Procedures are described whereby acoustical systems supporting one‐dimensional harmonic waves may be synthesized for a prescribed frequency dependence of the input reflection coefficient. An important design latitude is gained by allowing the nominal characteristic impedance of the structure to vary in a stepwise manner with the independent space variable. An initial simplified model is presented which assumes only propagating plane waves and neglects multiple reflections between discontinuities. These results are corrected to include the effects of multiple reflections by techniques from signal flow graph theory. Effects of local higher‐order modes existing at the discontinuities are represented as lumped shunt susceptive elements. Data obtained from measurements on an acoustic matching section design operating between a 5.76:1 impedance mismatch over a 50% bandwidth centered about 1333 cps confirm the theoretical predictions. Further data, obtained from standing‐wave measurements at the input to coupled resonator sections, indicate the importance of higher‐order nonpropagating modes when the discontinuities are large. It is found that effects of these modes upon the input reflection coefficient are readily obtained from Smith chart calculations.The synthesis procedures constitute a simple, rapid means for initial design of a stepped structure to yield a prescribed frequency dependence of input reflection coefficient by essentially graphical techniques. The most important unsolved problems are the direct inclusion of multiple reflections and susceptive discontinuity effects in the synthesis procedure and the establishment of general physical realization criteria. Finally, it remains to relate the response of stepped systems to a more general smoothly varying system.