Structure for capacitive micromachined ultrasonic transducer (CMUT) device or other vibrating membrane device having non-uniform membrane so that membrane mass and stiffness characteristics may be substantially independently adjusted. CMUT having trenched membrane and/or membrane with non-uniform thickness or density. Method for operating transducer or vibrating membrane device. Array of devices at least some of which have non-uniform membrane properties. CMUT comprising substrate, support for membrane, and membrane extending over support to create cavity, membrane having non-uniform membrane thickness resulting from at least one of: thickening on upper surface of the membrane outside of cavity, thickening on lower surface of membrane inside cavity, trench on upper surface of membrane, trench on lower surface of the membrane, and any combination of two or more of these. Method for fabricating CMUT or vibrating membrane device having non-uniform membrane. High mechanical sensitivity transducer for sensor, microphone, and/or transmitter.