In quantum mechanics, there exist reflectionless potentials, i.e. potentials whose transmission coefficient is one regardless of the incoming energy.
For a classical wave, such as a wave on a string, one could imagine creating a 'potential' by adding a term to the wave equation, $$\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial^2 x} + k(x)y.$$ Physically, this would correspond to attaching the string to vertical springs with strength proportional to $k(x)$. Do there exist reflectionless potentials for this problem?
