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## Progressive waves

Lesson for 16-19

Students may have had a range of experiences of waves before starting their post-16 level course. For example, some may be confident in handling symbolic equations such as v = f l. Others may have only seen this as speed = frequency ´ wavelength, and may not be confident with rearranging it.

Your specification may mean that you have already covered the topic of oscillations (simple harmonic motion), in which case many of the ideas should already be familiar.

The episodes presented here give you the opportunity to assess what your students already know; you may be able to move quickly through this material if you feel that your class is confident in particular areas.

The demonstrations suggested give plenty of opportunity for student involvement.

## Episode 308: Preparation for the progressive waves topic

Teaching Guidance for 16-19

- Level Advanced

Assess what your students already know; you may be able to move quickly through this material if you feel that your class is confident in particular areas.

#### Main aims of this topic

Students will:

- Distinguish between transverse and longitudinal waves by describing the motion of particles
- Describe and represent waves graphically in terms of wavelength, frequency, displacement and amplitude
- Derive and use the wave equation
*v*=*f*×*λ* - Use an oscilloscope to look at waves and deduce their frequencies

#### Prior knowledge

Students should know that waves transfer energy without transferring matter. They should know about the waves of the electromagnetic spectrum.

As discussed above, students’ knowledge of waves will depend greatly on the course they have followed previously, and this topic gives an opportunity to bring all students to the same point of understanding.

We have not assumed that students have already studied oscillations.

#### Where this leads

Ensuring that students have a firm grasp of these fundamental ideas will make it easier for them to deal with the idea of superposition and its consequences (standing waves, interference and diffraction).

### Up next

### Generating waves

## Episode 309: Generating waves

Lesson for 16-19

- Activity time 100 minutes
- Level Advanced

This episode introduces the idea that vibrations can give rise to disturbances travelling outwards, i.e. to progressive waves. It shows some of the basic properties of waves.

Lesson Summary

- Demonstration and discussion: Transverse waves on spring (15 minutes)
- Student experiment: Transverse waves on springs (15 minutes)
- Demonstration: Longitudinal waves on spring (10 minutes)
- Student experiment: Longitudinal waves on springs (15 minutes)
- Demonstration: Waves along a row of students (10 minutes)
- Demonstration: Ripple tank (10 minutes)
- Demonstration (optional): Waves along linked trolleys (15 minutes)
- Summary: Discussion of common wave properties (10 minutes)

#### Discussion and demonstrations: Transverse waves on spring

Here you are trying to establish these points:

- a wave is caused by a vibrating source, and travels outwards from the source
- the particles of the medium through which a wave travels move about their equilibrium positions; they do not move along with the wave
- energy is transferred from place to place by a wave i.e. a change at A can cause at change at B with nothing materially passing between A and B.
- you can demonstrate these points by showing waves on a stretched spring – there are two types of spring used in activities of this sort:
- The
slinky

type, diameter about 9 cm, can also be used to show longitudinal waves. It is easily damaged if it is let go when stretched, not stored carefully or allowed to fall on the floor. Keep a good one out of students’ reach for your demonstrations, as the slightest kink spoils the motion - The long narrow, tightly coiled spring, about 3 cm diameter shows transverse waves well and is relatively robust. Alternatively, use a rope or rubber tubing.

Fix one end of the slinky using a retort stand and large weight, keep it on the floor or bench, and keep hold of the other end yourself. Demonstrate how a pulse travels along the spring when you move the end from side to side. (You will have to move your hand sharply to get a good pulse.) Repeated pulses make up a continuous wave.

Mark one coil with a white sticky label so that the motion can be seen. Remind your audience that the coils represent particles of a medium such as water or air as they move to and fro giving rise to the wave. A little ball of crumpled paper hit by a transverse pulse moves convincingly at right angles to the motion of the pulse. This is a transverse wave.

Use questioning to draw out the points listed above.

