why cant a wave transmit angular momentum?
They can and do. Electromagnetic waves can carry both orbital and spin angular momentum. If you shine a beam of circularly polarized light on a plate, the plate will twist slightly. This has been http://prola.aps.org/abstract/PR/v50/i2/p115_1" [Broken].
You could do a similar experiment with waves on a string. Connect a rope to one edge of a disc mounted so that it is free to rotate about its axis. Hold the other end and send circular waves down the rope and you will be able to get the disc spinning. The angular momentum in the wave is imparted to the disc.
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