This episode introduces the basic idea of superposition of waves, explaining what happens when two or more waves meet.

Lesson Summary

• Discussion: Recapping wave ideas (10 minutes)
• Demonstration: Waves on a rope (15 minutes)
• Discussion: Ripples on a pool (10 minutes)
• Student questions: Adding waves graphically (30 minutes)

Discussion: Recapping wave ideas

Show a representation of a wave and rehearse basic knowledge about waves:

Frequency f is determined by the source of the waves (the transmitter)

Wave speed c is determined by the medium in or on which the wave propagates.

So wavelength follows from c = f ×  λ .

At boundaries (reflection, refraction etc), f must stay the same.

Demonstration: Waves on a rope etc

This is a useful attention grabber. Pass pulses through each other on a stretched rope, rubber tube or stick wave machine.

Set off a large amplitude pulse; just before the first pulse is reflected, chase it with a smaller amplitude pulse – observe that the pulses briefly combine then pass through each other, and carry on as before.

(For later comment, observe and note the phase change on reflection; that is, the pulse turns upside down when it reflects.)

Draw attention to this damage-less collision of waves passing through each other that is a hallmark of all wave behaviour. Colliding particles rebound and may suffer damage. (The other hallmark of wave motion is that the speed of a wave is constant and determined by the medium supporting the wave motion. Unlike particles, friction doesn’t reduce the speed; it reduces the energy (i.e. the amplitude) of the wave.)

When two waves arrive at the same point and at the same time, the resultant displacement is given by the algebraic sum of the two individual displacements. (Algebraic sum means that you have to take account of positive and negative values.)

Discussion: Ripples on a pool

Superposition is an everyday phenomenon. Show some images of ripples on a pool of water as they pass through each other.

Episode 320-1: Overlapping ripples (Word, 322 KB)

Student questions: Adding waves graphically

Drawing exercises reinforce the idea that, to find the displacement when two waves meet, you simply calculate the algebraic sum of the two individual displacements.

Episode 320-2: Phase difference and superposition (Word, 94 KB)

Episode 320-3: Superposition of waves: a drawing exercise (Word, 60 KB)

When two or more waves meet, we may observe interference effects. It is likely that your students will have already met the basic ideas of constructive and destructive interference.

Lesson Summary

• Demonstrations: Simple interference phenomena (20 minutes)
• Demonstration: Two sound sources (15 minutes)
• Demonstration: Young’s two-slit experiment (15 minutes)
• Discussion: Deriving and using the formula (20 minutes)
• Student experiments: Double slit analogues (30 minutes)
• Student questions: Using the Young’s slits formula (40 minutes)
• Demonstrations: For students to explain (20 minutes)

The following simple demonstrations could be used to introduce this section; don’t feel that you have to give detailed explanations at this stage.

• Laser speckle pattern. Shine a laser onto a screen. Move your head side to side and observe the dark and light speckles, due to the different path lengths to the eye from different positions on the spot of laser light (If the beam is too small to show the speckles, try expanding it by passing it through a low-power lens, either converging or diverging.)
• Observe the colours in soap bubbles or oil films. Light is partly reflected by the upper surface of the film, partly by the lower surface. Depending on the thickness of the film, these two light rays will superpose constructively or destructively, depending on the wavelength. Thus two paths giving constructive superposition at one wavelength will not give constructive superposition for other wavelengths – hence only the colour with the correct wavelength is seen.
• Tuning forks. Hold the vibrating fork with its prongs vertical and close to the ear. Twist the fingers so the fork slowly rotates about a vertical axis. The loudness of the sound will rise and fall, four times per complete rotation. (Each prong acts as a source of sound waves; twisting the fork alters the distance between each prong and the eardrum.)

Emphasise that, in each case, there are two or more sources of light or sound reaching the eye or ear. You are going to look at an experiment designed to have two sets of light waves meeting in a very controlled way, i.e. Young’s two-slit experiment.

Safety

In any work with lasers, it is worth pointing out to the class the label in the laser. It should say Class 2: do not stare down the beam. With such a laser, a momentary reflection of the beam into someone’s eye will not cause an injury.

Demonstration: Two sound sources

Because the wavelength of light is very small, it is worth setting up an equivalent experiment with sound waves. Use two loudspeakers connected to a single signal generator. At this stage, it is not necessary to make detailed measurements.

