An apparatus is presented for damping an undesired component of an ultrasonic signal. The apparatus includes a sensor affixed to a pipe. The sensor includes a transmitter and a receiver. The transmitted ultrasonic signal includes a structural component propagating through the pipe and a fluid component propagating through a flow in the pipe. The receiver receives one of the transmitted components. The apparatus includes a damping structure. The damping structure dampens the structural component of the ultrasonic signal to impede propagation of the structural component to the receiver. The damping structure includes one of a housing secured to the pipe to modify ultrasonic vibrational characteristics thereof, a plurality of film assemblies including a tunable circuit to attenuate structural vibration of the pipe, and a plurality of blocks affixed to the pipe to either reflect or propagates through the blocks, the undesired structural component of the ultrasonic signal.