A capacitive micromachined ultrasonic transducer (CMUT) includes a structured membrane which possesses improved frequency response characteristics. Some embodiments provide CMUTs which include a substrate, a first electrode, a second movable electrode, and a structured membrane. The movable second electrode is spaced apart from the first electrode and is coupled to the structured membrane. The structured membrane is shaped to possess a selected resonant frequency or an optimized frequency response. The structured membrane can include a plate and a beam coupled to the plate such that the resonant frequency of the structured membrane is greater than the resonant frequency of the plate. Furthermore, the ratio of the resonant frequency of the structured membrane over the mass of the structured membrane can be greater than the ratio of the resonant frequency of the plate over the mass of the plate. In some embodiments, the CMUT is an embedded spring ESCMUT.