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Multiple contributions - Physics narrative
Physics Narrative for 14-16
A Physics Narrative presents a storyline, showing a coherent path through a topic. The storyline developed here provides a series of coherent and rigorous explanations, while also providing insights into the teaching and learning challenges. It is aimed at teachers but at a level that could be used with students.
It is constructed from various kinds of nuggets: an introduction to the topic; sequenced expositions (comprehensive descriptions and explanations of an idea within this topic); and, sometimes optional extensions (those providing more information, and those taking you more deeply into the subject).
The ideas outlined within this subtopic include:
- Contributions of beams determined by trip times.
- Trip times set by geometry: difference in trip times is key.
- Resultant of contributions determines amplitude.
- Amplitude determines intensity.
- Revisit phenomena and redescribe to show unification: reflection, refraction, propagation/transmission.
- Start with beams, end with paths.
- 2 beams and 2 paths as paradigm.
- Extend to new phenomena: polarisation, beats, interference and diffraction.
Physics Narrative for 14-16
Clapotis: water waves pile up to make large peaks and troughs
What happens here affects what happens over there: that is the essence of radiating. An influence travels from source to absorber without stuff making the trip. The amplitude and frequency of the source partially determine the effects at the absorber. The nature of the absorber and the details of the trip (setting the fraction of the emitted photons striking the absorber) complete the description.
Most absorbers or detectors receive radiations from many sources and usually we think nothing of this: two appropriate sources illuminating one detector normally produces no special effects, except that the detector records a higher intensity. But there are very particular conditions where special effects can be noticed, even for very high frequency radiation, like visible light.
Rather spectacular effects are experienced when two sets of water waves meet, perhaps around a seaside headland.
As a large positive displacement from one set of waves meets a similar displacement from another, so a temporary
mountain of water is created: the detector (the water where the waves meet) has a very large displacement, often both positive and negative in a short sequence. Such
clapotis phenomena can be hugely impressive, although you are more likely to notice more moderate examples where reflected waves from a sea wall or cliff meet incoming waves.
These messy, transient, mountains and pits of water may well have analogous phenomena in light, but the changes are so much more rapid that no detectors can show them up (remember that the frequency of light is very high).
Trip time, amplitude and frequency all affect contribution
Both frequency and amplitude are defined by the source of the radiation. If you move away from the source then the trip time also determines how that source contributes to the vibration at a later point.
That is why it may be helpful to talk about the contributions of vibrations from different sources rather than waves or radiations meeting. Radiations and waves are both more general terms and do not spell out the details. Here we aim for the details, so that you can see how waves and radiating works.
In this episode you'll meet the phenomena that are particular to radiations, as these depend on frequency. They depend on something vibrating, and so were instrumental in the increasing acceptance of the wave theory of light in the 19th century as a series of experiments showed that light did indeed exhibit these phenomena. In this episode you'll see how these new phenomena can be accounted for by linking the frequency of a vibration with the trip time between the source and the detector. This trip time, the journey time from the source to the detector, is determined by the distance between the two and the speed at which the information about the changes travels.
Do like me – but later is still the idea that we suggest you build on.
In and out of step: a relationship between two vibrations
Two vibrations of the same frequency can be in or out of step as they are emitted from their respective sources. As they travel, so this relationship between them may alter as a consequence of the journey times (the trip times) between the source and wherever we place the detector.
If the sources are not of the same frequency then they will come into and go out of step without any movement of the detector. In fact this coming into and going out of step will happen wherever the detector is located.
Beating: an effect produced by superposition
Two vibrations arriving together at a point superpose. Each one contributes – it conveys information to whatever is vibrating: particles of water, air, steel; electric and magnetic fields. This information is a rather simple set of instructions about a set of displacements resulting in the same pattern of vibration that originated in the source. That is where the phrase
do like me, but later came from.
If the contributions from each of the pair of sources at the point of co-arrival are in step, then the resulting displacement at that point will be larger than either of the contributing displacements as the contributions just add. This is constructive superposition. The two vibrations are superposed – stacked one on top of the other – so that the resultant amplitude is the sum of the two contributing amplitudes.
If the two contributions are out of step then the resulting displacement is smaller than either. This is destructive superposition. The two vibrations are superposed – stacked one on top of the other – so that the resultant amplitude is the difference between the two contributing amplitudes.
