A fluid flow meter apparatus operable to be attached to a pipe and to meter fluid flow traveling within the pipe, the apparatus composing: a metering device having a plurality of ultrasonic sensors, each having a transmitter and a receiver, which sensors are operable to be mounted on an exterior surface of a wall of the pipe and to transmit ultrasonic signals orthogonally through the wall and into the fluid flow traveling within the pipe, which signals create secondary ultrasonic signals circumferentially traveling within the pipe wall at a frequency, and a plurality of damping devices attachable to the pipe wall, wherein each damping device comprises: one or more tuned bodies, each tuned body having a plurality of members (58, 86), which members (58, 86) are spaced from one another by a member-to-member separation distance, and each tuned body having one or more resonant structural modes, each mode with an associated natural frequency, characterized in that each of the plurality of members (58, 86) extends to the pipe wall at a contact point, wherein the associated natural frequencies are closely matched to the one or more frequencies of the circumferentially traveling secondary signals, and wherein the member-to-member separation distance corresponds to an integral number of wavelengths of the circumferentially traveling secondary signals; or to an odd integral number of half wavelengths of the circumferentially traveling secondary signals; or to less than half wavelengths of the circumferentially traveling secondary signals. a metering device having a plurality of ultrasonic sensors, each having a transmitter and a receiver, which sensors are operable to be mounted on an exterior surface of a wall of the pipe and to transmit ultrasonic signals orthogonally through the wall and into the fluid flow traveling within the pipe, which signals create secondary ultrasonic signals circumferentially traveling within the pipe wall at a frequency, and a plurality of damping devices attachable to the pipe wall, wherein each damping device comprises: one or more tuned bodies, each tuned body having a plurality of members (58, 86), which members (58, 86) are spaced from one another by a member-to-member separation distance, and each tuned body having one or more resonant structural modes, each mode with an associated natural frequency, characterized in that each of the plurality of members (58, 86) extends to the pipe wall at a contact point, wherein the associated natural frequencies are closely matched to the one or more frequencies of the circumferentially traveling secondary signals, and wherein the member-to-member separation distance corresponds to an integral number of wavelengths of the circumferentially traveling secondary signals; or to an odd integral number of half wavelengths of the circumferentially traveling secondary signals; or to less than half wavelengths of the circumferentially traveling secondary signals. one or more tuned bodies, each tuned body having a plurality of members (58, 86), which members (58, 86) are spaced from one another by a member-to-member separation distance, and each tuned body having one or more resonant structural modes, each mode with an associated natural frequency, characterized in that each of the plurality of members (58, 86) extends to the pipe wall at a contact point, wherein the associated natural frequencies are closely matched to the one or more frequencies of the circumferentially traveling secondary signals, and wherein the member-to-member separation distance corresponds to an integral number of wavelengths of the circumferentially traveling secondary signals; or to an odd integral number of half wavelengths of the circumferentially traveling secondary signals; or to less than half wavelengths of the circumferentially traveling secondary signals. to an integral number of wavelengths of the circumferentially traveling secondary signals; or to an odd integral number of half wavelengths of the circumferentially traveling secondary signals; or to less than half wavelengths of the circumferentially traveling secondary signals.