The recently revived interest in “acoustic resonances,” whose details are still not well defined or understood, points to a realization that a new look at some previously unrecognized findings is needed to explain problems encountered in operation of compressors and turbines. The purpose of this paper is to call the attention of the turbomachinery community to an important physical phenomenon of pressure waves in form of pulses, which reflect between blades of adjacent blade rows of turbomachines discovered more than 40 years ago, about whose existence and consequences there is little awareness today. The turbine test results which led the author in 1957 to hypothesize the existence of the phenomenon of reflecting pressure pulses are described. Subsequently, his 1966 ASME paper is discussed. In it, the author reported on the photographed observations of pressure pulses reflecting between stationary nozzles and moving blades of a water-table turbine at Lehigh University, on the description of the various types of such waves, and on an explanation of some of the resonant blade excitation frequencies observed by National Advisory Committee for Aeronautics (NACA) in a turbine of turbojet engine. This is followed by a description of his 1984 ASME paper, in which more general formulae were derived for the blade excitation frequencies caused by the reflections of pressure pulses between the rotor blades, and both upstream and downstream stator vanes. These equations were subsequently used to explain the blade excitation frequencies measured in an axial compressor stage. Finally, his 1992 AIAA paper is discussed, in which additional formulae relating to the reflecting pressure pulses were derived, and the process of formation of a pressure pulse was explained. To put this work in perspective, the author provided, in mostly chronological order, excerpts from reports on operational problems encountered with turbomachines in service and brief descriptions, from selected publications, of pertinent research work.