#### Student experiment: Transverse waves on springs

Pulses and continuous waves with a Slinky spring

#### Safety

- Insist that springs are kept on the floor
- Use eye protection
- Beware of the spring
whipping back

- if it someone lets it go when stretched
- Do not lean over the stretched spring

Undirected activity here can lead to chaos! So discuss a list of things to look for with the transverse pulses:

- Does the size of a pulse change as it travels along the spring? (Single pulses lose energy and so height but they don’t slow down.) What if there was no friction? (Pulses would stay the same height, i.e. no energy lost.)
- What happens when a pulse reflects at the fixed end? (It flicks over to the other side – there is a 180 ° phase change, but this doesn’t happen at a free end.)
- How does the speed change if you change the tension? (This is not easy to show with spring. In fact it always takes the same length of time for a pulse to travel the length of the spring, as the speed increases to compensate for the greater distance travelled.) It is easier to show with a stretched rope.
- What happens if pulses set off from opposite ends at the same time? (If students at both ends each make a pulse on opposite sides they will be seen to travel through one another.) Students may think that pulses bounce off one another and return. You may wish to demonstrate pulses passing through one another. Time lapse photography or frame by frame video might be used here or make sure one pulse has a larger amplitude than the other
- They should also draw the sudden change in phase upon reflection at a fixed end. This lets you check that phenomena have been observed correctly.

#### Demonstration: Longitudinal waves on spring.

Now use the slinky spring to show longitudinal pulses. Fix one end to a retort stand, and quickly push the free end back and forth, along the length of the spring. Watch the motion of the marked coil. It moves to and fro as the disturbance is passed along.

Identify compressions and rarefactions. Point out that *rarefaction* is not the same as *refraction*!

#### Student experiment: Longitudinal waves on springs.

Students can now find out whether longitudinal waves show the same behaviours as transverse waves; in this case, they send longitudinal waves along a stretched spring by pushing the end back and forth, along the length of the spring (rather than from side to side).

They should find that the crumpled paper ball is left untouched at the side when a pulse travels down. There is the same increase in velocity with tension.

#### Demonstration: Waves along a row of students

To emphasise the two types of waves, try simulating both types of wave using a row of students.

This can work well if students are cooperative. It is best to practise e.g. two steps forward, four back, then two forward brings them back to their equilibrium position. (You could use the Mexican wave for the transverse wave.)

Two points to bring out here:

- The meanings of displacement and amplitude
- Individuals in the wave are out-of-step with each other; this is useful for developing the idea of phase

This motion can then be compared with a standing wave later on.

Waves along a line of students

#### Demonstration: Ripple tank

In everyday life, waves

are something we see on water. In physics, the idea of waves has been greatly extended. Show some simple water wave effects using a ripple tank.

You may find it simplest to work as follows:

Place the ripple tank on an overhead projector, so that students see the waves projected on the screen. If this proves difficult, place the lamp under the tank and project onto the ceiling.

Do not use a motor to generate ripples; make short bursts of ripples by hand (by dipping into the water), or by dripping water into the tank.

Use a minute piece of paper to show that it just bobs up and down when a ripple goes past.

Submerge a flat glass plate, and show the change of wavelength when ripples arrive head on

to the shallower water.

Omit curved barriers at this stage.

#### Safety

Be very careful not to spill any water as you are using electrical equipment. If possible, have the low voltage power unit at least 1m away from the tank.

#### Practical Physics experiments:

Simple circular pulses in ripple tanks

Reflection of a straight pulse by a barrier

Reflection of a circular pulse by a barrier

You should be able to make these points:

- A disturbance travels outwards from a vibrating source
- Ripples pass through one another and carry on
- The ripples would not be circles if the wave speed was not the same in all directions. Extend to 3D and spheres with the inverse square law in mind
- Wave speed changes (and hence wavelength changes) when in shallower water above a glass plate. (Don’t show change in direction here, refraction proper comes later.)

c= *g* *h*

(Wave speed in shallow water so if h is smaller, (due to the smaller depth due to the shape), and then the waves go slower.)

Refraction of ripples entering shallow water

#### Demonstration: Waves along linked trolleys

Waves travel through matter; each particle interacts with its neighbours, and so the wave is passed along. With this demonstration, you can show that the speed depends on the strength of the links between the oscillating particles

.

A row of trolleys is linked by springs and the trolley at one end moved back and forth.

