(Issued Wed, Nov. 1)

Due Wed, Nov 1

Reading: Boas Ch. 2, Ch 7.1-7.5

1) Boas 2.16.4 (just find the speed and magnitude of acceleration, that's it)

2) Boas 2.16.10. If R=1, C=1, L=1, =2 (MKS units), what is Z (in standard form)?

3) Boas 7.2.18

4a) Boas 7.4.7 4b) Boas 7.4.15

5) Consider a particle, mass m, moving horizontally in 1-D in a viscous medium that exerts a drag force -cv (c is a "drag constant", v is the particle's velocity). The particle is attached to a spring (spring constant k, F=-kx), and is also being driven by an external force . (This is a "damped, driven oscillator")

a) Write down Newton's law (F=ma) for the particle. Write it in terms of x (the position), and its derivatives (and all the constants given.) Next use easy algebra to rewrite your equation in terms of the following (alternative) constants:

(this is still a measure of the drag coefficient).

(the natural frequency for the mass on the spring),

(a measure of the driving force).

(the driver frequency, you don't have to change this one)

Your new equation should be a differential equation for x, with only these 4 constants appearing. What are the units of the 4 constants?

b) The particle will oscillate with the angular frequency [[omega]] of the driver, so we can write x(t) as the real part of the fictitious complex function . Substitute this into your equation above (and also substitute the corresponding complex form for the driving force into the equation). (Note e.g. . What is ?) Now solve your equation for . What is the (real) amplitude of the resulting motion, as a function of ?

c) Sketch your result for the square of the amplitude . You may assume that is fixed, and the damping is weak, i.e. . What happens at low ? At high ? What is ? How about near ? How "wide" is the peak? (See my class notes if you need help with the last two)

6. We can use Mma to plot the amplitude squared from the last problem (this function is related to a Lorentzian, and is also known as a "response function".) Have Mma make 3 plots of (hopefully, this is what you got for #5!) as a function of [[omega]] for with , , and . Then show the three curves on a single plot (remember the Show[...] command?). Be sure to include appropriate, meaningful labeling of both axes. Note that the y-scale increases as the damping diminishes. Make the plot range run from 0 to 100 when you plot all three together (you can use PlotRange inside the Show command.).

7. AM radio signals have the form . The factor is called the carrier wave, and has a very high frequency, (typically 10^6 cycles/sec, i.e. kHz) The amplitude of the carrier wave is . This amplitude itself varies with time - hence the term "amplitude modulation", or AM. The audio frequency, f, is much smaller than the carrier, more like 10^2 cycles/sec. To see the general appearance of such a wave, use the following simple but unrealistic data: A=2.5, B=1, f=.5, =3, c=1. Let Mma plot a graph of y as a function of t for x=0 over exactly two periods of the amplitude function. Then plot y as a function of x at t=0, over exactly two wavelengths of the carrier function.

It would be nice to watch the time evolution like a movie - we can do that with Mma! (See Blachman p.98, but we won't need any extra packages... ) Create a series of plots like the last one (y(x)), for a range of t, running from 0 to exactly one period of the amplitude function. You can do this with a statement like

"Table[ Plot[y[x,t],{x,0,xmax}, <plot options>]

,{t,0,tmax,deltat}];

You must figure out appropriate values for xmax, tmax, and deltat. Choose options such that the plots ALL have the same, fixed y-range, otherwise Mma will autoscale and ruin the movie. (Use PlotRange) Choose deltat so you generate AT LEAST 20 plots. 30-40 or more make a nicer animation (but take a longer time) After they are generated, select them ALL by clicking on the appropriate side-bar on the right side of the window. (One bar "]" should be cleverly designed to pick all the graphs, and only the graphs. Otherwise, holding shift and clicking one by one works too) Select "Animate selected graphics" under the "Graph" menu - (or, just use option-y). The 1st of your selected graphs will start playing like a movie! The movie will be WAY too fast. There is a new status window now in the lower right corner, with double up arrows and double down arrows - clicking on the double down arrow will slow down your movie until it looks like a wave moving across the screen.

When you are done enjoying the movie, please save a tree and DELETE all the graph windows, don't print out 30 graphs! (But, DO leave the command line which generated them) However, describe carefully what you saw. Which way did the wave appear to move? What was its amplitude doing? Did it jive with what you had expected from your two "still shots"?

Extra credit: Using trig formulas, you can write y as a sum of three waves of frequencies , , and . Do this. (you can do it on paper with your regular homework, if you prefer) The first of these is called "the carrier wave", and the other two are "side bands".