“How do waves have momentum?” is a very good question, but like many good questions it seems to attract a lot of over-confident incomplete answers.
It is in fact true that many kinds of travelling waves do transfer momentum to anything that actually absorbs or reflects them, and the momentum transfer is often proportional to the energy density and speed of the wave; but just stating that something is true is not an explanation of why it is true, and if the mere fact of carrying energy explained why waves have momentum then a moving charged battery would have more momentum than an uncharged one.
Indeed, it is perfectly reasonable to not be immediately convinced that waves have any momentum at all in the direction of propagation. For transverse waves the primary motions are perpendicular to the direction of motion and for compression waves the motions forwards and backwards mostly cancel out. And the fact that we can get pushed inwards by a water wave doesn’t tell us anything about net momentum transfer, since anyone who has experienced that inward push has probably also experienced the outward suction of the receding wave; and although waves seem to bring flotsam in to the shore it is not obvious that this is due to the waves themselves rather than the wind that gives rise to them.
When it comes to the often mentioned pressure and momentum of electromagnetic radiation, while we can see the effect of light pressure on the tails of comets, the derivation from Maxwell’s equations is rarely given completely. Many sources (such as this one) explain how the perpendicular electric and magnetic fields lead to a force on any charged particle that is perpendicular to both of them, but don’t give any proof that this is in the forward direction of wave propagation rather than backwards; and even when such a proof is given it is usually shown just as a formal calculation without any physical motivation as to why it is working.
A google search for “wave momentum” is unfortunately overwhelmed by ads and reviews for a popular brand of volleyball shoes, but if we change the order and/or add words like “electromagnetic” or “water” we do get a lot of useful hits. The best I’ve seen so far is
https://as.nyu.edu/content/dam/nyu-as/as/documents/silverdialogues/SilverDialogues_Peskin.pdf
This gives pretty complete arguments for the momentum content of various kinds of waves, (and also includes examples of waves that carry energy but do not have momentum – which shows that your skepticism is not at all unreasonable). But it is at a fairly high mathematical level and so takes a pretty advanced reader to see the physical motivation for why its results are true.
So what I want to do in the rest of this answer is provide a bit of a handwavy argument to give some physical motivation for the momentum content of one particular kind of wave. It is not to be taken too seriously, but just as a hint of what might actually be shown by a proper detailed analysis.
Consider a rope tied to a wall or post at one end, with you holding the other end and moving it up and down to project a wave along the string. If the wave carries momentum then during at least part of the cycle your hand must be applying a forwards push (or at least a reduced amount of tension compared to the starting situation) – and I suspect that, even before thinking about this, it has indeed felt that way when you tried it. That may of course just be a psychological effect rather than anything real, but perhaps we can think of an actual physical reason for it. When your hand is at the extreme top of its motion the rope near the end you are holding is bent up a bit, and as you move it back down the tension in the rope tends to straighten it (even if you just let it go free rather than pulling it down). This pulls up the lower part of the bend, and to counter that pulls down the part nearest your hand. But this swinging down of the end would, if tension were maintained, cause it to project outwards a bit – and so maintaining the original distance from the far end would require a bit less tension (or equivalently a slight push forward relative to the starting level of tension). As I said, this is not a real argument, but it’s the best I can do short of a proper mathematical proof as given in the paper linked to above.
Source: (1000) Alan Cooper’s answer to How do waves have momentum? – Quora