I. Good Vibrations-Exercises

Instructions:  Complete the following exercises, using the functions defined in the tutorial.

1. Type your name here:

2. When you are finished, remove any graphs or sounds that you produced, BUT LEAVE THE INSTRUCTIONS THAT PRODUCED THESE GRAPHS OR SOUNDS.  (Please remove the graphs and/or sounds that came with the notebook as well.)

Exercise 1: Evaluating Sines and Cosines

(a)   Find the Cosine of  π/4 ?

(b)   Find  the Sine of  π/3 ?

(c)  Find the Sine of  2π ?

(d)   Mathematica is able to do symbolic math, and it knows about a lot of interesting properties of Sine and Cosine.  In an input cell below, evaluate Cos[x-π/2], Cos[x+2π], Sin[x+π/2] and Sin[x+2π].  Do you see any relationships between Sin and Cos?   By what phase angle do they differ? What is the natural period of a Sine or Cosine function?

(Here is an example of a Mathematica "Text" cell in which you can write things that are not meant to be evaluated or executed.  Copy/Paste this cell and edit it as you wish when you want to write comments that help to answer the questions.)

(e)  More along the lines of symbolic math,  Mathematica is able to solve equations.  The syntax to do this is like:

	Solve[ function1[x] == function2[x],{x}]

Note the use of the double equal sign!  

Solve finds a value of x that for which function1 is the same value as function2.

Can you find one or more values of x where Cos[x] = Sin[x]?

B: Examples:  Plotting the Graphs of Vibrations

1. In the following parts, find an expression of the form A Sin[2π f t ] or A Cos[2π f t ] that will reproduce the waveform shown in each image below.  (If you click your mouse on the graph, and then push the "apple" key, your cursor will become a set of cross-hairs.  The coordinates of the point that the cross hairs point at will show up at the bottom left part of the Mathematica window.)

(a)

Plot[]

(b)

Plot[]

(c)

Plot[]

(d)

Plot[]

C: Example:  Playing the Sounds of Vibrations

1. The frequency of this tone is some multiple of 25 cycles per second.  Can you find out what the frequency is?

2. Does the tone below sound different than the tone above?  If it is different, describe the difference?   Is it because the second tone has a different pitch?

D: Complex Tones:  Introduction to Superposition

1. Use Mathematica Lists to construct a complex tone based on a fundamental frequency of 220 Hertz.  The tone should have contain only the first 5 odd Harmonics (e.g. f, 3f, 5f,......) and the amplitude of each higher harmonic should be half as much as the prior one, starting with an amplitude of 1 for the fundamental 

Write the  "Play" function that will produce this tone.