Answer: Guitar New Problem: Whisper Dish

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

Where f_n is the fundamental frequency of the string at the n^th 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
Tension^1/2*density^-1/2*length^-1

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).
l_n+1 = 1/r*l_n
l_n+1 - l_n = (1/r - 1)l_n = (1-r)/r * l_n

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?