The two contributions can also be partially in or out of step, so leading to a range of different possible displacements from just these two contributing displacements, one from each vibrating source.
Being in step is often called being
in phase; being out of step,
out of phase. A fixed phase relationship between the two results in
beats, as shown here.
Fourier synthesis – adding together different contributions
Adding together different contributing vibrations can produce many different patterns of resultant vibrations. Making a cunning selection of simple contributions allows you to construct any resultant vibration you choose. This is Fourier synthesis – named after Joseph Fourier (1768–1830), who first did the mathematics.
But what is a
simple contribution? In this context it means simple to specify – that is, with no sudden changes in the time evolution, and so in the time trace, and also, partly as a consequence, with very few parameters needed to specify the vibration. That'll be just the usual minimum of two parameters to specify a vibration: the amplitude and the frequency. Feed these parameters into a sine or cosine function and you'll get a simple vibration. Vary the parameters to vary the vibration.
With just amplitude and frequency and the sine or cosine function, we can specify a whole range of simple vibrations that vary in systematic ways. Combine these varied contributions to build up a complex vibration. Unlike clapotis, maintain a regularity of input: then you'll get a very regular output. So you can build up time traces that are square or triangular, or indeed any chosen shape, given enough determination to find the correct contributions, specified only by their amplitude and frequency. This is both remarkable and powerful.
Fourier analysis – disassembling a vibration
You can also start with a complex vibration and split – or analyse – that vibration into its component contributions. This is Fourier analysis, and is the basis of voice-prints, optical spectra, earthquake analysis, and many other helpful descriptions of vibrations.
Each complex vibration can be defined exactly by listing the amplitudes and frequencies of is component contributions. These arrays of numbers can then be compared with other arrays to identify the vibration (whether it be starlight, an earthquake, or a performance of a piece of Mozart). You can even feed the numbers back into a Fourier synthesiser to recreate the pattern of the original vibration. Synthesisers do just this: whether for music or for voice.
Arranging for beams to meet
Physics Narrative for 14-16
Physical arrangements used to make beams meet
Radiations travel in beams: the power in a single beam, often related to the number of photons a second (emitted or absorbed), is a truly useful quantity. Quite often it is enough to describe the possible effects of the energy being shifted. However, when two beams meet, you observe significant new phenomena. Because, in the simpler cases, these depend on arranging for two separate beams to meet, it's a good idea to have these situations clear before attempting an analysis, concentrating on how we arrange for beams to meet.
After that it's worth extending the idea to situations where you cannot readily describe the situation in terms of two beams, but where two distinct vibrations do contribute: that is, where we can find two or more significant paths from the source to the detector.
Here are the four most common different ways of arranging for two beams to meet.
On being permanently in or out of step
You have seen that two beams meeting can cause instructions from two sources to arrive at a single point – two lots of
do like me, later. These usually cause transient effects because the two sets of vibrations come into and out of step rather often.
Might careful arrangements of the sources slow these transients right down, so that you can see the effects?
One way to slow the changing patterns right down is to ensure that the contributions from the different waves don't go into and out of step all the time. That ensures that the mountains and pits don't wander about all over the place. As we arrange things so that the two contributions remain in step for longer and longer, so the resultants become less transient and more stationary. If the two contributions are kept in step then they are said to be in phase. You can imagine that this is very hard to do unless they are also of the same frequency.
To get two vibrations arriving at the same detector in step usually means fixed trip times for both radiations delivering the vibrations and selecting the sources rather carefully as well: a couple of car headlamp bulbs won't do – and this is rather fortunate because the lighting patterns might be interesting where they overlap. You'll see why shortly.
Sources that are of the same frequency and in phase are called
coherent. Beams produced by them are also described as coherent. As a reasonably intense beam consists of a random hail of many photons, it is hard to arrange two independent sources to be coherent, and a more common arrangement is to take a single source and then split it in two, often by reflecting a part of the beam.
Stationary patterns: moving vibrations
Interference is a stationary pattern of varying intensities as a result of two or more colliding beams. These beams must be coherent, and there are a few simple ways of taking one vibrating source and producing two beams. One vibration can be used to drive two sources, as when a single electrical signal drives two loudspeakers. A single beam of photons can be split by a prism, or by reflection. This is rather harder to do for sound, because it does not travel in such well defined beams. Look out for the account of diffraction in this episode to see why that is so.
