What shape is optimal for echo? So at school we have a semicircle made of stone, and if you stand in the middle and face the semicircle and speak, then you can hear every word echoed perfectly

There's also a second (big) semicircular arc made of stone such that if one person stands at each end and whispers into the stone, the other person can easily hear it. Is a semicircle really the optimal shape for echoing though (in either the first case or the second)? I know that parabolas have nice reflective properties, but I'm not really sure what shape would be best.
 A: When you stand at the center of a semicircle you will hear your voice amplified, as the reflections coming from all sides will be exactly in phase. This also means that if you are addressing an audience that is sitting in front of you, they will hear you more clearly (if the wall is quite close to you... If it's further, high frequencies will be lost because of phase shifts).
If you want to communicate with someone far away, you want to stand at the focus of a parabola - and want them to be standing at the focus of a "receiving parabola". This is because the sound energy will be directed along the axis of the parabola, and will therefore travel further without losing quite so much intensity (for a while you will beat the inverse square law...). Note that a circular wall is a reasonable approximation for a parabola (but you have to stand closer to the wall, roughly halfway between the center and the wall, if you are trying to "transmit" the sound to a distant observer).
In other words - "it depends".
A: First of all you are not standing at the optimum position.  

You will note from the above diagram that it would be better to stand near the focal point of the mirror which is midway between the centre of curvature $C$ and the pole $P$ of the mirror.  
Although in theory a parabolic reflector will be better in this case you would probably be wasting your time as your ear(s) will not be exactly at the focal point and diffraction effects will possibly mask any potential improvement.   
