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
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:
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.
- Create a pattern from the second harmonic with amplitude A2=0.7 and the fourth harmonic with amplitude A4=0.3.
- 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:
***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.
- PATTERN 2 is produced by two waves.
- PATTERN 3 is produced by two waves and it resembles beats produced by two waves with close frequencies.
- CHALLENGE! PATTERN 4 is produced by three waves.
- 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