These intensities can vary from the sum of the intensities of the contributing beams to their difference. So two beams of light illuminating one detector can apparently produce a lower intensity than either one by itself, if the detector is placed so that the contributions are always exactly out of step, and so the resultant amplitude is, at all times, the difference in amplitudes.
A particularly interesting case arises when the amplitudes of the contributing beams are equal. Then the two beams produce, in some places, total darkness. This can only be explained if light does some vibrating, and it was crucial, historically, in persuading people that light could not be fully explained by a simple particle model. It's hard to see how two particles passing by a single point act as if there are no particles passing, and then, once past that point, act as if there are two independent particles again.
All radiations exhibit this phenomenon of interference: from very high-frequency photons, such as gamma radiation, to low-frequency photons, such as radio waves; through water waves, to very low-frequency sound waves. So it's a real fingerprint of a travelling vibration. If you suspect you'd like to use a wave description of a phenomenon, try to show interference from the appropriate source.
Vibrations account for interference
Physics Narrative for 14-16
Coherence is difficult to arrange
Now let's work out how the trip times and frequencies combine to predict the different fixed patterns of displacements. These depend on the geometry of the situation.
In each case here the vibrations at the sources are in step. This is a rather special situation, and it's often not easy to arrange. Remember that light, just as an example, has a frequency in the terahertz range. So keeping vibrations in step at this kind of frequency is a pretty stringent condition – not easy to realise in practice. Where there is a single source that provides both beams (for both cases where there are reflections, or for the pair of slits), it's a bit easier to see how this might be arranged.
The essential idea is rather simple. Each path is of a fixed distance, from the source to the detector. This introduces a fixed delay between the displacement at the source and the displacement at the detector. This delay is just the trip time, calculated, as ever, from the distance and speed of propagation of the waves. For two sources and one detector there will be a pair of trip times.
The difference between the the two trip times determines whether the two contributions, one from each source, are in step or out of step.
How interference comes about
Interference occurs when coherent beams both arrive at a point. The vibrations in the beams may be in or out of step.
If the vibrations of the two are in step then you'll see constructive superposition. The contribution from one beam adds to the contribution from the other beam to give a large resultant amplitude. If both beams have the same amplitude – so they are the same intensity or brightness – then the resultant amplitude will be twice the amplitude of either beam because the amplitudes simply add.
If the vibrations of the two are completely out of step, then the two contributing amplitudes will add to give a resultant amplitude of zero. This will lead to a point where there is no illumination.
As you scan across the possibilities, varying the trip times of the two beams systematically, so the two contributions move from being in step, through being partially in step, to being completely out of step. The two contributions combine in exactly the same way: they add. But the resultants are different. These resultants predict different brightnesses, or different loudnesses, or, more generally, just different intensities.
Here there are two distinct, and apparently physical, beams, both of which are modelled by paths. Yet, if you remember the work on paths from episode 01, paths can also be used to explain why certain rays are drawn. As you delve more and more deeply into the nature of radiating, you'll get more and more entangled with paths.
Photons and interference
Physics Narrative for 14-16
A difficulty is that we only know where photons are when they no longer exist
There is a lurking difficulty here, that cuts to the core of what physicists know about radiating.
Both paths contribute, and these contributions add. The resultant sets the power at the detector. Yet the power at the detector can also be figured out in terms of the number of photons arriving each second. At very low intensities there may only be a single photon emitted over the trip times. So how do the two contributions come about from a single photon? This is the central mystery of quantum physics: it turns out that our picture of the world needs re-imagining. Things at this very small scale are just not like things at the human scale, if the experimental results are to be trusted, and so you ought to be very careful about what is
Paths helped us out before, when reasoning about the rules for rays. So let's think again about this interference situation, but this time with paths. The advantage for making sense of the core difficulty is that the paths are not physical: paths are just a way of scanning across possibilities.
For reflection and refraction you're looking for paths where the trip time is lowest.
For two beams that does not make much sense, as we need to consider the connections between the contributions of the paths. So here you'll look at differences in trip times – as the next simplest thing, and building on the ideas of being in and out of step.