Episode 321-1: Hearing superposition (Word, 84 KB)

Demonstration: Young’s two-slit experiment

Young’s two-slit experiment is perhaps one of the most famous experimental arrangements in physics. It was inspired by Young’s discovery of interference that he related in May 1801: Given a pond with a canal connected to it. At two places in the pond waves are excited. In the canal two waves superpose forming a resultant wave. The amplitude of the resultant wave is determined by the phase difference with which the two waves arrive at the canal.

Shine laser light through a double slit on to a screen. You should see a series of evenly-spaced bright spots (fringes or maxima). Ask students to relate this to the sound experiment. (The bright fringes are the equivalent of the loud points in the sound field.)

Point to the central bright fringe. Emphasise that two light rays reach this point, one from each slit. They have travelled the same distance, so there is no path difference between them. They started off in step (in phase) with each other, and now they arrive at the screen in phase with each other. Hence their displacements add up to give a brighter ray.

The next bright fringes (on either side of the central one) represent points where one ray has travelled λ further than the other, so they are back in phase. Why is there a dark fringe in between? (One ray has travelled λ/2 further than the other, so they are out of phase and interfere destructively.)

If we could measure these distances approximately, we could determine λ .

Show the effects of:

• Using two slits with a smaller separation (the fringes are further apart).
• Moving the screen closer to the slits (the fringes are closer together).

It is clear that we might use this experiment to determine the wavelength of light, but how?

A modern version of the Young’s Two Slit experiment was voted the most beautiful experiment in physics in a Physics World readers’ poll in 2002. It still forms the basis of ongoing research into the fundamental quantum nature of matter.

If you have already covered the photon model for light, you may want to refer back to this. As early as 1909, it was established that fringes were found even if the source was so faint that only one photon at a time was in the apparatus. Fringes can also be seen using de Broglie (or matter) waves. The most massive particles used to generate fringes to date (March 2005) are fluorinated buckyballs C₆₀F₄₈ (i.e. 1632 mass units).

Discussion: Deriving and using the formula

Now derive or quote the formula (depending upon your specification). A good way to start is to ask your students to identify the important variables (they are all lengths), and to give their approximate sizes:

λ , wavelength of the light (~500 nm)

d, separation of the two slits (~1 mm)

s, separation of the fringes (bright to bright or dark to dark) (~1 mm)

L, distance between slits and screen (~1 m)

How can we make a balanced equation from four quantities that are so different in magnitude? The simplest solution is that the product of the biggest and smallest is equal to the product of the two in-between quantities. Hence:

λ L = sd

or

λ d = sL

Episode 321-2: Calculating wavelength in two-slit interference (Word, 79 KB)

Episode 321-3: Two-slit interference (Word, 51 KB)

Student experiments: Double slit analogues

Set up a circus of Young’s two-slit arrangements (depending upon the available equipment to hand) using light, microwaves, 3GHz radio waves, ultra-sound (less noisy than audible sound!) and a ripple tank, and get students to determine the wavelength of the waves being used in each case.

You may prefer to set some up as demonstrations.

Episode 321-4: Interference patterns in a ripple tank (Word, 48 KB)

Episode 321-5: Measuring the wavelength of laser light (Word, 42 KB)

Student questions: sing the Young’s slits formula

The first set of questions covers the principles of Young’s experiment.

The second set is questions for practice in using the equation.

Episode 321-6: Questions on the two-slit experiment (Word, 28 KB)

Episode 321-7: Two-source interference: some calculations (Word, 23 KB)

Demonstration: For students to explain

Here are two fun demonstrations to round off this episode. Demonstrate them, and ask your students to provide explanations.

1. A nice demo using audible sound is to fix two loudspeakers at each end of a longish piece of wood. Mount the wood on a suitable pivot (large nail) mid-way between the two speakers. Drive both in parallel from signal generator. Slowly scan the class. As the interference fringes sweep across the audience, they hear the regular change in volume.
2. Make a sound trombone. Mount a small loudspeaker in the wide end of a small plastic funnel. Tubing from the other end divides into two tubes: one takes a direct route, the other a route whose length can be varied by a U-shaped glass tube sliding trombone section. The two routes combine and are fed into another funnel that acts as an earpiece.
The loudness of the sound depends upon the position of the trombone slider. It is obvious that there are two paths by which the sounds reach the ear. There may be a path difference between them. If the path lengths differ by an exact number of wavelengths, constructive interference increases the volume; integral half wavelength path differences mute the sound due to destructive interference. Do not confuse this with beats; here, only one frequency is involved, whereas beating is an effect due to two close frequencies (see below).

The diffraction grating was named by Fraunhofer in 1821, but was in use before 1800. There is a good case for describing it is the most important invention in the sciences.

