Sound and Acoustics

Due: Monday Sept 25

[ Note: two small corrections to the original assignment are shown in red. ]

1. Suppose that a particular instrument is characterized as having odd overtones at an amplitude 1/n of the fundamental, where n=1 is the fundamental, n=2 is the first overtone, and so on.  Write this as a (closed) sum of sine waves, using f as the fundamental frequency.
    
2. Suppose that 20 Hz is the lowest note that a particular instrument can play.  If its sound can be completely characterized by its overtones, how many overtones are needed to model this instrument?
    
3. Suppose that a listener is sitting at one vertex of an equilateral triangle, and two speakers sit at the other two vertices.  The triangle is 20 feet on each side.  The same sound is played into each speaker, but the wires are reversed on one of them.  What happens?
    
4. Suppose that the listener is now 20 feet from one speaker, and 10 from the other.  At what audible frequency(ies) might the signals cancel out?
    
5. Suppose we are sitting in a large concert hall, but the only surface that reflects any sound is the wall in the very rear of the room.  The hall is 200 ft long, we are seated right in the middle, and the musicians are seated on stage at the very front.  How much delay will the reflected sound have, and how does that translate to phase shift compared to the original sound at 100 Hz?  1000 Hz?  5000 Hz?  If you are interested in hearing the least amount of phase distortion, would you rather sit at the very front, very rear, or very middle of the room?  Explain.
    
6. One way that humans localize sound is by the phase difference between the sound reaching one ear versus the other.  Compute that phase difference for your own head, and comment on the ability to localize sounds of differing frequencies.
    
7. The volume of sound drops off as the square of the distance.  A sound of 100 dB at 1 foot away from a speaker, has what level at a distance of 100 ft?
    
8. How many decibels correspond to the doubling of a signal's amplitude?
    
9. What do aliasing / fold-over have in common with old Westerns (i.e. movies)?
    
10. Although humans can hear as high as 20 kHz or even higher, the human voice can be easily recognized with only 2 KHz or even less of bandwidth.  Explain concretely how this helps Vonage.
     
11. Extra credit:  For a signal x(t) = sin(wt), show that X(f) (i.e. its Fourier Transform) is zero everywhere except for f = w/2pi and f = -w/2pi (I am using w here for the Greek letter omega).
     
12. Extra credit:  Prove that the RMS amplitude of a sine wave is 0.707 of the peak amplitude.

Solution.