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The exact details on many of the PDE designs currently being developed are rather sketchy -- mainly because they have the potential to be extremely valuable so most of companies researching in this field are not about to tell us what they're doing.
However, from the information that has been published, it appears as if most designs are using a two-stage ignition process to achieve detonation.
Once a fresh air-fuel charge has been drawn into the pipe, a much smaller amount of a very volatile fuel (such as hydrogen) and an oxidizer (such as oxygen) are injected into a trigger chamber at the closed end of the pipe. This mixture is then ignited by an intensely powerful electrical discharge and made to detonate by forcing it through some carefully designed passages which create high levels of turbulence in the burning mixture.
This tube is sometimes referred to as a DDT (Deflagration to Detonation Transition) tube and its job is to force the trigger charge to burn at a rate that creates a supersonic shockwave.
Once it detonates, the small charge in the trigger chamber creates a very powerful shockwave that then hits the main air/fuel charge in the engine's secondary combustion chamber.
It may sound odd that it is possible to compress the gas in a tube which has an open end -- but the incredible speed of the detonation shockwave means that the air/fuel simply doesn't have a chance to be pushed out of the tube before it is compressed.
As, or because it is highly compressed, the air-fuel is also detonated by the intense heat of the shockwave.