Lesson Summary

• Demonstration: Looking through gratings (10 minutes)
• Discussion: Deriving the grating formula (20 minutes)
• Student experiment: Measuring wavelength (30 minutes)
• Student questions: Using the grating formula (30 minutes)
• Student experiment: A CD as a grating (home experiment) (30 minutes)
• Discussion: The meaning of coherence (10 minutes)

Demonstration: Looking through gratings

Pass around some diffraction gratings or hand-held spectroscopes. Invite students to look through them at various light sources: a torch bulb, LEDs of different colours, a sodium or mercury lamp. They should see a continuous spectrum (red to violet) for a white light source, but multiple images of different colours for coloured sources.

Shine a laser through a grating (as in the Young’s slits experiment above). Explain that light rays are emerging from each slit. What condition must be met for a maximum on the screen? (All rays must be in phase with each other.)

Observe the effect of moving the screen further away (the maxima are further apart). Try a grating with a different slit spacing. (Closer slits give maxima that are further apart.)

Shine a bright white light through the grating and observe the resulting spectra.

Describe the central maximum as zero order. The next maximum on either side is then first order ( n = 1), and so on. You will probably observe two or three on each side, far fewer than with a double slit. (It’s a much more demanding condition to have many rays in phase than just two.)

It is hard to overstate the importance of the diffraction grating to the progress of the sciences. Some would claim it is the most important invention, used in investigations from the structure of matter including DNA (being used as its own grating) to the structure of the Universe (from Doppler shifted spectral lines), and the determination of the composition of everything from stars to chemical compounds, by analysis of their spectra.

Discussion: Deriving the formula

You may need to derive (or simply present) the grating formulafirst derived by Young 1801:

d sin( θ ) = n λ

where d is the grating separation (the distance between slits). NB gratings are often specified as so many lines per metre (often on old gratings as per inch, so beware if doing quantitative work!). Separations as small as 100 nm possible, so 10 million lines per metre (equivalent to about 40 000 lines per inch).

Note that this formula gives the directions for the maximum intensities ( cf the single slit formula which gives directions for the minimum intensity).

Point out that smaller wavelengths give smaller angular separations. (You should have observed that violet light is diffracted less than red light.)

Episode 322-1: A transmission grating (Word, 40 KB)

Student experiment: Measuring wavelength

Students can determine the wavelength of laser light using a diffraction grating. They could discuss the relative precision of this method, compared with the two-slit arrangement.

Episode 322-2: Measuring the wavelength of laser light (Word, 44 KB)

Student questions: Using the formula

Students can perform calculations relating to diffraction gratings.

Episode 322-3: Grating calculations (Word, 37 KB)

They can interpret some images made using gratings.

Episode 322-4: Using diffraction gratings (Word, 157 KB)

Student experiment: A CD as a grating (home experiment)

Use a CD as a reflection diffraction grating. This experiment can be done at home.

Episode 322-5: Using a CD as a reflection grating (Word, 27 KB)

Discussion: The meaning of coherence

Now that your students have had practical experience of observing maxima and minima, further discussion of the necessary conditions should make sense. For superposition effects to be observable, the conditions must persist for a time long enough for them to be observed and extend far enough in space for a reasonably-sized pattern. It is easier for this to be the case if the waves have a well-defined frequency or wavelength (they must be monochromatic).

Ordinary light sources emit light in overlapping bursts, are not usually monochromatic (so the number of wavelengths in a given path can vary) and no source is a true point (so a range of path lengths is inevitably involved).

Laser light is particularly useful for showing interference effects because it is intense, highly monochromatic, and emitted in long wave trains and so maintains a constant phase relationship over large distances. We say that the waves arriving along two or more paths are said to be coherent .

Episode 322-6: Coherence (Word, 30 KB)

Note the spelling – double ff. The first recorded observation of diffraction was by Grimaldi in 1665. The shadows cast by light sources were not quite the same size as the anticipated geometrical shadows. Furthermore, very close to the edges, the shadows were bordered by alternate bright and dark fringes or bands. What good observation – given the sources of light available in 1665!

Lesson Summary

• Demonstration: Observing the effect (10 minutes)
• Demonstration: Ripple tank and/or laser (20 minutes)
• Discussion: Deriving the formula (20 minutes)
• Discussion: Resolution (15 minutes)
• Student experiment: Resolution of spectral lines (20 minutes)

Demonstration: Observing the effect

A distant light bulb viewed through the gap between the prongs of a tuning fork shows clear dark and bright bands. The effective gap can be altered by turning the tuning fork. Simply making a narrow gap between an index finger and thumb can produce the same effect.

