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Waves AS 2010 – Revision Notes

OscillationsOscillations are movements back and forth about a mean position - e.g. the swingingof 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 otherside, 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

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f 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.

f a wave is fro!en in time, a graph of displacement against distance along the wavewill be as shown below:

The wavelength is the distance shown in the graph.There is also a relationship between the velocity v "in ms -#$ of the wave, itswavelength λ "in m$, and its fre%uency f "in &!$:

v = f REMEMBER!

Phase Difference

Phase difference is illustrated by the diagram below:

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A and ' could be ( different oscillators or ( 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$a fraction of a period "T*+ in the above example$a distance "in m$a fraction of a wavelength " λ *+ in the above example$or an angle " o or π * radians in the above example$

This last way of expressing phase difference is based on the fact that the graphs aresinusoidal - i.e they are of the same form as sin θ against θ as shown below:

Radians are an alternative way of expressing angles:o

π * radians/0 o π *( radians#+0 o π radians120 o ( π radians

etc etcn the example above, A has its peak at an earlier time "or smaller value of distance$

than '3s peak. Therefore A is said to be leading ' by π * radians etc. Alternatively, one could say that ' is lagging A by π * radians etc."In phase" means that there is !ero phase difference."Out of phase" means that there is a phase difference of T*(, λ *(, #+0 o or π radians.

Transverse and Longitudinal Wavesn 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 shownbelow:

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4ight "electromagnetic$ waves, water waves, waves on strings, etc are examples oftransverse waves.

n a longitudinal wave , however, the oscillations are parallel to the direction of travelof the wave. A longitudinal wave therefore )doesn3t look like a wave). t appears to be aseries 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 - 5ust as for a transverse wave. The only differenceis that positive displacement is )to the right) rather than )up) - and negativedisplacement is )to the left) rather than )down).

Sound Waves

ound waves are an important example of longitudinal waves . n the case of asound wave the particles would be air molecules "or molecules of whatever materialthe sound was passing through$.The graphs below show the variation of displacement and pressure for a sound wave:

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!ote that the displacement and pressure variations are out of phase by /0 o "π *($.

Su er ositionuperposition "or interference $ is what happens when ( waves are in the same place

at the same time. 6ssentially they add together , as shown below:

The above assumes that the ( waves both have the same amplitude. f theiramplitudes are different they will not completely cancel out when they are out of phase.

f the ( waves are not perfectly in phase "or out of phase$ they will superpose to give awave of the same wavelength as the originals, but whose amplitude is less than thesum of their amplitudes.

f ( waves arrive at the same place you will only get consistent superposition effects ifthey have the same fre%uency "or wavelength$, and there is a constant phasedifference between them. 7uch waves are said to be coherent . n practice, this means

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that it is more or less impossible to get interference effects using "say$ ( separatesources of light since, even if they had the same wavelength, there would be noconsistent phase relationship between them. To get interference effects with light youhave to split light from a single source into ( beams.

Standing Waves A standing wave is the result of the superposition of ( identical waves travelling inopposite directions, as shown below:

(The wave coming in the opposite direction is often the result of the first wave beingreflected back off some barrier or change in the medium. e.g. You get a standing waveon a long spring when you waggle one end, and the other end is tied to a post. Thewave 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 fixedplaces " nodes $ where its displacement is always !ero. &alfway between the nodes arethe antinodes - fixed places where the displacement varies through the maximumrange.

The distance between ad5acent nodes is λ *(.

The distance between ad5acent antinodes is λ *(.

Standing Waves in Pi esnwind instruments standing waves are set up in pipes of various kinds. 7ome

instruments "e.g. the flute and the recorder$ behave like pipes which are open at bothends, and others "e.g. the clarinet$ behave like pipes with one end open and one endclosed - 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,

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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 closedpipes. t is important to bear in mind, however, that these are longitudinal standingwaves , so the lines shown within the pipes are ust graphs of displacement againstdistance - the standing waves actually consist of compressions and rarefactions, andthe pressure nodes and antinodes are /0 o out of phase with the displacement onesshown in the graphs.

Pi e o en at !oth ends

Pi e closed at one end

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n order to understand the above it is important to bear in mind that v # f and v $thevelocity of sound% is a constant . Thus, if λ decreases by a factor of 1 "say$, then fmust increase by a factor of 1 to compensate.

