transver waves and conservation of linear momentum
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In general, if a wave carries energy density u with velocity v, it also carries momentum density u/v. I've seen this explicitly shown for electromagnetic waves and (longitudinal) sound waves.
However, I'm having trouble seeing how the momentum density of a transverse wave on a string can be anything besides zero. The string elements only ever move up or down, so they can't have longitudinal momentum. And if you compute the force on any small piece of string, its net horizontal component is zero. These suggest the momentum density and flow of momentum density are both zero.
I realize that accounting for higher-order effects, such as making the wave not purely transverse, or having nonuniform stretching of the string, can produce a longitudinal momentum. But if these effects are included, the waves won't satisfy the ideal wave equation. Maybe we have to account for funny nonlinear effects to get the right answer here, but we don't have to do this for other kinds of waves. So I don't think this approach is right, unless transverse string waves are somehow unique.
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