Demonstration: Ripple tank and/or laser

For a larger scale demo from which to draw qualitative conclusions use either or both of:

• a ripple tank
• a single slit with laser light

(The slit width can be changed by simply turning it about a vertical axis perpendicular to the beam.)

Episode 323-1: Ripple tank diffraction (Word, 33 KB)

Safety

Provided the laser is class 2 (less than 1 mW), the warning Do not stare down the beam is sufficient. Avoid specular reflections.

Summarise findings:

• A pattern of intensity maxima and minima is formed
• The pattern is symmetrical
• The central maximum is wider than the other maxima (unlike with Young’s two slit fringes)
• The maxima and minima are evenly spaced
• The intensity falls off with angle (distance from the central fringe)
• The spreading depends upon the wavelength and the gap width – l less than gap width gives less spreading out; l equal to gap width gives biggest effect; l greater than gap width results in the wave being almost blocked
• Energy in the incident beam has been redistributed

Discussion: Deriving the formula

For the minima in the diffraction pattern:

sin( θ ) = mld

where m is an integer and d is the slit width. To derive this, we have to imagine that each point across the width of the slit is sending rays to every point on the screen.

Episode 323-2: Diffraction at a single aperture (Word, 75 KB)

Discussion: Resolution

At this point, you could discuss the idea of resolution . Resolution refers to the ability to distinguish two objects that are close together. The light from an object is diffracted by the aperture of the viewing instrument. Two neighbouring objects can be resolved provided that the peak from the central maximum of one is no closer than the first minimum of the other (and vice versa). This is called the Rayleigh Criterion.

For light of wavelength λ passing through an aperture of width a, the central maximum and first minimum are separated by an angle λb. If d is the actual separation of the two objects and L their distance from the aperture then

dL =  λ b

Take care! This has an identical mathematical form to the Young two slits formula – so beware possible confusion.

Explore the dependence on λ and on b for two given objects (i.e. d and L given). Resolution improves if λ get smaller and or b gets smaller (i.e. shorter wavelengths and smaller apertures).

Student experiments: Resolution of lines

How far away from a student observer can the lines on differently coloured J cloths be distinguished?

Alternatively, rule pairs of differently coloured lines 1mm apart on some white card. Decide how far away from the card can you stand and just distinguish the pairs as two separate lines.

Assume the eye pupil is 2 mm diameter. You will need to decide on wavelength values for the different colours.

Check that the data is consistent with the Rayleigh Criterion.

Episode 323-3: Distinguishing lamps as separate (Word, 38 KB)

A slightly more sophisticated experiment could involve the use of a pin-hole camera (this also makes a good suggestion for a course work investigation). For circular apertures, the formula for the first diffraction minima is modified slightly to be

1.22 λ b , where b is now the diameter of the hole (rather than the width of the slit).

Most musical instruments depend upon standing waves, as does the operation of a laser. In a sense, a diffraction pattern is a standing wave pattern.

Lesson Summary

• Demonstration: Setting up waves on a rope (15 minutes)
• Discussion: How superposition results in standing waves (20 minutes)
• Demonstration: Melde’s experiment (20 minutes)
• Discussion: Stringed instruments (10 minutes)
• Demonstration: Standing sound waves (20 minutes)
• Student experiments: Measuring λ and c (30 minutes)
• Demonstration: Measuring c using a microwave oven (15 minutes)
• Student questions: On Melde’s experiment, and on waves in pipes (30 minutes)
• Demonstration: Standing waves in 2 and 3 dimensions (20 minutes)

Demonstration: Setting up waves on a rope

Standing waves do not travel from point to point. They are formed from the superposition of two identical waves travelling in opposite directions. This is most easily achieved by reflecting a travelling wave back upon itself.

Reflection from a denser medium gives a phase change of 180 ° ( π radian) for the reflected wave. Show this using a rope or stick wave machine held between two people– pull down and let go to make pulse. See that there is a phase change if reflecting from a fixed end, but no effect if reflecting off a less dense medium (e.g. a free end).

The incoming reflected wave superposes with the outgoing wave. The result (if you vibrate the one end of the rope at the correct frequency) is a standing wave. The mid-point of the rope vibrates up and down with a large amplitude – this is an antinode. Other points vibrate with smaller amplitude.

Double the frequency of vibration and you get two antinodes, with a node in between. The amplitude at a node is zero.

Discussion: How superposition results in standing waves

Discuss how two travelling waves superpose to give a standing wave. A node (where there is NO DisplacEment is a special point, where a positive displacement from one wave is always cancelled by an equal, negative displacement from the other wave.

