Doppler Effect: waves emitted from a moving source. Source travelling at the same speed as the waves propagate.
Approaching the Sound Barrier: this animation shows how a shock wave propagates from an aircraft travelling at the speed of sound. The aeroplane compresses the air immediately in front and creates a shock wave (shown as a sequence of cloudy spheres with their equators emphasised for clarity). The waves expand outwards at the same speed as the aeroplane moves. Consequently, the wave fronts all concertina together at the front of the aircraft, whose nose just keeps pace with the expanding wave. This wall of compression that cannot escape from the plane is called the sound barrier. If the aircraft exceeds the speed of sound it breaks through the sound barrier and becomes supersonic creating a conical shock wave (Mach cone).
Shock waves are subtly different from sound waves but the Doppler effect is the same. The airplane is keeping up with its shock wave and so creates a sound barrier. This is an unstable state and the aircraft is buffeted by turbulence.