When we considered waves on a string, one of the reasons we explored sinusoidal waves is that any shape wave signal can be constructed out of sums of sinusoidal waves. Analyzing a complex wave signal into its component sinusoidal parts is called spectroscopy. For light, it means separating the electromagnetic signal into the amount of each pure color it contains. For sound, it means separating the audio signal into the amount of each pure tone that it contains. These techniques are extremely powerful and find multiple uses in biology and chemistry.

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. Download this program (or run it from the PhET website). In this program we will just look at building up shapes of the form

In principle, we could have an infinite number of terms! The program shows 11 terms. Of course these are only functions of x. We can make them into traveling waves by simply replacing "x" with "x-vt" or "x+vt" wherever it appears.

The screen of the simulation looks like the figure shown at the right. Be sure it is set for "custom" so you can create your own combinations. Two contributions are shown: an amplitude of A₁ = 1.0 for the first term (j = 1) and amplitude A₂ = 0.58 for the second term. (We don't include any units since this mathematical analysis applies equally well to transverse waves on strings, electromagnetic oscillations -- light, pressure oscillations -- sound, or electrical voltage oscillations -- EKGs.

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.)

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.

Bonus game (Nothing to hand in for this part): Choose 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. 

B. 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)?

C. 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 list them.

D. What will this look like it 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.