Episode 324-1: Standing waves (Word, 51 KB)

Demonstration: Melde’s experiment

A stretched string or rubber cord can be made to vibrate using a vibrator (of the type illustrated) connected to a signal generator. This is known as Melde’s experiment.

Show that different numbers of loops can be formed; identify the pattern of frequencies (e.g. one loop at 40 Hz; two loops at 80 Hz; three at 120 Hz, etc.)

Point out that each point on the string oscillates with simple harmonic motion – if the frequency is high you just see the envelope. Freeze the wave using a stroboscope. To do this, place an electronic strobe between the students and the string, so that it illuminates the string. Adjust the frequency of the strobe until the string appears stationary. ( Safety precaution : warn about the flashing light in case you have any students present who may suffer from photo-induced epilepsy.)

The distance between successive nodes (or antinodes) is λ 2. Deduce the wavelength.

Notice that only certain values of wavelength are possible. If the string has length L , the fundamental has λ 2 = L , first harmonic λ L, second harmonic 3 λ 2 = L and so on. The allowed wavelengths are quantised .

(This experiment can be extended to look at how the pattern changes with length and tension of the string.)

Episode 324-2: Standing waves on a rubber cord (Word, 96 KB)

Discussion: Stringed instruments

Relate what you have observed to the way in which stringed instruments work. (Wind instruments are covered later.)

Episode 324-3: Standing waves on a guitar (Word, 29 KB)

Episode 324-4: What factors affect the note produced by a string? (Word, 51 KB)

Demonstration: Standing sound waves

You can observe the same effect with sound waves. To generate two identical waves travelling in opposite directions, reflect one wave off a hard board.

Episode 324-5: Standing waves in sound (Word, 57 KB)

Student experiments: Measuring λ and c

Standing waves make it easy to determine wavelengths (twice the distance between adjacent nodes), and hence wave speed c (since c = f ×  λ ). These experiments allow students to investigate wavelength and speed for sound waves, microwaves and radio waves.

Have a short plenary session for the different groups to report to the whole class.

Episode 324-6: Kundt's experiment (Word, 39 KB)

Episode 324-7: Standing waves with microwaves (Word, 30 KB)

Episode 324-8: A stationary 1 GHz wave pattern (Word, 32 KB)

Demonstration: Measuring c using a microwave oven

Use a microwave oven to measure the speed of light. This makes a memorable demo if you have a microwave oven to hand in the lab. Without using the turntable, place marsh mallows or slices of processed cheese in the oven. Observe that the heating occurs in definite places – the cooking starts first at the anti-nodes, where the standing waves have the maximum amplitude. The purpose of the turntable it to even out this localised heating. Measure the distance between nodes, deduce λ , and multiply by f to find c .

NB It is a widely repeated misconception that microwave heating is a resonance effect, i.e. that the frequency of the microwaves is chosen to be one of the vibration frequencies of the water molecule. It is not. Water molecules resonate at rather higher frequencies than the 2.5 GHz of microwave ovens (22 GHz is one such frequency for free water molecules). The frequency used in ovens is a compromise between too low a value, when no microwaves would be absorbed by the food at all, and a frequency too close to the resonant frequency, when the microwaves would all be absorbed in the outside layer, instead of penetrating a few cm. Microwaves are attenuated exponentially by foodstuffs with a half-depth of about 12 mm.

Student questions: On Melde’s experiment, and on waves in pipes

Some questions based on Melde’s vibrating string experiment.

Episode 324-9: Stationary waves in a string (Word, 106 KB)

Questions on standing waves in air in a pipe; note that you will have to explain that there is a node at the closed end of a pipe, and an antinode at an open end. (The air molecules cannot vibrate at a closed end.)

Episode 324-10: Standing waves in pipes (Word, 46 KB)

Episode 324-11: Standing waves in pipes questions (Word, 37 KB)

Demonstration: Standing waves in 2 and 3 dimensions

You can demonstrate standing waves in two and three dimensions. Mount a wire loop or a metal plate (a Chladni plate) on top of a vibrator.

A wire loop shows a sequence of nodes and antinodes around its length, and is an analogue for electron standing waves in an atom.

Episode 324-12: More complicated standing waves (Word, 42 KB)

Chladni plate: Fine sand sprinkled on the plate gathers at the nodes.

Jelly (make up a large cubical shape) on a vibrating plate can look fantastic.

Rubber sheet stretched over a loud speaker: the low frequency standing waves are clearly visible.