I'll stick with the SR-71 manual and other sources that the J-58 never quite got into a "ram jet" mode at all
The flight manual in fact makes absolutely no mention of the word "ramjet". Nor does it tell us whether the engine operates on an Ericsson cycle, Joule cycle, Otto cycle, or Brayton cycle.
Not all detail makes it into a flight manual, and that's why you have tech reps who understand what is/isn't going on, as the case may be.
A pilot does not need to know the nuts and bolts, or principles behind the engineering. What he needs to know is how to operate the machinery, and sort the problems as they arise. You will find the flight manual goes into immense detail on the operation, both normal and emergency, of the spike, front bleed, rear bleed, nozzle, afterburner, derich, EGT trim, shifting of the IGV, and compressor inlet temperature and pressure. An organ player doesn't have it so tough. What he is doing is managing the operation of the ramjet function and keeping it on song.
From,
Kerrebrock, Jack L. (1992). Aircraft Engines and Gas Turbines (2nd ed.). Cambridge, MA: The MIT Press. ISBN 978-0-262-11162-1
Heiser, William H.; Pratt, David T. (1994). Hypersonic Airbreathing Propulsion. AIAA Education Series. Washington D.C.: American Institute of Aeronautics and Astronautics. ISBN 1-56347-035-7
The air turboramjet engine is a combined cycle engine that merges aspects of turbojet and ramjet engines. Air passes through an inlet and is then compressed by an axial compressor. That compressor is driven by a turbine, which is powered by hot, high pressure gas from a combustion chamber.
The air compressed by the compressor bypasses the combustor and turbine section of the engine, where it is mixed with the turbine exhaust. The turbine exhaust can be designed to be fuel-rich (i.e., the combustor does not burn all the fuel) which, when mixed with the compressed air, creates a hot fuel-air mixture which is ready to burn again. More fuel is injected into this air where it is again combusted. The exhaust is ejected through a propelling nozzle, generating thrust.[
Perhaps
Lyman can expound on why compressor inlet pressure is of importance.