A damping device for a fluid flow meter is provided. The fluid flow meter is mountable on the exterior of a pipe to meter fluid flow traveling within the pipe. The flow meter has at least one ultrasonic sensor operable to transmit ultrasonic signals through a wall in a direction substantially normal to the pipe wall and into a fluid flow disposed within the pipe. The transmitted signals create secondary ultrasonic signals circumferentially traveling within the pipe wall at one or more frequencies. The damping device includes a tuned body attachable to the pipe wall at one or more contact points. The body has one or more resonant structural modes, each with a natural frequency, which natural frequencies are closely matched to the one or more frequencies of the secondary signals. The body is operable to dissipate energy upon excitation by the secondary signals.