describing waves / representing wave motion / wave equation
A "mechanical" wave is a disturbance in a medium that moves through the medium thus transferring energy from one place to another. It is important to note that while waves transfer energy from one place to another, they do this without transferring any of the particles in the medium.
Some common examples of waves include:
- waves on the surface of a liquid,
- sound waves,
- waves on stretched wires/strings/springs,
- electro-magnetic radiation, for example light, x-rays, radio waves etc behave, in many situations like waves, however, they differ from other waves in that they can travel through empty space.
There are two types of waves that are known as transverse waves and longitudinal (or sometimes compressional) waves.
The medium through which the wave is moving is the spring. The direction of motion of the wave (direction of propagation) is at 90° to the direction of the disturbance (or displacement) of the medium. If this were a water wave, the molecules of water would be moving up and down at right angles to the direction of motion of the wave. Electromagnetic waves are also a type of transverse wave.
A continuous wave involves a succession of individual oscillations. A wave pulse involves just one oscillation.
c - compression
e - expansion (or rarefaction)
In these waves, the direction of the disturbance of the medium is parallel to the direction of propagation. Sound waves are longitudinal waves. The molecules of air oscillate back and forth parallel to the direction of movement of the wave.
The velocity of propagation of a wave depends on the properties of the medium through which it moves. Waves travel faster through more dense media.
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Representing Wave Motion
Displacement of the Medium vs. Position Graphs for a Typical Wave
The maximum displacement of the medium from the equilibrium position is called the amplitude of the wave.
The wavelength, l, is the distance moved by the disturbance during one time period. It can be measured from the graph above by determining the distance between two successive troughs or crests.
This graph could have many different shapes; the graph shown here is similar to a graph of sine of angle against angle. For this reason, waves of this shape are called sine waves.
For a transverse sine wave, this graph can be considered to be a picture of the wave at a given instant in time. However, it should be remembered that the graph could represent a longitudinal wave because the y-axis shows the displacement of particles from their rest position.
Displacement of a Point in the Medium vs. Time Graphs for a Typical Wave
T, is the time for one "cycle" of the wave. This is called the time period.
The frequency, f, of the wave is the number of cycles per second. This is determined by the source of the wave.
The mathematical relationship between frequency and the period is
The units of frequency are per second (s-1) or Hertz (Hz) and the units for the period are seconds (s).
Suppose a stone is thrown into a pond and the waves spread out as shown on the left. The top of the wave is known as the crest, whereas the bottom of the wave is known as the trough.
Note that there are several aspects to this wave that can be studied. These aspects are important to all waves.
1. 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.
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The Wave Equation
There is a generalised wave equation for the relationship between the frequency, wavelength and velocity of a wave.
Where v is the velocity in metres per second (ms-1)
f is the frequency in Hertz (Hz)
l is the wavelength in metres (m)
This equation can be simply derived as follows:
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download activity 8.2.1 - oscillations
download worksheet 8.2.1 - describing waves
download answers to worksheet 8.2.1 - describing waves