Whisper Dish
The answer is that they're parabolas, which I know because I read it on the little sign by the exhibit, which said, "these are parabolas." But several people pointed out that ellipses would also work, which is quite true. You could test this in person by seeing whether a person standing directly in front of you or directly behind you was more of a hindrance to hearing incoming messages.
There's a simple explanation for why parabolas focus incoming plane waves to a point. A parabola is defined as all the points equidistant from a focus (point) and a directrix (line, or plane in 3D). Since an incoming plane wave would hit a directrix evenly along its length, then in traveling the same distance all parts of the plane wave can meet at the focus together.
A person standing in between the two parabolas could potentially act as something of a low-pass filter. The speed of sound is about 300m/s, so a tone of 300Hz has a wavelength of about a meter. Tones lower than this will easily diffract around the person, but tones higher than this can scatter off. So if you were to use the whisper dish to broadcast music, the "color" might change when a person stands in the way because more high-frequency pitches could be blocked.
Writhing Chain
A chain hangs from a wheel. The wheel turns at a certain, fairly slow speed of about one meter per second. It pulls the chain around with it at the same rate. The wheel is about 3m off the ground, and the chain hangs from the wheel down to the ground. If you tap on the chain, a second or two later the bottom of the chain will twist and turn in slowly-changing patterns, snaking out from the plane of the rest of the chain. Why?
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