A single element therefore has time varying momentum. The whole structure does not.

https://www.physicsforums.com/threads/wave-momentum.281995/

Vibration isn't really my area, so treat my comments with caution. Surely it's meaningless to talk about the momentum of a mechanical wave? You can talk about momentum for a photon because it is not actually a wave, but does display wave like properties (I'm sure something I just said there will upset 'real' physicists). Mechanical waves (flexural and longitudenal) are just vibrations - they are not 'things'. It's just a way to describe a type of movement. The waves themselves do not have a mass (a photon does... more gasps from physicists, I'm sure) so can't have momentum. If you look at the medium a mechanical wave is transmitted in, an element inside of it has a displacement (and the derivatives of displacement), a mass and forces acting on it, which vary with time (not the mass!). A single element therefore has time varying momentum. The whole structure does not.

Reference https://www.physicsforums.com/threads/wave-momentum.281995/