This takes a while to set up so will have to be done beforehand. You will need several trolleys and a lot of space if you want to do both types of wave at the same time. Double the number of springs on each link to change the interaction. The mass of each particle

is also easily doubled using trolley masses. Relate this to the speed of sound in solids which is much faster than in gases.

This demonstration may best be done on the floor to avoid having the trolleys fall off the bench.

#### Summary discussion: Discussion of common wave properties

Summarise the common properties seen with mechanical waves:

- Vibrating source
- Disturbance travels outwards from a vibrating source
- Transport of energy (but not matter)
- Reflection
- Waves pass through one another and carry on.
- Dependence of speed on conditions.

### Up next

### Wave graphs

## Episode 310: Wave graphs

Lesson for 16-19

- Activity time 40 minutes
- Level Advanced

Students often confuse displacement/distance graphs (from which wavelength can be deduced) with displacement/time graphs (e.g. on an oscilloscope, from which frequency can be deduced).

Lesson Summary

- Student Activity: Drawing longitudinal waves (10 minutes)
- Discussion: Extending the first activity (15 minutes)
- Discussion: Practice with phase (15 minutes)

#### Student activity: Drawing longitudinal waves

By plotting the displacements of individual particles through which a wave is passing, students can develop their ideas of the underlying process of a wave.

Provide graph paper. Students should use a particle spacing of 1 cm. The pattern is best seen by doing one particle at a time.

The completed diagrams can be used in discussion later. The same exercise can be done for transverse waves, but this might be overkill.

Episode 310-1: Graphs for a longitudinal wave (Word, 66 KB)

#### Discussion: Extending the first activity

Check the outcome of the graph plotting activity, particularly the final displacement/ position graph which gives the wavelength; discuss the link with pressure if this seems appropriate. If the particles either side of an undisturbed one are both displaced towards it, it will be in a compression; if they are displaced away from it, it will be a rarefaction.

Ask students to draw a displacement/time graph for particle 1 of the exercise above; it will be seen to be the same shape as the displacement / distance graph but starting at a different point.

Discuss the following quantities and their units, and how they can be deduced from the appropriate graphs:

Displacement *y*

Period *T*

Frequency *f*

Amplitude *A*

Point out that wave velocity *v* (or *c* ) cannot be deduced directly from a graph.

At this point, you might discuss the symbols (and units) for the various quantities related to waves. Warn your students that each textbook will be different. For example, amplitude may be represented by *a*, *A* , *x*_{0} or *y*_{0}. (Check your specification.)

#### Discussion: Practice with phase

Now for a mathematical interlude. The phase difference between two waves is the fraction of a cycle (in radians) by which one wave would have to be advanced or retarded to be in phase with the other.

It is best if students can work in both degrees and radians. If there is anyone who has not met radians before a swift detour will be necessary to introduce them.

Students should appreciate that out of phase

can mean by any amount. If they mean 180 ° they should say completely out of phase

, in opposite phase

or in antiphase

.

Check they understand that a phase difference of zero, *n* × 360 ° or *n* × 2 π is in phase, 180 ° or odd multiples of π is completely out of phase. Then try ⅓ or ¼ of a cycle out of phase; relate these to fractions of π radians, such as π 2.

A demonstration can help. Set a pendulum swinging by releasing it at *t* = 0; set another of the same length swinging exactly as the first goes through equilibrium position, travelling in the same direction. In this way you can illustrate 90 ° lag. (Note that, because the two pendula won’t have exactly the same period, they will move regularly in and out of phase; it is worth drawing attention to this.)

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### Up next

### Speed, frequency and wavelength

## Episode 311: Speed, frequency and wavelength

Lesson for 16-19

- Activity time 150 - 240 minutes
- Level Advanced

This episode considers how these three quantities are linked by the wave equation *v* = *f* × *λ* , measuring *f* using an oscilloscope, and measuring the velocity of sound in free air.

