cause them to diverge away from the axis.
At the moment, we are only concerned with converging lenses. Diverging lenses are
dealt with later in the course.
The principal focus F and focal length f of a converging lens are defined as
follows:
i.e. Parallel rays converge at the principal focus.
The power P of a lens is defined by the formula:
P = 1/f
If f is in metres (m), P is in dioptres (D).
Thus, if the focal length of a converging lens = +25 cm .....
..... its power P = 1/0.25 = +4 D
(Converging lenses, by convention, have +ve focal lengths and powers - whilst
diverging lenses have -ve focal lengths and powers.)
The Lens Formula
As we saw above, parallel rays coverge at F - i.e. they form an image at F. (There
would be a dot on a screen placed at F.) Parallel rays could be said to come from a
point source of light at infinity. Thus an object at infinity forms an image at the
principal focus.
If the object is not at infinity we can use the lens formula to find the position of the
image:
1/u + 1/v = 1/f
u is the object distance and v is the image distance - as shown in the diagram
below:
Ray Diagrams
All the ray diagrams above are for point objects - which form point images. If we
want to represent objects and images that are not points, and if we want to use the
diagram to find out where the image will be formed, we must draw a more
sophisticated type of ray diagram, as shown below:
Points to note:
 A horizontal scale must be chosen (e.g. 5 cm = 1 cm) and stuck to.
 The vertical scale should be such as to make the diagram as "square" as possible.
i.e. It needs to be larger than the horizontal scale. That way the diagram is more
accurate. However, you won't normally be given the size of the object, so you can
make it whatever size will give you the "squarest" diagram.
 There is a principal focus F on both sides of the lens. (Lenses work both ways!)
 There are, in reality, an infinite number of rays coming from the top of the object.
Out of these the above 3 are chosen and drawn as shown. They converge at the top
of the image. (in fact, only 2 rays are needed to define the top of the image. The 3rd
one is there as a check.)
 If you put the object between F and the lens you will find that the rays on the right
hand side of the lens diverge rather than converge. The image is then said to be
virtual - i.e. it cannot be cast on a screen - but can be seen by the eye - rather like
the image in a mirror. If you were to use the lens formula in such a case, v would
turn out to be -ve.
Critical Angle
Suppose a ray of light is passing from medium 2 (e.g. glass) into another medium 1
(e.g. air) in which its velocity is higher - i.e. the reverse of the situation we had when
we first considered refraction. The ray arrows will simply be reversed as shown
below:
Although "i" is no longer, strictly speaking, the angle of incidence - and "r" is no
longer the angle of refraction - it is simplest to keep the labels "i" and "r" (rather than
reverse them), since they are the same angles as before.
Suppose now that r is increased to the point where i becomes equal to 90 o, as
shown below:
r is now equal to C - the critical angle.
If r is made greater than C, total internal reflection takes place, as shown below:
Applying Snell's Law to the critical angle case:
12
= sin i / sin r = sin90/sinC = 1/sin C
The diagram below shows a ray of light being turned through 90 degrees by a 45-4590 prism.
This is only possible because the critical angle for air-glass is about 42o - which is
less than 45o.
Photons and Energy Levels
Atoms can only have certain values of energy - and nothing inbetween. These
values of energy are called energy levels. They are usually represented as
horizontal lines - with energy increasing from the bottom to the top of the page, as
shown below:
(The energy levels arise from different ways of containing the electron waves within
the atom.)
If an atom is given energy it said to have been excited. It will rise to a higher energy
level. During the course of time it will fall back to a lower level - either in one large
step or several smaller ones. As it does so it will emit light (or other electromagnetic
radiation), which can be thought of as either waves or particles. Both models have
their uses.
Particles of light (or other electromagnetic radiation) are called photons.
If the difference in energy between the higher and the lower level is E, the atom will
emit a photon of energy E when it fall from the higher to the lower level. The
frequency f of the light emitted can then be found using the formula:
E = hf
..... where E is in J, h is Planck's Constant (6.63 x 10-34 Js) and f is in Hz.
(Note that the above situation is reversible. An atom can absorb a photon of energy
E and be excited from level B to level A.)
Because there are many possible ways in which an atom can fall from a higher to a
lower level, each involving a different energy difference - photons of many different
frequencies will be emitted by a given atom. These sets of frequencies are
characteristic for the various elements and, if they are spread out using (for example)
a diffraction grating, they form a line spectrum, as shown below: