82A - Wave Model

Waves are everywhere and whether we recognise it or not, we encounter waves on a daily basis. Sound waves, visible light waves, radio waves, microwaves, water waves, sine waves, cosine waves, stadium waves, earthquake waves, waves on a string, and slinky waves and are just a few of the examples of our daily encounters with waves. In addition to waves, there are a variety of phenomena in our physical world that resemble waves so closely that we can describe such phenomenon as being wavelike.

Waves are periodic disturbances or variations that transfer energy progressively from point to point in a medium. The disturbance can be the movement of particles in the medium or a variation in pressure, electric or magnetic intensity, electric potential or even temperature.

The wave model is used to describe the characteristics of wave phenomena and includes the use of terms such as speed, frequency, period, amplitude and wavelength.

82A - Describing Waves

Here you will learn to:

Describe how mechnical waves transfer energy through a medium,

Distinguish between transverse and longitudinal waves,

Describe waves in terms of speed, wavelength, frequency, period, and amplitude, and

Use the wave equation to calculate speed, wavelength or frequency.

What Are Waves?

The wave model is used in physics to describe the transfer of energy from one place to another carried by the regular and ordered oscillation of particles in a medium (for mechanical waves) or electrical and magnetic fields in a vacuum (for electromagnetic waves).


YouTube - Introduction to waves

Mechanical waves are those waves that require a medium for the wave to travel through. Sound waves require air as a medium and seismic waves travel through the Earth's crust as a medium. Electromagnetic waves do not need a medium to travel through and propogate by the mutual oscillation of electric and magnetic fields. Light is an electromagnetic wave as it is able to travel through the vacuum (no air) of space to get to us on the Earth. Other electromagnetic waves include radio waves, microwaves and ultraviolet waves to name a few. While electromagnetic waves behave most of the time like mechanical waves, we will limit our discussion of the wave model in this section to mechanical waves that require a medium, even though the same principles can be applied to electromagnetic waves.

A familiar and concrete example of wave motion is the Mexican wave spectators create at sporting events by standing up and sitting down at appropriate intervals. Each person stands up just as that person's neighbor stands up, transmitting a form of energy all the way around the stadium. Mechanical waves like this are transmitted through a medium - the poeple in the stadium. The energy and the wave are both created by the successive action of people standing up and sitting down in a regular way. If there were no people in the stadium, no wave could exist and no energy could be transmitted. The people at the stadium are analagous to water at the beach for water waves or the air molecules transmitting sound for water waves. The medium itself is not propagated which is why the poeple don't move with the wave. For the wave to work, each person in the stadium only needs to stand up and sit back down. The wave travels around the stadium, but the people do not. Think of waves as a means of transmitting energy over a distance. One object can transmit energy to another object without either object, or anything in between them, being permanently displaced. For instance, if a friend shouts to you across a room, the sound of your friend’s voice is carried as a wave transmitted by the air particles. However, no air particle has to travel the distance between your friend and your ear for you to hear the shout. The air is a medium, and it serves to propagate sound energy without itself having to move. Waves are so widespread and important because they transmit energy through matter without permanently displacing the matter through which they move.


A transverse wave (a) and longitudinal wave (b) in a slinky spring.

There are two types of mecahnical waves known as transverse waves and longitudinal (compressional) waves.

Transverse Waves

A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil up and down (as shown in (a) in the diagram on the right). Energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced upwards and downwards. In this case, the particles of the medium move perpendicular to the direction that the pulse moves. This type of wave is a transverse wave. Transverse waves are always characterized by particle motion being perpendicular to wave motion.



In a transverse wave (above) the particle displacement is perpendicular to the
direction of wave propagation. The animation below shows a one-dimensional
transverse plane wave propagating from left to right. The particles do not move
along with the wave; they simply oscillate up and down about their individual
equilibrium positions as the wave passes by. Pick a single particle and watch its
motion.


In a longitudinal wave (above) the particle displacement is parallel to the direction
of wave propagation. The animation above shows a one-dimensional longitudinal
plane wave propagating down a tube. The particles do not move down the tube with
the wave; they simply oscillate back and forth about their individual equilibrium
positions. Pick a single particle and watch its motion. The wave is seen as the motion
of the compressed region (ie, it is a pressure wave), which moves from left to right.

Longitudinal(Compressional) Waves

A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction that the wave moves. Suppose that a slinky is stretched out in a horizontal direction across the classroom and that a pulse is introduced into the slinky on the left end by vibrating the first coil left and right as in (b) above. Energy will begin to be transported through the slinky from left to right. As the energy is transported from left to right, the individual coils of the medium will be displaced leftwards and rightwards. In this case, the particles of the medium move parallel to the direction that the pulse moves. This type of wave is a longitudinal wave. Longitudinal waves are always characterized by particle motion being parallel to wave motion.

Longitudinal waves are characterised by regular regions of high and low pressure along the wave. High pressure regions are called compressions and low pressure regions are called rarefactions. Check the wave on the right and notice the compressions and rarefactions moving to the right in the direction of the energy transfer.

Longitudinal waves can sometimes be represented in what looks like a transverse wave. There are two ways to think about this.

  1. If we plotted the distance along a longitudinal wave on the x-axis and the pressure at that point along the wave on the y-axis, we would see a wave pattern resembling a transverse wave.

  2. If we plotted the distance along a longitudinal wave on the x-axis and the displacement from the rest position on the y-axis, we would also see a wave pattern resembling a transverse wave.



Graphing a longintudinal wave so that it looks transverse. Note
the choice of y-axis.

It is important to check the axes when interpreting graphs of waves. A transverse wave will always have the displacement of a particle from the rest position on the y-axis. Longitudinal waves can have the same thing on the y-axis or pressure along the wave.

Describing Waves

Some common terms are used to describe the characteristics of waves.

Amplitude (A / m) - The maximum dispalcement of a particle from it's rest or equilibrium position.

Frequency (f / s-1 or Hz) - the number of oscillations of the particles in the wave each second. Somes defined as the number of cycles or revolutions per second.



The characteristics of a typical transverse wave moving from left to right across the page.

Period (T / s) - The time taken for one complete oscillation of the particles in the wave. One complete wavelength is produced during each oscillation.

Wavelength (λ / m) - The distance travelled by a wave during one complete oscillation of its particles. The wavelength can be determined by measuring the distance between two similar points on a wave, e.g. crest to crest or trough to trough.

Wavefronts - Suppose a stone is thrown into a pond and the waves spread out as shown below right. The top of the wave is known as the crest, whereas the bottom of the wave is known as the trough. There are several aspects to this wave that can be studied that are important to all waves:



  1. A diagram showing a wave front produced by
    dropping a stone is a pond.

    The movement of the wave pattern - the wave fronts highlight the parts of the wave that are moving together.

  2. The direction of energy transfer - the rays highlight the direction of energy transfer.

  3. The oscillations of the medium. It should be noted that the rays are at right angles to the wave fronts in the above diagrams and this is always the case.

Anton Paar eLearning - What are waves
Video showing the characteristics of waves

 

 

 

 

 

 

 

 

 

 

Wave Equation

As was discussed above, a wave is produced when a vibrating source periodically disturbs the first particle of a medium. This creates a wave pattern that begins to travel along the medium from particle to particle. The frequency at which each individual particle vibrates is equal to the frequency at which the source vibrates. Similarly, the period of vibration of each individual particle in the medium is equal to the period of vibration of the source. In one period, the source is able to displace the first particle upwards from rest, back to rest, downwards from rest, and finally back to rest. This complete back-and-forth movement constitutes one complete wave cycle.

In a time of one period, a wave will move a distance of one wavelength. Given that speed is equal to distance/time, it can be said that the speed of a wave is also equal to the the wavelength/period. Since the period is the reciprocal of the frequency, the expression 1/f can be substituted into the above equation for period. Rearranging the equation yields a new equation of the form speed = wavelength x frequency (v = fλ). The equation shown below is known as the wave equation. It states the mathematical relationship between the speed (v) of a wave and its wavelength (λ) and frequency (f).

We can deduce three important relationships from the wave equation:

  • Speed is directly proportional to wavelength,
  • Speed is directly proportional to frequency, and
  • Wavelength is inversely proportional to frequency.

Stuff to Do

TutorialTutorial 82A - Wave Model
TutorialTutorial 82A - Wave Model - Answers
Activity 82A - Graphing Longitudial Waves
MovieBozeman Science - Waves
MovieKahn Academy - Introduction to Waves
MovieKahn Academy - Amplitude, Period, Frequency and Wavelength