Thursday, July 24, 2008

Answer: Guitar
New Problem: Whisper Dish

The frets are designed so successive frets are a constant interval. An "interval" is a certain ratio of frequencies.
fn+1 = r*fn
fn+1 - fn = (r-1)fn

Where fn is the fundamental frequency of the string at the nth fret, and r is some number representing a ratio.
The frequency is inverse proportional to the length of the string. One way to see this is to find the differential equation describing waves on a string and solve it for the fundamental node. An easier method (although not a derivation) is to assume that the frequency of a vibrating string can depend only on the tension in the string, its linear mass density, and its length. The only way to combine these quantities and obtain a number whose dimension is frequency is

Then going back the equation for the change in frequency between frets, the change in lengths must also be geometric, but the factor is now a reciprocal (to double the frequency you must halve the length).
ln+1 = 1/r*ln
ln+1 - ln = (1/r - 1)ln = (1-r)/r * ln

So the distance between frets is proportional to the length of the string. As the string gets shorter, so does the distance between frets.

Whisper Dish

Two people can talk normally if they're positioned at large, hard dishes like the one in the picture. The dishes may be 30 meters apart, but the speakers can hear each other well if the dishes point towards each other and the speaker sit and speak at the right location.

What is the ideal shape for such a dish? Why does it still work if a third person stands in the middle to try to block the sound? Why can't that third person eavesdrop on the conversation?

1 comment:

kangway said...

I believe the ideal shape would be a parabola rotated around an axis for form a paraboloid. Each parabola has a focal point, the point at which point the speaker's head is positioned. The sound waves would effectively travel parallel and be received by the other "whisper dish" and then focused at the other focal point of the other parabola. A third party standing in the middle would not have all of the sound focused in such a manner and thus would not be able to hear this.

I think this might additionally be possible if you just stood within an elipsoid at its focal points, no?