# Interference of Waves: Fourier Series

### Purpose: To investigate how various wave patterns can be represented as a sum of harmonics.

Simulation Used: Fourier from the PhET at the University of Colorado.
• Theory and Settings.

The idea behind Fourier Series is that any curve can be represented as a sum of sin functions with a definite set of amplitudes. Mathematically, the function describing the curve is written as:
.         (Eq.1)

The Fourier series can be applied to interference of physical waves. If we have several waves with different frequencies traveling through space, the resultant displacement of a given particle (x=0) will be given by:
(Eq.2)
Here, the physical time, t, in Eq.2 corresponds to the independent variable x in Eq.1.

Though, mathematically there are infinite number of terms in the Eq.1 and Eq.2 above, in this simulation, the terms are limited to eleven. In real life experiments, the number of terms is determined by the desired accuracy of the final result.

The first term in the Eq.2 corresponds to the wave with the lowest frequency, the fundamental frequency. The next has a frequency twice the lowest, thus corresponds to the second harmonic. The third has a frequency three times the lowest - the third harmonic, etc.

• Activity 1: Become familiar with harmonics and their series (Fourier Series)
• Open the simulation Fourier: Making Waves.
• Create eleven different one-wave patterns, that is only one wave has a non-zero amplitude. For each pattern, record of how many wavelengths it consists, or in other words, how many times the pattern repeats within the window. For example, the first pattern is formed by the first wave with amplitude 1.00, all others have zero amplitude. We see one wavelength in the window and so we record "1" in the table below. When only the second wave has a non-zero amplitude, the pattern repeats itself twice on the window, thus we write down "2". Fill in the table for all patterns.
 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Multiple wavelengths
• Create a pattern from the second harmonic with amplitude A2=0.7 and the fourth harmonic with amplitude A4=0.3.
Does the pattern repeat itself? If yes, how many times?
What is the wavelength?:   m
• From the top menu on the right, select "Function: Triangle", and record the amplitude of the eleven harmonics creating the triangular pattern
• Repeat for "Square"*** and "Wave Packet". Record your data in the following table:
 Function A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Triangle Square Wave Packet
***Note. The "Square" wave is not exactly square - the top portion is not straight. It can be made arbitrarily close to a square if an infinite sum is considered.
• Activity 2: Produce a wave pattern by combination of waves.
• Open the simulation Fourier: Making Waves.
• From the top menu on the right, select "Function: Custom"
• For each of the following patterns, by the process of trial and error, determine the harmonics and their amplitude. Record your results in the table below.
• PATTERN 1 is produced by only one wave.
 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Pattern 1
• PATTERN 2 is produced by two waves.
 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Pattern 2
• PATTERN 3 is produced by two waves and it resembles beats produced by two waves with close frequencies.
 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Pattern 3
• CHALLENGE! PATTERN 4 is produced by three waves.
 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 Pattern 4

Acknowledgements.
• The java applet comes from the PhET Interactive Simulations at the University of Colorado, Boulder.
• Funding was provided by the VCCS Paul Lee Professional Development Grant Program.

Created: Thu May 28 06:40:53 Eastern Daylight Time 2009 Last modified: Thu Nov 05 16:42:48 Eastern Standard Time 2009