About ramjet efficiency in supersonic speed. Is there an optimal speed? I know there is an optimal speed for ramjet.

This are the results of my research.

The reason the efficiency increases with supersonic speed to an optimal value then decreases is related to the combustion speed.
The burn flame in the combustion chamber propagates at subsonic speed.
When the air is compressed the speed of intake airflow falls subsonic while flowing into combustion chamber. However, when the air speed in the combustion chamber matches the flame advance speed, then the flame remains stable inside the combustion chamber and the top efficiency is achieved.
At higher speeds the airflow still needs adjusted to match the flame propagation speed. This means the compression has to reduce the speed to the airflow to the same subsonic value, while dealing with even stronger effect generated by shockwaves. There is a maximum flow rate through engine that still maintains the maximum subsonic speed of the flame propagation. Anymore air resulted from higher speed will push the front of the flame into the exhaust nozzle which in turn pushes the expanding exhaust gas outside of the nozzle, with loss of thrust.
Well, that would depend on the design parameters, but for example this ideal ramjet would, with stoichiometric combustion process,  provide best thrust per unit mass flow at a speed of approx. 2.4 Mach


  ...the best operating range of a hydrocarbon-fueled ramjet is $ 2 \leq M_0 \leq 4$ . The parameters used are $ \tau_\textrm{max}
= 10$ , $ a_0 \approx 300\textrm{ m/s}$ in the stratosphere, $ f_\textrm{stoich} = 0.067$ for hydrocarbons, such that $ a_0/g
f_\textrm{stoich}
\approx 450\textrm{ s}$ .