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Physics 20: Oscillatory Motion & Mechanical Waves

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Wave Interference

Purpose:

To explore constructive and destructive interference between two waves.

Background:

Unlike physical objects, two waves can occupy the same space at the same time, which gives rise to the phenomena of interference.  If the two waves are in phase, that is to say their crests match, then complete constructive interference results.  It the waves are 1/2 a wavelength out of phase, then complete destructive interference results; any other combination results in partial constructive and destructive interference.

How to graph interference:

One complete wavelength of a wave is one cycle.  This can be considered as one oscillation or rotation of the particles of the medium.  One complete cycle corresponds to 360°, the number of degrees in a one complete rotation.  Portions of a complete wave can be represented as portions of one complete revolution.

The diagram below shows one complete wave.  Portions of the wave are expressed in two ways:

1. as a fraction of the wavelength (i.e., 1/4, 1/2, 3/4, and 1)
2. as a number of degrees (i.e., 90, 180, 270, 360)

Computing the sine of all angles from 1° to 360° plots a wave. To plot a second wave that is out of phase with the first, we simply start plotting the second wave at 1/4 of a wavelength (90°), 1/2 a wavelength (180°), or 3/4 of a wavelength (270°) out of phase with the first.

Instructions:

1. Open the accompanying Excel spreadsheet.

Choose a value (between 0° and 360°) for cell B4 (90° is already entered for you).  This cell represents the angle that the two waves are out of phase with each other.  The spreadsheet’s graphing function is set to plot the graph of Wave 1, Wave 2 and the resultant wave that occurs because of the interference of the two waves.  Experiment with 90°, 180°, and 270°, as well as other amounts you feel would be instructive.

Analysis:

1.

Plot the graph of the interference that occurs when two waves are out of phase by ¼ of a wavelength (90°).  Print this graph.

a) Through what angles is the resultant wave’s amplitude in between waves 1 and 2?  What can you say about the type of interference that occurs in this region?

b) What can you say about the type of interference when the resultant wave’s amplitude is a greater than either wave?  At what angles does this occur?

c) Find the position on the graph where the angle is 145°.  Add the amplitude of both waves together and compare it to the amplitude of the resultant wave.  How do these amplitudes compare?
2. Plot the graph of the interference that occurs when two waves are out of phase by 1/2 of a wavelength (180°).  Print this graph.

a) Can you suggest a reason why the amplitude of the resultant wave is zero?

b) What type of interference is this?

c) At any position on the graph, add the amplitudes of waves 1 and 2.  What value do they equal?
3. Using the Excel spreadsheet or a graphing calculator, plot the graph of the interference that occurs when two waves are out of phase by 3/4 of a wavelength (270°).  Print this graph.

a) Through what angles is the resultant wave’s amplitude in between waves 1 and 2?  What can you say about the type of interference that occurs in this region?

b) What can you say about the type of interference when the resultant wave has a greater amplitude than either wave? At what angles does this occur?

c) How is this graph similar/different than the interference that occurs when the waves are 90° out of phase?
4.

Use the up and down control arrow buttons in cell D5 to change the phase angle between the waves.  If you hold down one of the arrow buttons you can see how the resultant wave changes with respect to the change in the phase angle.

Experiment with the two waves and see what angles will result in total constructive interference.  When you have found an angle where total constructive interference takes place, plot this graph and label it.
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