Differences in trip times for paths is the key to figuring out contributions
This interactive focuses on difference in trip times for paths. The very first pane shows that where the difference in trip time is small, the number of photons arriving per second is large – that is a high intensity. You can connect this to the
real wave behaviour: where the difference in trip time was small, there the contributions from each source superposed constructively. Many photons are detected where the contributions add to give a large resultant amplitude. Where the resultant amplitude is close to zero, few photons are detected.
Yet there is nothing to suggest that we need to have only one path per slit. So the second pane of the interactive shows that three paths per slit predicts exactly the same result. If you are not convinced, you can of course do everything here with a piece of A3 paper and a sharp pencil: it's only geometry, trip time, and a bit of
do like me, but later to figure out contributions (which have both frequency and amplitude). So now you're working with a real stripped-down picture of what a wave is – a real mathematical model, applying to any physical wave.
The third pane shows something a little more interesting. It predicts that, if you have only one slit, the beam will spread. That's where we're going next, to look at the phenomenon of diffraction.
Ripples from pebbles thrown into ponds and torch beams provide different patterns
Throw a pebble into a still small pond. What happens in one place pretty soon affects what happens in all. The up and down movement triggered by the pebble radiates uniformly across the pond until it influences all parts of the pond. Looked at from on top or underneath (a bird's eye/newt's eye view), there is a radial symmetry to the spread of the influence: it does not matter in which direction you set out from the point of impact of the pebble to track the spread of the influence, the results are identical. The ripple propagates radially: what happens
here now happens
over there, but later.
Do like me – but later is the fundamental characteristic of a radiation.
A candle placed in the middle of a room does nearly the same in three dimensions. Why nearly? Well, it's not quite uniform in all three dimensions because the flame is not symmetrical in all three planes, and then there are the wax and wick that make up the candle. But we hope you can imagine a simpler candle, with none of these annoying physical limitations in place. Again note the move from the physical to the conceptual, while remaining rooted in the physical.
natural action of such a source, an ideal point source, is to radiate in all directions. But notice how many simplifications you've had to make to get to this
natural (so unconstrained) state of affairs. This very simple universe (one point source radiating, no obstructions) does have significant value: you can add complexity and then predict what happens on the basis of what you know about the simple situation.
Simple arguments based on symmetry suggest that there is an equal chance of a photon being detected in all directions, so you'd guess that they are emitted equally in all directions. In such a simple universe, there is no reason to prefer any one direction over another. An equal supply of photons to any detector, placed anywhere on the surface of a sphere, centred on the source, predicts equal brightness in all directions.
Beams of photons are altogether different. A detector placed anywhere except in the beam has a truly tiny chance of detecting a photon, whereas a detector placed in the beam has a large and constant chance of detecting a photon, however far from the source it is placed. Such a beam has a constant brightness, at all distances from the source, but it is as much of a leap of the imagination as the point source. Once again, it is an idealisation, but yet again, a very useful one. To see why it is an idealisation, we'll compare the two kinds of propagation: radial and beam.
Point sources and sources radiating beams are quite different
Rays were used in the SPT: Light topic, to predict beams. They were also used to predict shadows from point sources. The patterns made by these two sets of rays are very different: one set spreads radially and one set are parallel. Yet a long way from the point source it is very hard to tell whether the rays are still spreading: they appear nearly parallel. This is simply a consequence of the geometry.
So, the further you are from a point source, the more nearly the rays model and predict a beam. Here is a summer-time example: Sun-beams selected by holes in a tree canopy and made visible by mist or smoke do appear as beams, not apparently diverging. This is very clear to the eye, even though the Sun is a pretty good approximation to an ideal point source.
Back to the pond, looking for the same process. The vibrations can be both channelled, and a long way from the source, so they are constrained to be a beam. But what happens when you remove the constraint? Now the vibrations do not have to propagate down a fixed route and so can propagate
naturally again. The vibrations spread out, reverting to behaving as if there is a point source. The more you constrain them, the more they behave like a point source when that constraint is removed. The narrower the channel, the more the spreading at the end of the channel.
Diffraction: from beam to points source
The fundamental story of vibrations is that influence travels – and that what happens here now (the source) happens over there (the detector) later. The fundamental command from source to detector is
Do like me – but later. To that we now add that the natural movement for these vibrations, where they are unconstrained, respects the symmetry of space – there is no special direction.