Lesson Summary

- Discussion and worked examples: Deducing and using the wave equation (30 minutes)
- Student questions: Practice with
*v*=*f*×*λ*(30 minutes) - Demonstration: An exploration of sound waves (20 minutes)
- Student experiment: Measuring frequency using a CRO (30 minutes)
- Demonstration: Measuring the speed of sound using double beam oscilloscope (10 minutes)
- Student experiments: Exploring waveforms (30–120 minutes)

#### Discussion and worked examples: Deducing and using the wave equation

Use a simple approach to deduce the wave equation

Justification/deduction of the wave equation *v* = *f* × *λ* . For example: the coaches of a train are going past; you count how many coaches go by in a second and you know the length of one – so you multiply the two together to get the train’s speed. Apply this to waves: count the number of waves passing each second ( the frequency), and multiply by the length of each ( the wavelength) to find the speed.

If your syllabus says *deduce* then you will have to present the algebra of

speed = distancetime

*v* = *l**T*

*v* = *f* × *λ*

Work through three examples:

A simple example, perhaps for sound in air, with values in Hz and m.

An example involving electromagnetic waves with frequency units such as MHz or GHz, to show how to deal with powers of ten; emphasise that *c* = 3 × 10^{8} m s^{-1} for all em waves in free space.

An example in which the equation must be rearranged, to find *f* or *λ* .

#### Student questions: Practice with *v* = *f* × *λ* .

You may wish to make a selection from these.

Episode 311-1: Speed, wavelength and frequency (Word, 47 KB)

Episode 311-2: Using the wave equation (Word, 42 KB)

#### Demonstration: An exploration of sound waves

An exploration of sound waves.

Turn the volume down each time you change the frequency range because of the differing sensitivity of the ear. Keep the overall volume low.

Emphasise that the oscilloscope trace represents displacement against time. You will have to hammer home that the peak-to-peak separation is not the wavelength.

It is advisable to perform this activity as a demonstration, as a room full of signal generators and loudspeakers in operation can be very noisy!

#### Safety

If you are going to use a stroboscope to show up the vibrations of a loudspeaker cone, you must check whether there is anyone in the class who may be affected by it. (You might wish to omit the use of the stroboscope.)

Episode 311-3: Exploring waveforms with an oscilloscope (Word, 49 KB)

#### Student experiment: Measuring frequency using a CRO

Students can measure the period, *T*, and hence calculate *f*. Each group will need a signal generator and a single beam oscilloscope.

Remind students to make sure that the oscilloscope is on the calibrated setting.

With a clear trace, note the time base setting and determine *T* (over several cycles if possible, an important technique). Then calculate *f* and compare with the setting on signal generator.

This is a good opportunity to check they are confident with the controls on the CRO and they can also explore the different waveforms from the signal generator.

(If you really have time for fun, issue low-voltage AC power supplies, switch off the timebase and input the 50 Hz signal to the X plates instead (often at the back) and give them the treat of Lissajous figures.)

#### Demonstration: Measuring the speed of sound using double beam oscilloscope

Connect two microphones to a double-beam oscilloscope. Set up a signal generator and loudspeaker to give sound waves of frequency 1 kHz. (Their wavelength is thus about 0.3 m.)

Place one microphone close to the loudspeaker, and observe its trace. Place the second microphone further from the loudspeaker, in the same straight line. Observe its trace. Move it back and forth, noting the changing phase difference between the two traces as you move through the sound waves.

Measure the wavelength (with a ruler) by finding how far the microphone is moved between adjacent positions where the signals are in phase. Calculate the speed of sound.

Note that, if you don’t have two microphones, you can link the signal generator and loudspeaker to one input. Then find two consecutive positions of the microphone which are in antiphase with the signal. Antiphase is easy to see when the traces are superimposed on the screen.

If you don’t have a double beam oscilloscope, wait until a lesson on standing waves and then use a single beam one.

#### Student experiments: Exploring waveforms

These experiments make use of the CD-ROM Multimedia Sound

. You could make a booklet from the relevant pages, so that students can investigate unsupervised in their own time.

Episode 311-4: Exploring waveforms with multimedia sound (Word, 37 KB)

Episode 311-5: Multimedia Sound: studying a pre-recorded sample (Word, 131 KB)

Episode 311-6: Multimedia Sound: recording your own sound sample (Word, 54 KB)

Episode 311-7: Superposition (Word, 63 KB)

Episode 311-8: Multimedia Sound: combining sounds (Word, 64 KB)