Numerical exercise.
Updated 3/12/08
Fourier Series
Use a graphing program (for example graphing
applet ) to perform the following.
1) Provide a plot of the first term, the first two terms, the first
three terms, the first four terms etc. of the following series until
you
have enough terms to see what wave function you are creating with the
series.
-
sin(x)+ [sin(3x)]/3+[sin(5x)]/5+[sin(7x)]/7+.....
On a separate bar chart (the only bar chart you will see in this
physics
class!) plot the amplitude (y-axis) against the frequency (x-axis) for
each term in the series (this should look very similar to the FFT plots
you saw in the lab which analyzed sound samples.
2) Make a single plot of each of the following series to see what
shape
the series represents:
-
sin(x)+ [sin(3x)]/3+[sin(5x)]/5+[sin(7x)]/7+.....
-
sin(x)-[sin(2x)]/2+[sin(3x)[/3-[sin(4x)]/4 +.....
-
-sin(x)-[sin(2x)]/2-[sin(3x)]/3- [sin(4x)]/4 -.....
3) See if you can find a series which will create a non-repeating
pattern
(note that in a Fourier series you can only add sine waves with
different
amplitudes and frequencies; no powers or multiplication allowed).
4) Go to the Waves: Dispersion
simulation page and do the exercises there.
Questions:
-
What is a Fourier series?
-
What are harmonics? How are they different from overtones?
-
In air or a vacuum the components of a Fourier series representing an
electromagnetic
wave of any shape travel at the same speed (so for example if the first
series above represented a digital electromagnetic signal each of the
individual
terms in the series would travel at the same rate and so stay together
over time). In a wire or fiber optics cable, however, each Fourier
component
may travel at a slightly different speed.
-
What would happen over time to the shape of the wave in the first
series
if each term traveled at a different speed?
-
Why and how does this affect data transmission of a digital (square
wave)
signal in a fiber optics cable?
-
Is there any periodic shape which cannot be created with a Fourier
series?
-
Explain why we only examined the sine and cosine solutions to the wave
equation in class. Why didn't we worry about more complicated waves
such
as square waves or triangle waves?
Return to the IUS
Physics Top Page.
Contact Dr. Forinash,
for comments/suggestions/corrections.