In this problem, we will take a first look at building up more complex wave shap
ID: 1503768 • Letter: I
Question
In this problem, we will take a first look at building up more complex wave shapes out of sine waves using the PhET simulation from the University of Colorado, Fourier: Making Waves.
http://phet.colorado.edu/en/simulation/fourier
The default setting shows a sine wave for a symmetric region of the x axis -- positive and negative. We'll actually only focus on the positive side. You change the value of the amplitude for each term by grabbing the black bar in the top frame (Amplitudes) and pulling it up or down. Each term is shown in the second frame (Harmonics) and their sum is shown in the third (Sum).
Make sure that your controls are set at:
custom
space
sin
Be sure that the "auto scale" box is checked on the Sum frame so the entire sum is visible. (You may have to click it off and on again to make it take effect.)
Questions to Answer
A. The default version shown in the figure gives the first term going to 0 at x = 0.39 meters. Can you figure out what k is from this? If you can, find it. If you can't explain why not.
B. Chose the middle blue tab at the top that says "Wave Game." In this, you are shown a function and have to adjust the components in order to match the total. Play this game a couple of times at levels 1 and 3 until you get the hang of how it works. (These will typically only to ask you to adjust a single term.) Once you are comfortable with that, pick level 4 or 5 in which you have to use two or more terms. Get a fit and print your result and include it with your homework.
C. Go back to the first blue tab (the one that says "Discrete"). You can make a lot of different shapes by putting different amounts of different terms in. Can you find a property that characterizes your result if you only use odd terms (1, 3, 5, 7, 9, 11)? If you only use even terms (2, 4, 6, 8, 10)?
D. Suppose you want to make a narrow peak on the right hand side. How would do you it? My best effort is shown on the right. Find a set of coefficients that makes as narrow a peak as you can and include them along with a printout of your result.
E. What will this look like if you extend it to larger (positive) values of x? Look at it by pressing the plus button in the upper box (with the <> arrows). Explain what it looks like and why it does so using the mathematical expression given above.
D. Suppose you want to make a narrow peak on the right hand side. How would do you it? My best effort is shown on the right. Find a set of coefficients that makes as narrow a peak as you can and include them along with a printout of your result.
E. What will this look like if you extend it to larger (positive) values of x? Look at it by pressing the plus button in the upper box (with the <> arrows). Explain what it looks like and why it does so using the mathematical expression given above.
Discrete Wave Game Discrete to Continuous Preset Functions Function: custort Harmenics: 11 i511 Function with infinihe number of hanmonic Graph controls Function of space x O sin O cos Measurement Tools wavelength 0001 x (m) Period ool: Math Mode Math form: Wavelengh D) Expand sun. Sound corn x (m) Auno scale 9 HelptExplanation / Answer
A) value of k = 2 * 3.14/0.39
= 16.102 m-1
B) Wave is of wave number = 16.102 m-1 and wavelength = 0.39 m.
C) property that characterizes result if only use odd terms - odd harmonics of waves are seen .
property that characterizes result if only use even terms - even harmonics of waves are seen .
D) set of coefficients that makes as narrow a peak = 0.1 , 0.15, 0.2 , 0.25 , 0.3 .
E) For , larger (positive) values of x , it will give a much broader peak .
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