Waves Tutorial 2 - Transverse and Longitudinal Waves

We will assume that all waves are sinusoidal.  A sinusoidal wave-form (sine wave) is the simplest kind of wave.  Sound waves of sinusoidal form are rather boring to listen to.  The waves made by musical instruments are more interesting, but more complex.  However it can be shown that even the most complex wave-form can be broken down into sine waves.  Wave motion can be analysed in terms of circular motion and simple harmonic motion (SHM).  Since these topics are in the A2 unit 4, we will not attempt to do that in these tutorials.  (Originally the topic on wave properties was in the A2 syllabus, so waves could be discussed in terms of SHM.)

A transverse wave is one in which the displacement of the particles is at 90o to the direction of travel.  In a water wave, the particles move up and down while the wave travels horizontally.  All electromagnetic waves are transverse.

We can show the features of a transverse wave in the diagram below:

 

The equilibrium position in the diagram is the position that the material would take if the wave motion stopped.  We could also call it the rest position.  Both terms are used in SHM.

 

Strictly speaking, water waves are not transverse.  They are a type of wave called a roller and are only transverse when the amplitude is much less than the wavelength. When the amplitude is large, the wave is no longer transverse; it takes on the characteristic shown below. 

 

A cork on the surface does not go up and down; it takes on a circular motion.

As the amplitude gets bigger compared with the wavelength, the crest then breaks.  This often happens near the shore because the bottom part of the wave is travelling more slowly than the top.  Surfers use the rolling nature of water waves.

 

In longitudinal waves, the displacement is parallel to the direction of travel of the wave.  There are regions of high pressure, compression, and regions of low pressure, rarefaction.  In a sound wave the air molecules move forwards and backwards; where they are squashed together, a compression results, where they are forced further apart, there is a rarefaction.  Like all mechanical waves, a medium or material is required.  The speed of sound in air is 336 m/s, in water 1400 m/s, in steel it is 6000 m/s.  Other examples of longitudinal waves include some kinds of earthquake waves (the pressure or P-wave).  We can see the features of a longitudinal wave in the diagram below.

Question 1

Write down two similarities and two differences between transverse and longitudinal waves.  Give one example of a transverse wave and one example of a longitudinal wave.

Answer

 

 

Polarisation of Transverse Waves

Polarisation is a feature of transverse waves only.  Longitudinal waves are never polarised. We say that a wave is plane polarised if all the vibrations in the wave are in a single plane, which contains the direction of propagation of the wave.  Suppose we have a rope and make waves down it.  We could make waves in any direction we liked.  But if we made waves through a narrow vertical slit, we would find that the waves would only pass through if they were vertical. This would be a vertically polarised wave.

Light waves are easily polarised using polaroid filters.  Light waves, like all electromagnetic waves, consist of an electric field component perpendicular to a magnetic field component, which are always in phase.  We normally consider only the electric field component in polarisation, because the electrical effects are those that dominate.  The unpolarised waves are normally oriented in any direction.

If two polaroid filters are mounted such that they are parallel, the light will pass through both the first at which point it is vertically polarised, and then through the second.

If the two filters are crossed, so that the transmission planes are at 90o to each other, the vertically polarised light gets blocked, because it cannot pass the horizontal transmission plane. No light passes.

Crossed polaroids are found in liquid crystal displays on calculators and petrol pumps.

Question 2

Radio aerial rods must be in the correct plane, vertical or horizontal in order to work properly, otherwise the signal is weak.  Use the information above to explain why this is the case. 

Answer

 

 

Graphical Representation of Waves

We can show that both longitudinal and transverse waves can be represented by a displacement-distance graph.  If we take a snapshot of a wave at any instant, we see:

For a transverse wave we see that the graph looks very similar to the actual wave.

For a longitudinal wave the graph is not so easy to see.

Let us look at the air molecules in their undisturbed positions and compare them as a sound wave passes by.

If we plot displacement on the y-axis and distance on the x-axis, we get the same graph to what we had before.   The shape is a sine wave.

If we plot a displacement-time graph for a single particle we see:

This is true whether we have a longitudinal or transverse wave.  If we connect a microphone to a CRO, the CRO displays a displacement-time trace.  It is important that we do not confuse the displacement-distance with the displacement-time graph. The latter tells us nothing of the wavelength, only the period (hence the frequency) of the wave.

The simplest shape of graph we see is the sine wave.  The sine wave equation links wave motion with simple harmonic motion.   Sine waves in sound are very boring to listen to.  The quality of the sound gives us important clues as to the source.  Sounds made by different musical instruments have very different wave patterns, even if the notes sounded and the volumes are the same.

Question 3

The diagrams show the variation with time t of the displacement x of the two identical cones of loudspeakers A and B in air.

Calculate:

(a)    the frequency of the vibration of the speaker cones.

(b)   the phase difference between the speaker signals.

(c)    What kind of wave is being produced in the air by each speaker?

(d)   Which speaker produces the loudest sound?  Explain your answer.

(e)    The speed of sound in air is 340 m/s.  What is the wavelength of the sound waves?

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Summary

  • Waves transfer energy without transferring material.

  • Waves have a wavelength and amplitude.

  • The movement of particles is at right angles to the direction of movement in a transverse wave.

  • The movement of particles is parallel to the direction of movement in a longitudinal wave.

  • Waves can be represented graphically.

  • Graphs of two waves can show phase difference.

  • Transverse waves can be polarised, longitudinal waves cannot.