Physics of Music Physics 121

Assignment 6

1. The following assignment will give you experience in adding two sine
waves.  The waves are constructed such that the first has frequency
400 Hz and the second may be 'close' to 400 Hz, close to 3/2 of 400 Hz
and close to twice 400 Hz.  Let's see what happens when we add these two waves 
together.  Go to Jones 113; once you're logged in, use the right mouse
button and select xmaple.  

a) Type: plot( sin(2 * Pi * 400 *  t), t=0..0.01);
What do you observe?  Measure the period from the graph.  Is it what
you expect?  The horizontal axis will be time in seconds.

b) Type: plot( {sin(2 * Pi * 400 *  t)  ,  sin(2 * Pi *  (620* t + 0.) ), 
sin(2 * Pi * 400 *  t)  +  sin(2 * Pi *  (620* t + 0.) )} , t=0..0.01);
What color is the 400 Hz sine wave?  The 620 Hz sine wave?  Describe
the sum.  Is it regular?  What happens when you change 620*t + 0. to
620*t + 0.25?  (i.e. shift the second wave by 1/4 wavelength).  Print
this graph on the laser printer (look in the File menu for Print).

c) Type the following in turn.  You don't have to retype, you can 'edit'
the existing line by clicking the mouse at a location and erasing with
the backspace key or typing in more text.  
	What happens to the sum as I shift the phase of the second
	wave, also at 400 Hz?
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (400 * t + 0) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (400 * t + 0.25) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (400 * t + 0.5) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (400 * t + 0.75) ), t=0..0.01);
	What happens to the sum when 400 and 420 Hz are mixed?  Determine
the beat frequency from the graph.  Print this graph too.
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (420 * t + 0.) ), t=0..0.1);
	What happens to the sum as you shift the phase of the second
	wave when this frequency is double the first?
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (800 * t + 0.) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (800 * t + 0.5) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (800 * t + 0.25) ), t=0..0.01);
	Compare the long-time behavior of an octave, with an interval
	slightly larger than an octave (400 and 820 Hz).  What
	is the beat frequency in the later case?
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (800 * t + 0.) ), t=0..0.1);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (820 * t + 0.) ), t=0..0.1);
	Compare the long-time behavior of a fifth, with an interval
	slightly larger than a fifth.  What is the beat frequency
	in the later case?
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (600* t + 0.) ), t=0..0.1);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (600* t + 0.) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (620* t + 0.) ), t=0..0.01);
plot( sin(2 * Pi * 400 *  t) + sin(2 * Pi *  (620* t + 0.) ), t=0..0.1);


2. Read your email on the computer science machines as follows:  Use
the right mouse button and select Tkmail.  A window will pop up on
your screen.  If you have new mail, click on 'incorporate' to get
the message to show up.  Clicking on the message of your choice will
display it on the screen.  From now on, assignments for the course
will be emailed to you and listed on mosaic.

3.  Hall, Chap. 5, pg. 86, Exercise 1.

4.  Hall, Chap. 5, pg. 86, Exercise 3.

5.  Hall, Chap. 5, pg. 87, Exercise 5.

6.  Hall, Chap. 5, pg. 87, Exercise 6.