One can re-describe the same process in terms of photons: there is no preferred direction of emission – photons are emitted as a random hail over both time and angle.
Where the passage of the vibrations is constrained then this natural spreading of the vibrations cannot take place, but as soon as that constraint is removed, the spreading is reinstated. The more like a point source the end of the constraint appears, the greater the spreading: the tighter the prior constraint; the greater the reaction.
That is the phenomenon. The explanation follows.
Vibrations account for diffraction
Physics Narrative for 14-16
Multiple paths all related: waypoints across one slit
The explanation for diffraction already presented might seem a little vague: we suggested that passing through a slit allows the waves to spread because that is their natural motion, if unconstrained. That's what a point source is. A classical development of this argument, replete with complex diagrams, is Huyghen's construction. But we don't suggest that this is simple, or for students of this age.
Instead, we suggest that you think about this in a thoroughly modern way, going back to the paths, as these allow us to combine possibilities in ways that predict where photons will be detected. The physical situation of radiations passing through a slit can be modelled by considering the contributions of several paths starting on the slit. Of course, we could start the paths earlier, but if it's a beam arriving at the slit, the contributions will be in step as they leave the slit. It's only the differences in trip time that count, not the actual trip time.
Again, here you're only using geometry: the differences in trip time arise because of the different path lengths and the constant speed of propagation. Where the differences in trip time are small, there the contributions will combine so as to predict the arrival of large numbers of photons per second. No surprises: this is the direction in which the beam ploughs straight on. No, the surprises come for off-centre directions, where some photons are detected.
Making more connections
Physics Narrative for 14-16
Four interwoven strands in an approach to understanding waves
So far there have been four parallel, but interwoven strands in the explanation of waves. Here we'll show how they are interconnected, and point out how they continue to grow together as the explanation of radiations and radiating deepens beyond study at this level.
Firstly there has been the modelling of phenomena as vibrations that travel. So a source vibrates with a particular frequency and amplitude. After a delay set by the trip time (distance from sourcepropagation speed), the detector mimics that vibration. This passage of information without matter we call waves.
Secondly the idea that beams can be predicted by drawing rays has been revisited. These rays follow seemingly arbitrary rules; but it turns out that these rules can in turn be predicted if you select, from a whole range of possible paths from source to detector, the path with the minimum trip time.
Thirdly the importance of the minimum trip time itself has begun to be explored. Minima of time occur where the differences in trip time between adjacent paths don't vary by much. These are paths for which the contributions add up. As the contributions rely on the source vibration this begins to draw together the first two strands.
Finally, the world at the smallest scale is a quantum world. All the radiating – starting with emission at the source, and absorption at the detector – is grainy. So the power emitted or absorbed is measured in the number of photons per second. The resultant amplitudes from the contributions predict where, on average, these photons will be detected.
Multiple paths for reflection, refraction and propagation and least time paths
For reflection, refraction and propagation, the least time path predicts the ray. Here we consider a triplet of least paths, and the difference in time between these paths. If the difference in trip time is small, then the contributions from the three paths will superpose constructively, and there will be a large chance of detecting a photon there. This triplet will always be centred on the actual path of least time. So the least time path is the path that, if blocked, would make the greatest difference to the number of photons arriving at the detector, as it is the path that makes the greatest contribution to the resultant amplitude. It's all about considering possibilities, and these are constrained by geometry.
How to engineer mirrors and lenses with photons and paths
Here are found more complex systems, probably most easily thought of as being optical, but there is nothing in the model that demands this. Any travelling vibration, or waves, is predicted to behave in the same way.
The power detected – the activity – is predicted by the sum of the contributions. The sum of the contributions – the resultant amplitude – depends on the trip times.
So, finally, it all comes together:
- trip times
Waves and radiating are not best thought of as a list of independent rules about different phenomena. Then reason that they're connected into a single topic is a theoretical reason, and here you have reached one level of explanation as to why this unity is so compelling.
The phenomenon: vibrations in a plane
Radio waves can be both emitted and detected with long stick-like electrically conducting metal aerials. It is not too much of a stretch for the imagination to think of the electrically charged particles moving up and down the emitting aerials, and so being the source for the vibrations that spread to all the surrounding detectors. So you can picture the transverse nature of the vibrations: the electrons move up and down and the influence travels radially away from the aerial.