8ote also that the possible fre%uencies for an open pipe are: f o, (f o, 1f o, f o, f o .....

And for a pipe closed at one end: f o, 1f o, f o, 9f o .....

..... where f o is the fre%uency of the fundamental mode.

Standing Waves on Stringsnstringed instruments transverse standing waves are set up on the strings. The (

ends of the string are normally fixed in some way - so they have to be nodes. 7ome ofthe the possible modes of vibration are shown below:

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Thus, for a string fixed at both ends, the possible fre%uencies are: f o, (f o, 1f o, f o, f o etc .....

..... where f o is the fundamental fre%uency.

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Su er osition and Path Differencehen ( waves are out of phase it may be because they have have come from the

same source, but travelled via ( different routes - one longer than the other. Thus the

path difference between the ( routes will determine the phase difference.

The &' is a good example of this phenomenon. The tracks on a ;D consist of ase%uence of pits and bumps which, in cross-section, might look as follows:

(aser light is used to read a ;D, and the )umps are a quarter of a wavelength ofthe light $ *+% high. f the laser beam hits the edge of a bump as shown above, part ofthe beam "A$ will hit the top of the bump and part "'$ will hit the bottom. Thus there is apath difference of λ *( between waves following path A and those following path '. tfollows that these ( waves will be out of phase after reflection, and will superpose"interfere$ destructively. i.e. <ero light intensity will be reflected from the edge of abump.The reason for using laser light is that it is monochromatic "of a single wavelength$and coherent "random phase changes occur far less fre%uently than with )normal)

light$.

The Do ler "ffecthen you move towards or away from a source of waves "light, sound, etc.$ - or the

wave source moves relative to you - there is a change in the wavelength and thefre%uency of the waves. This phenomenon is called the 'oppler effect .

The diagram below illustrates the effect .....

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n summary ....

hen the source moves towards the observer - or the observer moves towards thesource .....

decreases and f increases hen the source moves away from the observer - or the observer moves away from

the source .....

increases and f decreases

(The diagram above explains the Doppler effect for a moving source. f the observer ismoving, the explanation is as follows! when the observer is moving towards the source,more wave"crests are encountered per second " so the fre#uency appears higher andwavelength appears shorter. $ice versa if the observer is moving away from thesource.)

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Polarisation4ight is part of the electro#agnetic s ectru# . That is to say, it is a transverse waveconsisting of sinusoidally varying electric and magnetic fields at right angles to eachother. t is )normally) un olarised . n other words, the oscillations "of the electric and

magnetic fields$ are in every plane at right angles to the direction of travel - as shownbelow:

n lane olarised light, the oscillations are confined to one plane - as shown below:

"7trictly speaking, the above wave 5ust represents the variation of the electric field - sothere should be another wave at right angles to it representing the variation of themagnetic field. &owever, it is usual to omit the magnetic wave for the sake of clarity.$

=npolarised light can be polarised by passing it through a olarising filter "such as)>olaroid)$. 4ight reflected off glass, water and many other surfaces is also partiallyplane polarised.

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Polari#etr$;ertain materials "sugar solution being an example$ rotate the plane of polarisation oflight. n the case of sugar solution, the higher the concentration the greater is the angleof rotation. This fact can be used to measure the strength of a sugar solution - using a

olari#eter , as shown below:

"see diagram overleaf$

The cylinder is filled with pure water to begin with, and the top polarising filter is rotateduntil the plane of light that it allows through is at right angles to that allowed through bythe bottom filter - i.e. until it blocks the polarised light produced by the bottom filter. Thecylinder is then filled to a certain depth with the sugar solution. 7ince the sugar solutionrotates the plane of polarisation, the light is no longer completely blocked, and it isnecessary to rotate the top filter in order the block the light again. The angle throughwhich the top filter has to be rotated is the angle through which the plane of polarisationhas been rotated. A graph of )angle of rotation) against )sugar concentration) can thusbe plotted, as shown below:

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"8.'. ?ou have to use the same depth of sugar solution each time because the angleof rotation also depends on the depth.$

The rotation angle of an unkown solution can be measured, and the sugarconcentration read off the graph.

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