11th or 12th
Time: 75 minute block
or two 50 minute periods (better)
- Students will be able to manually
add two sine waves together.
- Students will be able to describe
the conditions that cause a beat or pulsing pattern.
- Students will be able to attribute
the characteristics of a summation graph to each of the individual terms.
- Students will be able to demonstrate
the significance of adding sound waves together for nonstandard waveforms
(i.e. triangle, square, etc.)
- Students will have previously
studied properties of sine waves, and should be able to graph them and describe
- Students will know terms such
as period, frequency, amplitude and phase and how they relate to a graph of
a sine function.
Worksheets (these will open in a new window)
- Computers with Internet access
- 3 different colored pens or thin
- We know how to graph trigonometric
functions. We know about their properties. Now what happens if we want to
combine two trig functions?
- Examples of y = x and y = 2x,
and y = sin x and y = 2 sin x, point out grouping like terms.
- Cannot always group like terms.
- What is the relevance? Music.
Guitars. Radio. Etc., Etc.
- Student worksheet 1 - manual
addition of sine waves. Constructive and Destructive interference examples
should be straightforward. Emphasize using the table to get some data points.
The rest should be filled in by looking at the original graphs. Final example
will be more difficult because the pattern is not regular. Be sure to encourage
the table, and model following the two original graphs point by point. (I.e.
two positives = a bigger positive point. One positive one negative = somewhere
around the axis.)
- Student worksheet 2 - using the
Java applet to find patterns adding sine waves. Be sure to watch how they
set things up. Look for double sine wave pattern in the first activity. For
the second activity, the coefficient a should be near the coefficient for
the other term, but not equal. May need to review period, wavelength, frequency
- Student worksheet 3 - sounds and
sine waves. Wave files might be slow to download. First applet does traveling
sine waves, find the beat pattern. What would it sound like? Second applet
simulates fourier series additions (big scary words) to find out what kind
of sine waves add up to a triangle wave. May need to work through terminology
for this one.
- Where else might wave addition
come in useful?
- AM vs FM radio as an example
- We don't make these functions
up for our health!
- Have students explore and explain
what happens when you add sin (x) and cos (x) in the same fashion as the problems
on worksheet 1.
- Have students explore a square
wave that is symmetrical over the y-axis in the same manner that the triangle
wave on worksheet 3 is explored.
- Collect students manual plots
and their answers for worksheet two. This should give a good idea as to their
technical skills at adding the functions.
- Engage in discussion with the
students about the sound waves. Ask groups about their predictions before
they listen to the sounds. Review their answers to the triangle wave problem.
- For a more formal assessment,
students should be able to manually add two trig functions together on a test
or quiz. When asked, they should also be able to talk about using sines and
cosines to form other functions, especially in relation to sound.