This experimental study is focused on a rectangular synthetic (zero-net-mass-flux) jet impinging on a wall. The working fluid is air. A novel variant of a synthetic jet actuator (SJA) is proposed. Following biomimetic principles, the actuator incorporates a nozzle in which the cross-sectional area periodically oscillates during the driven cycle. Two independent actuating systems are used to drive the main SJA diaphragm (as seen in common SJAs with rigid nozzles) and the oscillating nozzle walls (unique to this study). Namely, the diaphragm is driven electrodynamically, and the nozzle walls are driven using a pair of piezoelectric transducers. The operating frequency of 62 Hz is selected by tuning both systems at their resonances. The nozzle slot width oscillates around an average value of approximately 6 mm. Four experimental methods were used: phase-locked visualization of the oscillating nozzle lips, direct measurement of the jet momentum flux using a precision scale, hot-wire anemometry, and an evaluation of the heat transfer coefficients on the exposed wall using heat flux sensors. The local distribution of the heat transfer coefficient on the wall (Nusselt number in a dimensionless form) indicated the three-dimensional character of the flow field with an axis-switching effect. It was concluded that the momentum flux and the heat transfer rate could be enhanced by the phase delay between the diaphragm and nozzle cycles. The reason is the fact that the nozzle cross-sectional area during the extrusion stroke is smaller than the area during the suction, thus the SJ velocity and momentum flux are promoted during extrusion while the losses are reduced during suction. The maximal effect was found at the phase shift of 270° for the ratio of the cross-section areas during the extrusion and suction of 0.74.