After visiting the Whispering wall where a whisper can be clearly transmitted between two locations on the surface of a dam wall over more than 100 metres, I was puzzled as to how this occurs. I decided to try to solve the following problem: What must be the shape of a curved wall such that any sound emitted towards the wall from one corner will be reflected directly to the opposite corner as in the following diagram (co-ordinates are $x$ and $y$)?
From this I derived the following differential equation (derivation below) but myriad substitutions and changes of variable have still left me puzzled as how to put it into a form which can be solved:
for $-a<x<0$ with $y(\pm a)=0$ and $y’(0)=0$, but I’m not sure that this is right. Can solve this equation or else find a mistake in it and find the correct equation to solve my geometrical problem?
Derivation (note that from the answers it would appear that this is slightly incorrect): Taking the following horrible paint diagram where $D$ is the point at which the sound wave is reflected and $AB$ is the tangent to the curve at that point, where we must have that the angles $BDC$ and $ADF$ marked $\theta$ are equal since a specular reflection is assumed to occur:
(Note that this is for negative $x$ only so $-x>0$; the vertical line is through the maximum of the curve). I then take the triangle $DBC$ and divide the lengths of the sides by $a-x$ to find that (using the relation between the tangent and the derivative):
I then take the triangle $ADE$ and divide the lengths by $y$ to obtain:
Setting these equal and taking $\tan$ of both sides and using the law for the tangent of a sum of angles to obtain two fractions which I cross-multiply and then collecting terms I get the differential equation shown above.
(Note that interesting facts about similar walls which might better explain the Whispering wall I originally mentioned can be found here, here and here, but I am now specifically looking for a solution to my problem. Also not that the Wikipedia page in my very first link mentioned a ‘parabolic effect’, but I could not see how that would be relevant to my problem)