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8.5 - MODELLING WITH COMBINED FUNCTIONS

A variety of real-world situations can be modelled using combined functions. To develop a model consisting of a combined function, consider

• The component functions that could be combined to form the model

• The nature of the rate of change of the component functions

• The other key features of the graph or equation that ﬁt the given scenario

Example

1) Combining Musical Notes

Musical notes are distinguished by their pitch, which corresponds to the frequency of vibration of a moving part of an instrument, such as a string on a guitar. The table lists one octave of the frequencies of commonly used notes in North American music, rounded to the nearest hertz (Hz). This is called the chromatic scale.

The graph of a pure note can be modelled by the function , where I is the sound intensity; f is the frequency of the note, in hertz (Hz); and t is the time, in seconds. According to music theory, notes sound good together when they cause constructive interference at regular intervals in time.

a) Graph the intensity functions for C and high C. Then graph the combined function for these two notes struck together.

Step 1) Determine the waveforms for C and C high by substituting their frequencies into the intensity function.

Step 2) Use a graphing calculator to graph both waveforms. Ensure that the calculator is set to Radian mode. Note: It is important to experiment with the window settings to get a clear view of the waveforms.

High C and C have the same waveform, however high C is compressed by a factor of 1/2, resulting in regular occurrences of constructive interference.

(Their crests and thoughts occur at nearly the same points.) à This suggests that these notes will sound good together.

http://everythingscience.co.za/grade-10/08-transverse-waves/08-transverse-waves-03.cnxmlplus

Step 3) To view the combination of these two waveforms, apply the superposition principle. To graph this on a graphing calculator, you can use the equation to add the graphs together. Turn off the functions for the two component waveforms for clarity.

http://www.greschner.wiscoscience.com/worksheets/adv%20phy/ap%20chpt%2011/notes/superposition,standing%20waves.pdf àSuper position principle

http://www.youtube.com/watch?v=3d2gfk1ih5E à Super position principle

When the superposition principle is added, we can see that the graph is periodic, also meaning that the two independent waveforms meet at regular intervals.

Graph of super imposed waves: http://www.youtube.com/watch?v=owQrC6BER5c

b) A C-major triad is formed by striking the following notes simultaneously:
C E G

Graph the combined function for these notes struck together. Explain why these notes sound good together.

To graph the C-major triad, enter the waveforms for the C, E, and G notes, and then graph their sum using the superposition principle. Use the frequencies given in the table.

This waveform is considerably more complicated than a simple sinusoidal waveform. However, that there is still a regular repeating pattern to the wave, which suggests that these notes will also sound good together.

The multiple variations along the cycle actually give the chord its interesting character.

c) Graph the intensity functions for C and F# (F-sharp) and the combined function for these two notes struck together. Explain why these notes are discordant (do not sound good together).

Use graphing technology to view the waveforms for the notes C and F#.

Notice that the crests and troughs of these two notes do not coincide at regular intervals, suggesting that these two notes will not sound good together. View the graph of their sum using the superposition principle to conﬁrm this.