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Wave Trains and the Wave Equation

Transmission of Waves

The frequency of a wave is defined as the number of crests or troughs, or complete cycles that pass a given point in the medium per unit of time. Once a wave is produced, its frequency never changes, even if its speed or wavelength does. This behaviour is a characteristic of all waves.

Example: Consider a continuous set of waves with l = 2 cm such that 2.5 complete waves pass by a fixed point every second. How fast is the wave travelling?

Wavelength

The wave traveled 5 cm in 1 second
and the frequency is 2.5 Hz.
Therefore, 5 cm/s = speed.

NOTE: 5 = 2.5 * 2 cm

Therefore,
v = f l



The Wave Equation is an expression which gives the speed of the wave. The Wave Equation is stated below:

Wave Equation

Frequency and wave length are inversely related to each other as the two digrams below show:

Low frequency == Long wavelength

Low Pitch

High frequency == Short wavelength

High Pitch

During a period the wave travels 1 l. Therefore, an alternate expression for the speed of a wave is as follows:

Velocity of a Wavelength

Standing Waves

A standing wave interference pattern results when two sets of waves with the same amplitude and wavelength travel in opposite directions in the same medium and travel through each other.

A node or nodal point (N) is a point that remains at rest throughout the interference. The region between the nodes are areas where double crests and double troughs occur. These areas are called loops or antinodes.

1

1) The point X represents a point of total cancellation of a wave. The X also represents a node or nodal point.



2

2) The point X represents a momentary cancellation of a wave.




3

3) The dots represent nodes (stationary points) and the D are antinodes (loops).



To find the wavelength of a loop, you can see that 2 loops = l. Therefore, 1 loop = 1/2 l.

Fixed-End Reflection

If the reflection is by a fixed-end reflection it is called fixed-end reflection. If the fix-end reflection the reflection pulse is inverted. An example of this is when a crest is reflected by a trough. Such wave patterns are called standing waves. Such waves are found in all string instruments such as the piano and violin.

A free-end reflection has no inversion. When this occurs there is no phase change. Waves of this nature are found in open end air columns as found in brass or horn instruments such as the tromebone and trumpet.

When two or more waves act simultaneously on the same particles of a medium, it is referred to as an interference of waves.

A constructive interference occurs when two or more waves interfere to produce a resultant displacement greater than the displacement that would be caused by either wave itself.

An example of this occuring is as follows

You can see that the so called superwave has an amplitude of 70 cm, the sum of both wave's amplitudes.

A destructive interference occurs when the resultant displacement is smaller than the displacement that would be caused by either wave itself.

The Principle of Superposition states that the resultant displacement of a given particle that is equal to the sum of the displacements that would have been produced by each wave acting independently.

Note to be Made from this Section

  1. Frequency and wavelength are inversely related. Therefore, if the frequency is doubled, what does the wavelength become. If you use the equation the wave's velocity is a constant and its value is not important.
  2. If a wave train has a speed of 50 m/s and a period of 2.0 x 10-2s, what is the wavelength and frequency of this wave?
  3. Know what an node and an antinode is and how they relate to crests and troughs from the previous section.
  4. Constructive and destructive interference is based on the Principle of Superposition. Know the terms.
    An exercise to show this graphically is to add a sine wave to a cosine wave.

    Lab Exercise

    A graph is to be drawn, using a calculator to obtain the values for a sine wave from 0o to 540o. This will genetate a wave and a half. Select a colour for this function.
    On the same graph sheet sketch a cosine function from 0o to 540o. This will also produce a one and a half wavelenghts. Select another colour when drawing this function.
    The two waves are to be added and the new wave (sine + cosine) is sketcked on the same graph sheet again using a different colour.
    Mark areas on the diagram where maximum constructive interference and desructive interfence occurs.
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