These vibrations are at right angles to the direction of travel between source and detector, and they are even more constrained than that – the electric component is moving up and down only.
That is why you can significantly alter the effectiveness of the detection by varying the angle at which the small domestic aerial is pointing. The vibrations may pass it by because they are not in the correct plane. Rotate the aerial far enough and it will be at right angles to the vibrations from the source, and you'll get no signal at all. The vibrations are polarised.
Exploiting polarisation by taking care with orientation
Remembering that light is one of a family of electromagnetic waves, you might set out to look for similar effects in other members of the family, even where the source is not obviously vibrating at right angles to the direction in which the vibrations are radiated. Such a quest might find you fishing by a lakeside, at sunset, trying to see into the water. You'll find it hard due to the glare from the surface. But as every complete angler knows, slipping on a pair of polarised glasses helps greatly. These block nearly all the vibrations in one direction and pass nearly all the vibrations at right angles to this direction. The beam reflected off the water is mostly vibrating from side to side, parallel with the surface, while still transverse to the direction of travel.
The beam reflected off the fish, enabling us to see them, is not constrained to one plane as it has not bounced off the flattish surface of the water, which acts like the source aerial in the first example. In that example the source was polarised; in this example the beam becomes polarised by reflection.
A third way of polarising a beam is by filtering. That is what polarising sunglasses do – they remove all the vibrations except those in one plane. So the beam reflected from the fish in the previous example is less intense than it would be without the polarising filter. However, the effect on the polarised beam is much greater, as the polarising filter is arranged to be vertical and so at right angles to the plane of vibrations of the incoming beam. Much of the glare that the sunglasses are designed to eliminate comes about from reflections off nearly horizontal surfaces, such as water and snow.
Polarisation is most easily understood as if there were two beams of vibrations. But in unpolarised light there are are vibrations in all possible planes at right angles to the direction of motion. So there is a bit of a challenge. But you can get a vibration in any direction you choose by combining fractions of the up and down vibration with the left and right vibration. In this way you can compose a vibration of any amplitude, in any direction. So it's a useful model to decompose these multiple vibrations to two planes at right angles to each other, either by adding the appropriate components from each of these planes (synthesis) or by resolving the desired vibration into these two components (analysis).
Polarisation has uses beyond increasing the density of radio broadcasting. Signals from adjacent stations can be at right angles and so not affect each other, even if they are broadcasting at the same frequency, or seeing more clearly over snow or water, or maybe even seeing through shop windows, but you'd expect to have to turn your head sideways to the ground for this to work, wouldn't you?
The sunlight is polarised, and bees use this to navigate. Photographers exploit this, using polarising filters to accentuate cloudscapes. Some molecules rotate the plane of polarisation, so chemists exploit this to measure concentrations; more molecules are assumed to lead to more polarising.
LCD screens such as those on mobile phones, calculators and GPS systems use polarised light to switch on and off different parts of the screen – so making it hard to read these devices with polarising sunglasses on. Try using these gizmos with polarising
sunglasses on to check that this is so.
Polarised light arises as one plane of vibration is selected – by reflection, by filtering, or simply because the source is designed that way. Remember the radio transmitter?
One can even obtain circularly polarised light by having the two components out of step.
A further polarising filter can be used as a detector to show the plane of polarisation. The filter will permit the largest transmission intensity when aligned with the incoming vibrations. The minimum is permitted when it is at right angles to those vibrations.
Superposition: amplitude and frequency
Physics Narrative for 14-16
The story so far… power, mimicry, paths, contributions
In these first three episodes of this topic, the idea of
waves as an explanation for some phenomena has been taken apart and put together again. There have been several strands to the argument, and these are all related, although you'd have to delve into all the expansion topics along the way to see the full range of the connections.
So here, presented in four panes, is the essence of waves:
The essence of a wave model
Power: energy is shifted by accumulated remote working, and, on the small scale, at a greater rate by more quanta per second.
Mimicry: information travelling from one vibrator to another. Paths: these are possibilities, some of which predict rays, and thus beams.
Contributions: where several beams or paths contribute, superposition occurs.